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Chapter 8: Managing in Competitive, Monopolistic, and Monopolistically Competitive Markets In this chapter we characterize the optimal price, output and advertising decisions of managers under three market structures: (1) perfect competition; (2) monopoly; and (3) monopolistic competition. PERFECT COMPETITION The key five assumptions for perfect competition are: 1. There are many small buyers and sellers in the market. 2. Firms’ products are homogeneous (identical or perfect substitues). 3. Buyers and sellers have perfect information of output, price and quality. 4. There are no transaction costs (traveling costs from one store to another). 5. In the long run there is free entry and exit in and from the market. The first four assumptions imply that single sellers are too small to have a perceptible influence on the price. Each seller is a price taker and the price or inverse demand equation for the firm is a constant. The second assumption implies that the products are perfect substitutes because they are identical. Since in the 4th assumption there are no transaction costs (e.g.: cost of traveling to a store), then if one firm charges a higher price consumers would not shop at that firm. Assumption (5) implies if the industry experiences a positive profit, new firms will enter the market and the market price drops and the economic profit shrink until it becomes zero (profit pays the opportunity costs for the owner). Similarly, if there are sustaining losses in the market firms are free to leave and price would move up, losses shrink and the firms earn zero profit. This implies that in the long run the perfectively competitive firm earns zero or normal economic profit. An example of perfect competition that fits the five assumptions above is agriculture (e.g.: corn, wheat, pork, beef, etc.). Another example is the catfish farm industry in the US. There are 2,000 small catfish farmers in the US. Another example is the T-shirt retailers in the US. Demand at the Market and Firm Levels The (output and demand) for the firm and the industry are represented by (Q, Df ) and (Qm, D), respectively, as shown below: 1 P P D S Horizontal line Df $4 $4 Q Qm (Typical FIRM) (Corn INDUSTRY) Market (industry) demand for corn shows how much corn all consumers will buy at each possible price in the market. Market demand (for corn) is downward sloping because consumers as a group buy more (corn) at each lower price. The individual firm sells additional corn at the same price (i.e., it is a price taker and the price is constant or the firm’s demand curve is a horizontal line). Short run output decisions: (One decision) To maximize profit in the short run, the manager must take fixed costs as given and use the market price and variable cost to determine the optimal output level. (Q*). Perfect competition is the easiest market structure for mangers to make decisions. They only have to determine the optimal output level Q*, given the market-determined price. Maximizing Profit in the Short-Run This leads to determining the profit-maximizing output Q*. The plant size (K) is fixed and there is a fixed cost because this is the short run. Let R be total revenue which is defined by P*Q where P is constant. Then profit is Profit = R – TC (where TC = VC + FC) or π = R – TC = total profit The marginal profit per additional unit of output is: ∆π / ∆Q = (∆R/∆Q) – (∆TC/∆Q) = MR – MC 2 If MR > MC then firm should increase output (Q↑) If MR < MC then firm should decrease output (Q↓) If MR = MC then there is no change in Q. This output is called equilibrium output (or the profit-maximizing output) and will be referred to by Q*. This rule MR = MC is called the first profit-maximizing rule (output choice Q*). We can examine profit maximization under perfect competition using two approaches: the total approach and the marginal approach. The total approach As noted above, total profit is given by п = total revenues – total cost = P*Q – C(Q). Fig 8-2 Revenue, Costs, and Profits for a Perfectly Competitive Firm In Fig. 8-2, total revenue under perfect competition is a straight line originating from the 3 origin because the price is constant (R= P-*Q). The cost function is generally a cubic equation. In this figure, the profit or loss at any output level is the vertical difference between sales revenues(R) and the cost function (C(Q)). The maximum vertical difference or maximum profit is located where the slope of the cost function equals to the slope of the total revenue or MR = MC (or slope of TR = Slope of C(Q)). This profit maximization rule (output choice) determines the firm’s equilibrium level Q* that maximizes profit. This is the 1st profit-maximization rule. Under perfect competition, it can be rewritten as P = MC because total Revenue is linear. That is, ∆R / ∆Q = ∆(P*Q)/ ∆Q = P*∆Q/ ∆Q = P. The Marginal Approach An alternative approach to the total approach is the marginal approach as depicted by Fig. 8-3. This approach applies the same 1st profit maximization rule but also uses the average and marginal costs instead of the total cost because in the short run part of the cost is fixed and that does not influence optimal decisions. Under this approach we will look at three cases of profit maximization. Case 1: Firm earning a positive profit in S/R. First draw the two average cost curves and the MC curve going through the minimums of the averages. Then determine output Q* where MR = MC o P MC d-Curve A Pe Pe = AR ATC * B AVC AT C q 0 Q* Fig. 8-3: Profit Maximization under Perfect Competition 4 S/R Profit Maximization Rule: MC = MR but as mentioned before because MR = P then this rule can be rewritten as MC = P Pe A π = rectangle = Profit ATC* B This rectangle gives the maximum (total) profit. It is given by its base (Q*) times the height [Pe –ATC*] which is the profit per unit, where ATC* = TC / Q* or C (Q*) / Q*. That is, this profit area equals to Q*[Pe - {(C (Q*) / Q*}] = Pe *Q* - C(Q*)= total revenue – total cost Note again that [Pe – ATC] is the profit per unit of output. TR = Pe A TR 0 Q* TC = ATC* B TC 0 Q* Example 1: A watch-making firm. Suppose: TC = 100 + Q2 → MC = ∆TC / ∆Q =0 + 2Q2-1 = 0 + 2Q = 2Q (MC is a straight line starting from the origin). FC = $100 and VC = Q2 and AVC = Q2/Q = Q (AVC is also a straight line but with a lower slope than MC). Note ATC = 100/Q + Q2/Q = 100/Q + Q Pe = $60 (the firm is a price-taker working under perfect competition) First profit-maximization rule: P = MC $60 = 2Q* o Q* = $60/2 = 30 units. Profit = R - TC = P*Q* - TC= ($60)*(30) – {100 + (30)2} = $800. Profit = $800. In this example, MC and AVC are linear (see graph below) PS = Profit + FC = $800 + $100 = $900. (NOTE: PS = TR-VC= TR - (TC-FC) Or PS = TR-TC + FC = Profit + FC, which is PS = TR - VC. 5 Price Costs MC =2Q ATC = (100/Q) + Q Pe = 60 ATC* AVC = Q Profit q Q* Example 2 VC = 3Q + Q2 (FC is unknown and is not needed for determining Q* but we cannot determine profit) MC = ∆VC / ∆Q = 1*3Q1-1 +2Q2-1= 3 + 2Q Pe = $9 (constant for perfect competition) Set Pe = MC $9 = 3 + 2Q* or 6 = 2Q* Q* = 6/2 = 3 units Then PS = TR –VC = $9*3 – (3*3+ (3)2) = $27 - $18 = $9 Demonstration 8-1 (maximizing profits) Suppose the total cost function of a firm operating under perfect competition is given by C(Q) = 5 + Q2 And the market price is $20 per unit. What price should the manager charge? What’s the level of output that maximizes profit (Q*)? How much is the profit? (Hint: MC = 0 + 2Q2-1 = 2Q) Answer: The firm’s price is the market price ($20) because the firm is a price-taker. Set P = MC (1st profit maximization rule) and solve for Q: $20 = 2Q* Q* = 10 units. The maximum profit is π = P*Q* - TC = (20) (10) – (5+102) = 200 – 5 – 100 = $95 6 Case 2: Firm Earning a Loss in S/R. should it shut down? MC ATC AVC B ATC* A Pe d-curve AVC* C MinAVC Loss q 0 Q* At point A, set P = MC → Q* π = rectangle ATC* B Pe A TC > R 0 Q* 0 Q* Loss = ATC* B loss e P A In case 2, the firm produces at a loss in the short run. Should this firm shut down? Here, Loss < FC. The firm covers part of the fixed cost (FC = Q**AFC). Since, ATC* B ATC* B FC > loss e AVC* C P A If it produces, it will cover part of the fixed cost (how much is covered?), if it shuts down it will incur all of the FC. Since Loss < FC then there is no shut down. 7 Demonstration 8-2 (minimizing losses with linear MC and AVC equations) Suppose the cost function of a perfectly competitive firm is given by C(Q) = 100 + Q2 where FC = $100, VC = Q2 and the market price is $10. What level of output Q* should the firm produce to maximize profits or minimize losses? What’s the level of profit or loss? Should the firm produce or shut down? Answer: Equilibrium condition: P = MC 10 = 2Q* Q* = 5 units. Profit = P**Q* – TC = ($10) (5) – (100+52) = -$75 (loss) The firm should not shutdown because Loss < FC $75 < $100 or ALTERNATIVELY Pe ≥ AVC = VC / Q* = Q2/Q or 10 ≥ (52) / 5 = 5 (which is equivalent to Loss < FT). The firm should not shutdown. Case 3. Losses with Shut Down Rule: If P < min AVC or loss > FC → Q* = 0 the firm should shut down. The firm should shut down because the loss is greater than FC (that is, Loss > FC). How to show that those two shutdown conditions are equivalent? Let P < min AVC. Multiply both sides by Q: Q*P < Q*AVC. Substitute in revenue and VC: R < VC. Substitute for VC as the difference between TC and FC: R < TC – FC. Rearrange: FC < TC - R FC < Loss or 8 Loss > FC (shut down), which is equivalent to P < min AVC. The Short-Run and Industry Supply Curves The supply curve for a firm describes how much output a firm will produce at each price level during a given period of time. This can be derived from the two profit max rules. P S Competitive Firm’s S/R Supply Curve Q q How? See below. For a perfectly competitive firm the price P* is determined in the market by intersection of market supply and demand. The firm’s equilibrium output Q* in the short run is determined by the two rules: 1. The profit-maximization rule; P = MC 2. The shutdown rule if there is a loss; whether P ≥ min AVC or loss < FC (no shutdown and Q* is positive) If P0 = min AVC then Q* can be positive or zero. If P < min AVC then Q* = 0 or shutdown. (Loss > FC). If P1 > min AVC then; Q1* > 0 no shutdown. (Loss < FC). If P2 > min AVC then; Q2* > 0. (positive profit). 9 MC = S/R Supply curve P2 ATC AVC P1 P0 Min AVC Q 0 Q0 Q1 Q2 q Fig. 8-6: Short run supply curve for a perfectly competitive firm The firm’s supply curve in the short run is the cross-hatched portions of the vertical axis below P0 and the marginal cost curve above min AVC. This is the supply graph in the previous graph for the perfectly competitive firm’s supply in the short-run. It is a cost curve. The market (or industry) supply is closely related to the supply curve of the individual firms in a perfectly competitive industry. The market supply is the horizontal sum of the marginal costs (above min AVC) of all firms and it determines how much total output will be produced at each price. 10 Fig 8-7: The Market Supply Curve Fig 8-7 illustrates the relation between a typical individual firm’s supply curve (MCi) and the market supply curve (S) for an industry that has, say 500 firms. Suppose when the price is $12 each firm produces, say, 1 unit of output. At $12 the industry total quantity is 500 units. What would be the industry output if the price is $12 and each firm produces 2 units? Suppose when P =$15, each firm produces 2 units. The market supply is much flatter than the individual firm’s supply, depending on the number of firms in the industry. Long-Run Decision [normal or zero economic profit) In the long-run there is a free entry into the competitive market if there is a positive profit. There is also an exit if losses exist. In the case of free entry, the market supply (S0) shifts to the right to (S1) if there are more firms entering. It shifts to the left to (S2) if 11 Fig 8-8 Entry and Exit: The Market and Firm’s Demand firms exit. At the firm level, the horizontal demand curve will also shift. In the case of positive profits, the firm’s demand curve (Df) shifts from P0 to P1. This decline in the price will shrink economic profit to zero in the case of entry. In the case of exiting the market as a result of negative profit, the increases in the price reduce losses to zero. Thus, under either way economic profit under perfect competition in the long run is zero (normal economic profit). That is, (Pe –AC)*Q* = Profit per unit * Q* - 0 (which is equivalent to Total Revenue = Total Cost). For economic profit to be zero, P = AC (or R = TC) as well as Pe = MC. For those two conditions to be satisfied the demand line or price line which is horizontal must be tangent to the min AC curve. Thus the long-run competitive equilibrium is characterized by the following conditions: 1. Pe = MC 2. Pe = min AC ( or zero econ profit) See Fig 8-9. 12 Fig. 8-9: Long-Run Competitive Equilibrium Monopoly A monopolist is a sole producer who sells a product that does not have a close substitute. When one thinks of a monopoly, it is important to specify the relevant market. Is the market local, regional or national? A utility company is a local monopoly in a city. People in this city must buy their electricity from this company or move to another city. Monopoly does not mean a large firm. A gas station in an isolated small town is a “small” monopoly. Because the monopolist is the sole seller, it has a monopoly power over the price. It can restrict output to increase the price over MC. Moreover, the demand curve for the monopolist’s product is the market demand. That means Df = Dm (firm’s demand = market demand) and thus the demand curve has a negative slope (see Fig 8- 11). If the monopolist sets the price too high, consumers do not have to buy the product. 13 Fig 8-11: The Monopolist’s Demand Sources of Monopoly Power: There are sources of monopoly power that constitute a barrier to entry in the market. These sources include economies of scale and scope, cost complementarities, patents and other legal barriers. Economies of Scale: This means average cost decreases when output increases. For many companies there is a certain range of output like [0 - Q*] in Fig 8-13 where economies of scale exist. Any output above Q* generates diseconomies of scale. If one firm exists in this market and produces say Qm to meet the demand, the ATC is ATCm. In this case the sole firm is making a profit because P > ATCm. If another firm enters and both firms share the output (Qm / 2 for each) then ATC for each one at Qm / 2 is [ATC(Qm / 2)] which is higher then the price. Both firms will earn a loss. This will deter the second firm from entering the market. 14 Fig 8-12 Economies of Scale and Minimum Prices. The number of firms that are able to fully exploit economies of scale depends on the size of the total market demand and the technology of the product. A study found for a plant to fully exploit economies of scale, it should produce at its minimum ATC. This study examined this issue for twelve industries in six countries. It was found that this number of firms varies from industry to another and from one country to another. For example, there are many plants that can fully exploit the economies of scale in the shoe industry, giving rise to a more competitive market in this industry. In the refrigerator industry the number is very small, implying an oligopolistic market structure. 15 Economies of Scope: If economies of scope exist, then it is easier and cheaper to produce two outputs Q1 and Q2 jointly in one firm than to produce them in two separate firms. Efficient production requires that the two outputs be produced in one firm. In this case the existence of economies of scope encourages building larger companies instead of small ones. This in turn gives greater access to capital markets, otherwise large capital can be a barrier. Cost Complementarity: When the marginal cost of producing one product decreases when production of another product is increased, then this encourages the establishment of multi-product firms. Such firms have large capital requirements which, discourages other firms from entering the market. This cost complementarily can be a barrier to entry. Patent and other Legal Barriers: The above sources of monopoly power are technological in nature. This legal source has to do with government regulations and policies which, for example, may grant a monopoly power for only one public utility in a specific city. Other examples include patents, trademarks and copyright protection. See INSIDE BUSINESS 8-3. 16 Maximizing Profits under Monopoly in Contrast to Perfect Competition: The manager under monopoly maximizes profit by choosing both equilibrium price Pe and equilibrium quantity Qe, knowing that it has monopoly power (i.e., P > MC). The monopolist maximizes profit by setting MR = MC. Then it determines Qe and Pe. Marginal Revenue: Formula The general relation between price and MR is given by MR = [Px(1+E)]/E, where E is the direct price elasticity of demand, %ΔQ / %ΔP = (ΔQ / ΔP)*(P / Q), (and E must be elastic). This relationship shows that MR is less than the price. For example if demand is elastic, MR is positive but less than P (e.g., E = -2 then MR = [P{(1-2)/-2}] = ½ P). If the elasticity is unitary, (EC = -1) then MR = [Px{(1-1)/1 }] = 0 and TR is at its maximum, which is less than the price since the price is positive. When demand is inelastic (say E = -1/2) and the price is positive then MR = [Px{(1-1/2)/-1/2}] = -P. This is less than the positive price. We can summarize the relationship between the price and MR in Fig. 8-13 as follows. When demand is relatively elastic, MR is positive, and when it is inelastic MR is negative. Moreover, when demand is unitary elastic, MR is zero. The implication of this relation is that the monopolist will not operate in the output 17 range where demand is inelastic because this means the contribution to the total revenue is negative or MR < 0. When demand is elastic an increase in output and a decrease in price are associated with an increase in total revenue. On the other hand, when demand is inelastic, an increase in output and a decrease in price are associated with a decline in total revenue. Finally, when demand is unitary total revenue is at its maximum and MR = 0 (TR maximization). When demand is zero ( P = 0), total revenue R is zero. Since the price changes when quantity changes, then total revenue (= P*Q) is not linear but is concave. 18 Fig 8-13 Elasticity of Demand and Total Revenues Formula: Deriving MR from Linear Inverse Demand Equations: Since demand has a negative slope under monopoly, that is, changes in quantity affect the price, and then the price is a function of quantity. P = P (Q). This is called an inverse demand function where the price is a function of output and is not a constant like under perfect competition. The most common form of this inverse demand function is the linear inverse demand. P = a – bQ, where (a) is the constant and (-b) is the (inverse) slope = ΔP / ΔQ. It can be shown that MR for the inverse linear demand can be written as MR = a – 2bQ. That is, Slope of MR = 2 * slope of (inverse) demand. Graphically, this result implies that MR curve divides the interval on the horizontal axis between zero and where the demand curve hits this axis into half. Demonstration 8–4: Determining type of price elasticity from MR. Suppose the inverse demand function is given by P = 10 – 2Q. What is the MR equation? MR = 10 – 2 * 2Q = 10 – 4Q What is the maximum price a monopolist can sell if output = 3 units? P = 10 – 2*(3) = $4. What is MR associated with 3 units of output? MR = 10 – 4*(3) = -2. The third unit reduced total revenue by $2. Is demand elastic or inelastic at Q = 3? Since MR is negative, then demand is -------. The Output Decision: Both the price and total cost are functions of output under monopoly. Then profit can be written as: п = R(Q) – C(Q). where R(Q) is total revenue and C(Q) is total cost. 19 The monopoly profit-maximization rule is MR (Qm) = MC (Qm), which can be solved for monopoly profit-maximizing output Qm. Output Qm can then be inserted in the inverse price equation, giving rise to the monopoly price rule: Pm = P(Qm). Demonstration 8-5: profit maximization under monopoly Suppose TC = 50 + Q2 → MC = 2Q is a straight line. Suppose P = 40 – Q (price is a function and not a constant) → MR = 40 – 2Q (twice the slope of P). R = PQ = (40 – Q)*Q = 40Q – Q2 (that is, multiply P by Q) 1. For equilibrium, set MR = MC 40 – 2Qm = 2Qm QM = 40/4 = 10 units → Plug Qm into Pm = 40 – 10 = $30. Calculate profit? Profit = Pm* Qm – TC = $30*10 – [50 + (10)2] = $300 - $150 = $150 The General Case for Profit Maximization: The Marginal Approach Here we skip the total approach for profit maximization to concentrate on the marginal approach. As mentioned above, for monopoly one sets MR = MC and solves for Qm. Then it substitutes Qm into the inverse demand equation to solve for Pm. In the graph below, total profit is: Output* unit profit Profit = Qm *(Pm – ATCm) where (Pm – ATCm ) = unit profit, or Profit = TR – TC = Pm*Qm -TC 20 Fig 8-15: Profit Maximization under Monopoly Absence of Supply Curve under Monopoly Monopoly does not have a supply curve because this curve is usually derived from equilibrium points formed by equating P and MC. Under monopoly, equilibrium is determined from having MR = MC and P > MR. Monopolistic Competition Examples: fast-food, toothpaste (see handout), soap, shampoo, cold medicine, etc. Characteristics: Monopolistic competition has three key characteristics: 1) Each firm competes by selling differentiated products. The differentiated products are highly substitutable but are not perfect substitutes like under perfect competition (i.e. the cross price elasticity of demand between the products of the firms is positive and high but not infinite). Crest is different from Colgate, Aim, and Close-up… etc. Therefore, because of differentiation there is consumer loyalty on part of some consumers. Consumers are willing to pay 25¢ to 50¢ more (but may be not a 1$). Therefore, Proctor & Gamble has some but limited 21 monopoly power. However, some of the customers may move to the substitutes. Therefore, advertising is important under monopolistic competition. 2) The demand curve is downward sloping but is fairly price elastic. The demand elasticity for crest is –7. Thus, because of its limited monopoly power, P&G charges a price that is higher than marginal cost but not much higher. 3) There is free entry and exit. It’s easier and cheaper to introduce, new brands of toothpaste than to start new models of cars. The latter requires large capital and technology to realize economies of scale. The free entry and exit implies that economic profit under monopolistic competition is zero (normal). Equilibrium in the short run and the long run: Like in monopoly, firms under monopolistic competition have monopoly power and, thus, they face a downward sloping demand curve. Therefore, MR < P. The profit maximization rule is MR = MC. In the short run the firm can earn a positive economic profit as shown in Fig. 8-18. MC ATC P* Profit $ ATC* DSR MR = MC Q*SR MR Fig. 8-18: Profit Maximization under Monopolistic Competition If there is a positive profit, there will be an entry into this market and prices should drop. This will shift both demand and MR curves of the individual firm down, and 22 profit will shrink until it becomes zero ( TR= TC or P = ATC) as shown by the tangency between the new inverse demand P and ATC curve (Fig. 8-19). Fig. 8-19: Effect of Entry on Monopolistically Competitive Firm’s Demand Like in perfect competition, because of free entry and exit firms under monopolistic competition earn zero economic profit in the L/R. The point where MR=MC should correspond to the point where the demand curve is tangent to the ATC curve to realize zero profit. The Long run The positive profit will induce entry by other firms who introduce competing brands. The incumbent firm will lose some market share and the demand curve will shift down. ATC and MC may also shift when more firms enter the market. Assume no shift in those cost curves. The DLR will shift down until it becomes tangent to the long 23 run AC corresponding to where MR=MC. In this case the profit is zero. We have two rules for the long run under monopolistic competition: 1. MR = MC (The 1st profit-max rule) 2. P = ATC > min ATC zero profit (because R = TC). See fig.8-20. This condition is different from the long run condition for perfect competition P = min ATC. MC AC ACLR = P*LR DLR Q*LR MRLR Fig. 8-20: Long-Run Equilibrium under Monopolistic Competition Implication of Product Differentiation: Advertising As mentioned above, monopolistically competitive firms differentiate their products in order to have some control over the price. In this case, the products are not perfect substitutes, and this makes the demand less than perfectly elastic. The implication of this is that some consumer won’t switch when the prices go up within a limit, while others are willing to switch. To keep the other consumers from switching to the substitutes, firms under monopolistic competition spend a lot of money on advertising. There are two kinds of advertising under monopolistic competition. 24 1) Comparative Advertising: This involves campaigns designed to differentiate a given firm’s brand from brands sold by competing firms. Comparative advertising is common in the fast–food industry, where firms such as McDonalds attempt to simulate demand for their hamburgers by differentiating them from competing brands. This may induce consumers to pay a premium for a particular brand. This additional value for a brand in the price is called brand equity. 2) Niche Marketing: Firms under monopolistic competition frequently introduce new products. The products could be totally “new” or “new improved”. Firms can also advertise a product that fills special needs in the market. This advertising strategy targets a special group of consumers. For example “green marketing” advertise “environmentally friendly” products to target the segment of the society that is concerned with the environment. The firm packages a product with materials that are recyclable. These advertising strategies can bring positive profits in the short–run. In the long–run other firms will mimic their strategy and reduce profits to zero. Optimal Advertising Decisions Optimal advertising is determined by the following formula Formula: The profit maximizing advertising-to-sales ratio. A/R = [(EQ, A) / - (EQ, P)] > 0, where A is expenditure on advertising and R is sales revenue. Note: A/R is a positive fraction because (EQ, P) is already negative and multiplied by a minus). EQ, A = %ΔQ / %ΔA = (ΔQ / ΔA)*(A/Q) is advertising elasticity of demand, and EQ, P = %ΔQ / %ΔP = (ΔQ / ΔP)*(P / Q), is the own–price direct elasticity of demand, which is negative. If EQ, P = - ∞ (demand is perfectly price elastic under perfect competition), then A/R = 0. That is, the optimal advertising-to-sales ratio is zero for the perfectly competitive firm. 25 The more elastic the demand with respect to own price (i.e., products are less differentiated and more substitutable), the lower the optimal advertising-to-sales ratio. This is a case of more competition than less, and there is not much need for advertising. The more elastic the demand with respect to advertising, the higher the optimal advertising- to-sales ratio. Demonstration 8-8 Suppose Corpus Industries operates under monopolistic competition and produces a product at a constant MC. Suppose the demand for its product is estimated with a log linear equation and the elasticities are: EQ, P = - 1 (price elasticity of demand) EQ, A = + 0.2 (advertising elasticity of demand) To maximize revenue what portions of revenue should this firm spend on advertising? Answer: A/R = EQ, A / - EQ, P = [+0.2 / - (-1.0)] = (+0.2 / +1.0) = 0.2 = +20% of total sales. 26 Chapter 9: Basic Oligopoly Models This chapter discusses managers’ decisions under five different oligopolistic market structures: Sweezy, Cournot, Stackelberg, Bertrand and Collusion. Comparison of the outcomes in these different oligopolistic situations reveals the following. The highest market output is produced under Bertrand oligopoly, followed by Stackelberg, then Cournot, and finally collusion. Profits are highest for the Stackelberg leader and the colluding firms, followed by Cournot, then the Stackelberg follower. Bertrand oligopolists earn the lowest level of profits. CONDITIONS FOR OLIGOPOLY Examples of Oligopoly: Steel industry, airline industry and auto industry. An Oligopoly is a market structure where there are few large firms in an industry. No explicit number is required. However, the number is usually between two and ten firms. If there are two firms, then the market structure is called duopoly. The product under oligopoly can be homogeneous (steel) or differentiated (airlines travel). The manager has a more difficult job in making decisions under oligopoly than under other market structures. Under oligopoly there is firm rivalry and interdependence in decision making. A manager, before it lowers the price of its product, it should consider the impact of the lower price on the other firms in the industry. THE ROLE OF BELIEFS AND STRATEGIC INTERACTIONS The optimal decision whether to increase or decrease the price depends on how the manager believes other managers in the industry will respond. If other managers lower the price in reaction to this firm’s lowering the price, this firm will not increase its sales much. In Figure 9.1, the reference point is B where the price is Po. The demand curve D1 is the demand when other firms match any price change. If the manager of a certain firm lowers his/her price, and the other firms in the market match this price decrease, then the quantity will not increase much as given by D1. But if they don’t match the price decrease then the manager can sell more as given by D2. Thus, the match D1 is more inelastic than the no-match D2 , or D2 is more elastic than D1. 27 If the manager increases the price and the other firms match, the firm’s sales will not decline much. So the matching demand curve will be D1 .But if they do not match the price increase, the firm will lose some market share and its demand will be the non- matching D2. The only difficulty for the firm manager to make decisions is determining whether or not rivals will match price changes. Demonstration 9-2 (The kink Demand): Thus if, for example, other firms match price reductions (D1) and do not match price increases (D2) then the oligopoly effective demand is kinked as given by ABD1 as in Fig. 9-1. This assumption gives rise to what is known as the kinked demand curve ABD1. Fig 9-1: A Firm’s Demand Depends on Actions of Rivals Then the kinked demand is given by the two segments defined by A, B and D1. 28 Price D2 No Match B P0 Match D1 Q Q0 The Kinked Demand Curve PROFIT MAXIMIZATION FOR OLIGOPOLY SETTINGS: We will examine profit maximization under four alternative assumptions on how rivals respond to price or output changes. Sweezy Oligopoly: An industry is characterized as Sweezy oligopoly if 1. There are few firms serving many customers. 2. The firms produce differentiated products. 3. *Each firm believes that rivals will respond to price reductions (effective D1) but will not respond to price increases (effective D2) (ABD1 is kinked demand as in Demonstration 9-2). This assumption represents the kinked demand curve. 4. Barriers to entry exist. In Fig. 9-2, the kinked demand curve that fits assumption 3 is given by ABD1. If the price is below P0 then the demand is the match demand D1, while if the price is above P0, then the demand is the no-match D2. The corresponding MR to the kinked demand is ACEF. 29 Fig 9-2: Sweezy Oligopoly Profit maximization occurs when MR = MC. Let us for simplicity assume that MC is linear (or straight line). If marginal cost is MC1 then profit maximization occurs at point E and the price is P0. If MC is MC0 the profit maximization occurs at point C and the price is P0. Note that if MC moves between points E and C (called the MR gap) there will be no change in the equilibrium price P0. This model is good in explaining that firms avoid price wars and thus prefer price stability by keeping the price at P0 even if MC changes (however, within a limited range). This model is criticized for not explaining how the firm arrived at point B in the first place. Nevertheless, the Sweezy model shows that strategic interactions among firms in terms of prices and the managers’ beliefs on how other firms would react to their price increases and decreases has a profound effect on pricing decisions. 30 31 The kinked demand is given by ABC and beyond as shown above. The corresponding gapped MR curve is depicted below. If the MC curve passes through the MR gap, modest shifts, upward or downward, in this curve will not change the industry price or the firms output. The Figure below (the cost cushion) shows the shifts in the MC1 curve to the MC2 and MC3 curves without a change in output or price (price stability). Recall, the 1st profit maximization rule requires that MR = MC q* p* Example: if the Match D1 is given by P1 = 15 – 2.5Q1 and the no match D2 is given by P2 = 10 – 0.5Q2, how do you determine the current or reference Q0 and P0 at point A of the Kink? Can you derive MR1 and MR2? Can you calculate the MR gap ? Answer: Set D1 = D2 and solve for the current or reference Q0 (=2.5) and P0 (=$8.75). Then substitute Q0 in the respective marginal revenues (MR1 = 15 – 2*2.5Q1 (=$2.5?) and MR2 = 10 – 2*0.5Q2 (=$7.5) to calculate the MR gap. Recall, the slope of the MR equation is twice the slope of the inverse demand equation. To find MC in the gap and profit maximization point, substitute Q0 into the MC equation. Cournot Oligopoly An industry is a Cournot oligopoly if 1. There are few firms serving many customers. 2. The products are either differentiated (e.g. automobile) or homogenous (steel). 3. *Each firm believes that rivals will hold their outputs constant if it changes its own output (naïve belief). Note that decision variables are outputs and not prices. 4. Barriers to entry exist. 32 Thus, in contrast to Sweezy oligopoly which uses prices, the firm under Cournot oligopoly believes that its output decisions have no effect on rivals output. Reaction Functions in Cournot Oligopoly To make matters easier suppose there are two firms. In this case, the market structure is a duopoly. To determine the optimal output level, firm 1 will equate its MR1 to its MC1, and firm 2 equate MR2to its MC2. The MR1 and MR2 equations are derived from the inverse market demand equation. P = a – b(Q1 + Q2)= a – bQ1 -b Q2 (note: output is homogenous there is one P) MR1 is derived by multiplying the slope b of Q1 by 2. MR1 = a – 2*bQ1 - b Q2 Firm 1’s marginal revenue MR1 is affected by firm 2’s output (Q2), as well as by its own Q1. The greater firm 2’s output, the lower is the marginal revenue of firm 1. In this case, firm 1’s profit-maximizing output depends on firm 2’s output level Q2 and its Q1. Set MR1 = MC1 and solve for Q1 as a function of Q2. This relationship between firm 1’s profit-maximization output Q1 and firm 2’s output Q2 is called a reaction function of firm 1. The same applies to firm 2 setting MR2 = MC2 where MR2= a – bQ1 – 2*b Q2 and deriving its reaction function which specifies Q2 as a function of Q1. Therefore, a reaction function for firm 1 is its profit-maximizing output (Q1) as a function of firm 2’s output (Q2). That is, Q1 = r1(Q2), where r1 is a “reaction function of”. 33 Similarly, the reaction function of firm 2 is its profit- maximizing output as a function of firm 1’s output. That is, Q2 = r2(Q1). Graphically, the reaction functions for a duopoly are given in Fig 9.3 where firm 1’s output is measured on the horizontal axis and firm 2’s output on the vertical axis. Q1 = r1(Q2), and Q2 = r2(Q1). Fig 9.3: Cournot Reaction Functions and Adjusting to Equilibrium If firm 2 produces a zero output, then firm 1 is a monopoly and its profit- maximizing or optimal output is Q1M. The greater firm 2’s output in Firm 1’ reaction function is, the lower firm 1’s profit-maximizing output. For example, if the firm 2’s output is Q*2 then the profit-maximizing output for firm 1 is Q*1. Similarly, if firm 1’s output is zero, then firm 2 is a monopoly and its profit- maximizing output is Q2M. Firm 2’s profit maximizing-output will go down if firm 1’s output in firm 2’s reaction function increases. What is the firm 2’s profit maximizing output when firm 1’s output is Q*1? It is Q*2. 34 Equilibrium in Cournot Oligopoly Graphically, we will describe how the duopoly reaches the equilibrium point (E) based on movements along the two reaction functions. Suppose firm 1 produces Q1M. Inserting this output into firm 2’s reaction function (by assumption 3), then this firm’s profit-maximizing output corresponds to point A on the r2 reaction function. On the other hand, given the positive output for firm 2 in the reaction function of firm 1, then firm 1’s profit maximizing-output will correspond to point B. Given firm 1’s output corresponding to point B in firm 2’s reaction function, then firm 2’s profit- maximizing output will correspond to point C. Given this output in firm1’s reaction function, firm 1’s output corresponds to point D. Then this will continue until it leads to point E. where the two reaction functions intersect. Therefore, equilibrium in Cournot oligopoly is determined by the intersection of the two reaction functions which determine Q*1 and Q*2. Formula: Marginal Revenues for Cournot Duopoly Suppose for a Cournot duopoly with a homogenous product, inverse demand function is P = a – b(Q1 + Q2) (we sum up the two outputs because the product is assumed to be homogeneous). Since the slope of MR is twice that of price then MR1 = a – bQ2 – 2bQ1 (only slope of Q1 is doubled) and MR2 = a – bQ1 – 2bQ2 (only slope of Q1 is doubled) Marginal products depend on own and the other firm’s outputs. Formula: Reaction Functions for Cournot Duopoly Suppose the inverse demand function is linear P = a – b(Q1 + Q2), and the cost functions with no fixed costs are 35 C1(Q1) = c1*Q1 (the cost function is linear starting from the origin and c1 is MC1) C2(Q2) = c2*Q2 (where c2 is MC2) To derive the reaction function for firm 1, set MR1 = MC1 and solve for Q1 as a function of Q2. a – bQ2 – 2bQ1 = c1 (divide both sides by 2b and solve for Q1), we have a/2b – 1/2Q2 – c1/2b = Q1.(combine the two constant terms a/2b and – c1/2b) Q1 = r1(Q2) = (a - c1) / 2b – 1/2Q2 [please remember this formula] Similarly for the reaction function of firm 2, set and solve for Q2 as a function of Q1. MR2 = MC2. a – bQ1 – 2bQ2 = c2 (divide both sides by 2b and solve for Q2) Q2 = r2 (Q1) = (a - c2) / 2b – 1/2Q1 [please remember this formula] To find the Cournot equilibrium (Q1*, Q2*) for this duopoly, substitute Q2 into the reaction function Q1 = r2(Q2) and solve for Q*1. Then substitute Q*1 into Q2 = r2(Q1) and solve for Q*2. The Cournot equilibrium is (Q1*, Q2*). SEE THE SOLVER TEMPLATE FOR THE SOLUTION OF LINEAR Cournot case on the website. Demonstration 9-4. (Remember in this example c2 = 0 and c1 =0) Suppose: The inverse market demand function is: P = 10 – Q1 – Q2 where a =10 and b =1. The firms’ cost functions are: C1(Q1) = 0 where C1(Q1) is total cost and MC1 is assumed to be c1 =0 C2(Q2) = 0 where c2 = 0. Same as above The long way for both firms: Then derive the two marginal revenues MR1 = 10 – Q2 - 2Q1 (twice the slope of inverse demand for Q1) MR2 = 10 – Q1 - 2Q2 ((twice the slope of inverse demand for Q2) 36 Long way for Firm 1 Next for firm 1, set MR1 = MC1 10 – Q2 - 2Q1 = c1 10 – Q2 - 2Q1 = 0 (MC1 or c1 is assumed to be zero in this example. Please keep in mind that c1 = 0 is a special case) and then solve for Q1 = r1(Q2) which implies that Q1 = (10-0)/)– 0.5Q2 (Remember: Firm 1’s reaction function which is Q1 = [(a - c1) / 2b – 1/2Q2 ]. The Formula way for Firm 1: use the above formula and P = 10 – Q1 – Q2 where a =10 and b =1. Here c1 is assumed to be zero. Q1 = (a - c1)/2b – 0.5Q2 = (10 - 0)/2 – 0.5Q2 = (10/2) – 1/2Q2 , where a=10, b = 1 and c1 = 0. Long way for Firm 2. Similarly, for firm 2 set MR2 = MC2 (the long way) 10 – Q1 - 2Q2 = c2 10 – Q1 - 2Q2 = 0 where c2 = 0 and divide both side by 2 and then solve for Q2 = r2(Q1): Q2 =(10-0)/2 – 1/2Q1 (Firm 2’s reaction function). Formula way for Firm 2: Use the formula Q2 = (a - c2)/2b – 0.5Q1 = (10-0)/2 -1/2Q1 where c2 = 0. To find the Cournot equilibrium point, substitute Q2 into Q1 Q1 = 10/2 – ½*Q2 = 10/2 -1/2(10/2 -1/2Q1) = 10/2 -10/4 +1/4Q1 Q1 = (20 – 10)/4 + 1/4Q1 . Then move the last term to the left, 3/4 Q1 = 10/4 solve for Q1* = (10/4)*(4/3) = 10/3= 3.33 units. 37 To solve for Q2*, substitute Q1* into the Q2*reaction function Q2 = (a - c2)/2b – 0.5Q1 Q2 =10/2 – ½(3.33) and solve for Q2*. Q2* = 10/3= 3.33 units. The result is Cournot equilibrium (Q1*, Q2*) = (3.3, 3.33) Calculate the market price where the output is homogenous: P* = 10 – Q1* – Q2* = 10 -10/3 -10/3 = 10 -20/3 = (30 -20)/3 = $10/3 Calculate market quantity Q* = Q1* + Q2* = 20/3 units. See the detailed continuation of the example solved above in the pages below (you may skip the second part because it is redundant): 38 5. To calculate profits for firm 1 define π1 = P**Q1* - c1Q1* = ($10/3)* (10/3) – 0* (10/3) = $100/9 Profit of firm 2 can be defined similarly. π 2 = P**Q2* - c2Q2* = $100/9 39 Profit in Cournot oligopoly: Isoprofit curve. Each firm has its own isoprofit curves given by the equation P*Qi – TC = πi where πi is constant. Each level πi of gives an isoprofit curve. Each curve includes combinations of outputs of both firms. For firm 1 the closer the curve to Q1M the greater the profit. In Fig 9-4, the points F, A and G for example have the same profit because profit is constant along the single isoprofit line πi0 and so on. The formula for the isoprofit line for firm 1 is: (a –bQ1 –bQ2)*Q1 = π1 which is a constant. Solve for Q1 as a function of Q2. Repeat it. 40 Fig. 9-4: Isoprofit Curves for Firm 1 The Isoprofit curve π2 is associated with greater profit than π1 and so on. (Why?) The chosen point should be on the intersection of the isoprofit curves with the respective reaction function line because the reaction functions come from profit maximization. This is also where the isoprofit curves reach their peaks, given the outputs of the other firm. For example, for given output of firm 2, say Q*2, if we move horizontally away from the peak point C in Fig. 9.5 we will be on lower and lower isoprofit curves along the way compared to ΠC1 . For a given output of firm 2, say Q*2, compare isoprofit curves associated with points A, B and D with that associated with point C which lies on the reaction function of firm 1 in Fig 9-5. Fig 9-5 Best Response to Firm 2’s Output 41 Similarly, Fig. 9-6 illustrates the isoprofit curves increase in value as they approach QM2. Fig. 9-6: Firm’s 2 Reaction Function and Isoprofit Curves Now we can bring Figures 9-5 and 9-6 together in one graph to determine Cournot equilibrium and profits for the two firms. The two isoprofit lines ΠC1 and ΠC2 through point C, where the two reaction functions intersect, represent the maximum profits for firm1 and firm 2 as in Fig 9-7. Thus, the equilibrium in a Cournot duopoly is given by point C which defines (Q*1 and Q*2). At this point, the two isoprofit lines also intersect. 42 Fig 9-7 Cournot Equilibrium and Profits Changes in Marginal Costs under Cournot In a Cournot oligopoly, the effect of a change in marginal cost is very different than in a Sweezy model. Suppose the firms are initially in Cournot equilibrium (Q*1 and Q*2) at point E in Fig. 9-8 below. Now suppose that Firm 2’ MC declines. Then for this firm 2 MR2 = MC2. The reaction function of firm 2 is Q2 = (a - c2)/2b – 1/2Q1 This means that this function will shift up and intersect firm 1’s reaction function at a higher output for firm 2 and lower output for firm 1. What will happen to profits of both firms? (hint: compare new profit level to that of monopoly or add the isoprofit curves). 43 Fig. 9-8 Effect of a decline in Firm 2’s Cost under Cournot Stackelberg Oligopoly The industry under this oligopoly has the following assumptions: 1. There are few large firms serving many customers. 2. The products can be differentiated or homogenous. 3. *In an oligopoly there is a leader (firm 1) and a follower (firm 2). 4. There are barriers to entry. In this oligopoly, the leader acts first and determines its output, knowing the reaction of the follower to its output decision. It has the first mover advantage. In this case, the leader maximizes profit, given the follower’s reaction function which depends on the leaders output. The follower maximizes profit given the leaders output Q1 as is the case in Cournot oligopoly. Thus, the follower’s reaction function is given by Q2 = r2 (Q1). For example, suppose the inverse market demand equation is given by the linear function P = a - b(Q1 + Q2) where output is homogenous. and firm 2’s cost function is C1= c1Q2 and C2 = c2Q2 where c2 is MC2. 44 Firm 2, the follower, sets MR2 = MC2. That is, a - bQ1 – 2bQ2 = c2 (move 2bQ2 to the left-hand side) 2bQ2 = a - bQ1 – c2 (next divide both sides by 2b) Q2 =( a-– c2)/2b - 1/2Q1 (Remember: follower’s reaction function). This follower firm solves for Q2 as a function of Q1, which is its Cournot reaction function: Q2 = r2(Q1) = (a - c2)/2b –[(1/2)Q1. Next Firm 1, the leader, knows this reaction function and plugs it into its profit equation in place of Q2 after substituting the inverse demand equation for P below: π1 = P.Q1 – c1Q1= [a - b(Q1 + Q2)]*Q1 – c1Q1 (substitute follower’s reaction function for Q2 below) π1 = {a – b[Q1 + (a - c2)/2b –(1/2)Q1]}*Q1 – c1Q1 (multiply things out) π1 = aQ1 – bQ12 - b(a - c2)/2b)Q1 + b(1/2)Q12 – c1Q1 π1 = aQ1 – bQ12 – (a - c2)/2)Q1 + b(1/2)Q12 – c1Q1 It then maximizes profit with respect to its own output Q1 by taking the derivative of π1 with respect to Q1. dπ1 / dQ1 = a – 2bQ1 – (a -c2)/2 + bQ1 – c1 = 0. Combine the constant terms together by combining a and c2: dπ1 / dQ1 = (a + c2)/2 – c1 – 2bQ1 + bQ1 = (a + c2)/2 – c1 – bQ1 = 0. Solve for Q1 by dividing by b. That is, Q*1 = (a + c2)/2b - c1/b At the end for this linear case, Firm 1 has the following value for its output Q1: Q*1 = (a + c2 - 2c1)/2b [remember this formula for the leader] The final step is to plug Q1* into the reaction function for Q2 above (see page 335 of the text and the Solver template on my Website or in your CD). Example on Stackelberg Oligopoly (Demonstration 9-6) Suppose that the inverse demand function for a homogenous Stackelberg Oligopoly is given by: P = 50 –Q1 - Q2 where a = 50 and b = 1. And the cost functions are given by 45 C1(Q1) = 2Q1 (where MC1 = c1 = 2) C2(Q2) = 2Q2 (where MC2 = c2 = 2) 1. Determine firm 2’s reaction function. This is a Cournot. Set MR2 = MC2 which is 50 – Q1 - 2Q2 = 2 and solve for Q2 as a function of Q1. OR use the formula for firm 2’s reaction function directly: Q2* = (a - c2)/2b – 1/2Q1*: (follower’s reaction function) Then Q2 = [(50 - 2)/2] – 1/2Q1 = 24 – ½ Q1 2. What is Firm 1’s output Q1* that maximizes profit Q1* = (a + c2 - 2c1)/2b (Leader’s reaction function) Q1* = (50 + 2 - 4)/2 = 24 units 3. Derive the follower’s output Q2* Q2* = (a - c2)/2b – (1/2)Q1* = (50 – 2)/2 – (1/2)Q1* = 24 -1/2(24) = 12 units 4. Calculate the market price P* = 50 – Q1* – Q2* = 50 -24 -12 = $14. 5. DETERMINE FIRM 1’S PROFIT. π1 = TR1 - TC1= P*Q1* – c1Q1* = $14*24 – $2*24 = $336 - $48 = $288 (leader’s profit) Firm 2’s profit can be determined the same way. π2 = P*Q2* – c2Q2* = 14*12 - $2*12 = $168 – $24 = $144 (follower’s profit) Bertrand Oligopoly 1. There are few large firms selling to too many customers. 2. The products can be identical or differentiated. 3. *The firm sets the price (not the output) that maximizes profit, given the price of the rival firm. (This is different from the kinked demand curve of Sweezy). 4. *Consumers have perfect information and there are no transaction costs. 5. There are barriers to entry. Suppose first firm 1 charges the monopoly price (initially one firm). The consumers have perfect information, there are no transaction costs and the products are identical. Firm 2 enters. If firm 2 slightly undercuts the monopoly price and since consumers know all 46 prices, they would switch to firm 2’s output (because of identical product, perfect information and no transaction costs). In this case firm 2 would capture the whole market. Therefore, firm 1, finding itself with no customers, would retaliate by undercutting firm 2’s lower price, thus recapturing the entire market. Then there is a price war under homogeneous product Bertrand with perfect information and no transaction cost. When would this “price war” end? When each firm charges a price equal to MC, P1 = P2 = MC. No firm would choose to lower this price below MC because it would make a loss. This is know as the “Bertrand Trap”. This is like perfect competition but the solution variable is the price (not output) and the profit is zero. In short, this type of Bertrand oligopoly, would lead to a situation where firms charge a price equal to MC and earn zero economic profit. Then the equilibrium is found by setting P1 = P2 = MC and then solving for Q1* and Q2* and P*. In any oligopoly with differentiated products including Bertrand, each firm has monopoly power over its brand loyal customers and it can charge a price higher than MC and earn positive economic profit. Fig 9-14 illustrates Bertrand equilibrium with differentiated products. 47 Fig 9-14: Reaction Functions and Equilibrium in a differentiated Bertand Then, in contrast to Cournot oligopoly, the reaction functions of Bertrand oligopoly with differentiated products are for prices and have positive slopes. When P2 is theoretically zero, the minimum price for firm 1 is P1min. This is a (high) price that firm 1 charges to its brand loyal customers who won’t switch to firm 2’s product despite firm 1’s higher price. As P2 increases, so does P1. Bertrand equilibrium in this case is given by point A. Reminder: Bertrand oligopoly is different from Sweezy which has match and no match demand curves. Collusion Finally, we will determine the collusive outcome, which results when the firms choose output to maximize total industry profits. This model is similar to the monopoly model as explained in Fig. 8-15 in chapter 8. When firms collude, total industry output is the monopoly level, based on the industry or market inverse demand curve. Since the inverse 48 market demand curve, which is result of summing up horizontally the outputs of all firms in the industry at each price, is P = 1,000 – Q = 1,000 – (Q1 + Q2), where Q = Q1 + Q2 (sum means homogenous output). The associated industry or market MR is MR = 1,000 – 2Q = 1,000 – 2(Q1 + Q2 ) (double the slopes for both Q1 and Q2) Notice that this MR function assumes the firms act as a single profit-maximizing firm, which is what collusion is all about. Assume total cost for the ith firm is: TCi = ciQi = 4Qi, where ci = $4 = MCi and i = 1,2 (assume identical MCs). Setting industry MR equal to MCi (which is equal to $4 as shown above) yields Market MR = MCi 1,000 - 2Q* = 4, where Q = Q1 + Q2 1,000 - 4 = 2Q* Then total industry output Q* is: 996 = 2Q* Q* = 996/2= 498 units. Thus, total industry output under collusion is 498 units, with each firm producing half of the market share: Q1* = 0.5Q* = 249 units Q2* = 0.5Q*= 249 units. 49 The industry’s price is: P* = 1,000 – Q* = 1,000 – 498 = $502. Since each firm had 50% of total output. Each firm earns profits = TRi – TCi = P**Q1* – c1Q1 = P** 0.5Q* – $4*(0.5Q*) = $502*249 - $4*249 = $124,002. 50 Chapter 10 Game Theory: Inside Oligopoly In this chapter, we will continue the discussion on managerial decisions in presence of strategic interaction and interdependence. We will develop tools using game theory that will assist future managers in making decisions in oligopolistic markets. Summary There should be a distinction between one-move games and repeated games. There also should be a difference between one-move, competing games and one-move, coordination games. 1. If each of the two players in a simultaneous-move, one-shot game has a dominant strategy, those strategies constitute a Nash equilibrium. 2. If player 1 has a dominant strategy, while player 2 does not, then the optimal strategy for player 1 is his dominant strategy. The best strategy for player 2 should be the strategy with highest payoff given player 1’s optimal strategy (both in the same cell). 3. If the simultaneous-move game is a one-shot game and there is no tomorrow, the collusion will not be sustained as a Nash equilibrium. Each player will cheat. In this case. Nash equilibrium will not have the highest payoffs. 4. If player 1 has a dominant strategy and player 2 does not, player 2’s secure strategy should correspond to player 1’s dominant strategy (in the same cell). 5. Suppose the simultaneous game is a one-shot game. Suppose each of the two diagonal cells of the two players is identical but, those numbers in each cell are not. Suppose the two off-diagonal cells have identical cells but their numbers are lower than the numbers in the diagonal cells. Then the game has two Nash equilibriums, which are the diagonal cells. If the game is infinitely repeated then 51 it is possible for collusion to be the Nash equilibrium (check the condition for sustainable collusion). Overview of Games and Strategic Thinking In a game, managers are players and the plans of managers are strategies. The payoffs are the profit or losses that result from the strategies. Due to strategic interdependence among firms, one player’s payoff depends on this player’s strategy and those of the other players. In a simultaneous-move game, each player makes decisions without the knowledge of other player’s decisions (an example of this game is the Bertrand duopoly game). In a sequential move game one player makes a move after observing the other players’ move (e.g.: chess, Tic-tac-toe, checkers and Stackelberg oligopoly). If the underlying game is played once, it’s a one-shot game. If the underlying game is played more than once, it’s a repeated game. First, we will study the foundation of game. We will begin with the study of simultaneous-move, one-shot games. Simultaneous Move, One Shot Games Such games are important to managers operating in an environment of interdependence. Let us examine the general theory which is used in analyzing managers’ decision in these games. First, strategies are decision rules that describe players’ actions. Second, normal-form representation of a game includes the players, the players’ possible strategies and the possible payoffs. To understand these concepts let us look at Table 10- 1. There are two players: A and B who are engaged in a situation of strategic interaction. You could think of the two players as managers of two firms competing in a duopoly. Player A has two possible strategies: “Up” and “Down”; while B has also two possible strategies: “Left” and “Right”. Table 10-1: A Normal Form Game: Dominant Strategies Player B Player A Strategy Left Right Up 10,20 15,8 Down -10,7 10,10 52 Each cell in the matrix above represents payoffs for the two players. For example, the cell “Up” for player A and “Left” for player B contains player A’s payoff equal to 10 and player B’s payoff equal to 20. The game is a simultaneous move, one shot game, the players make only one decision and they make it at the same time without any conditions. One shot implies that there is no future between the two players What is the optimal strategy for a player in a simultaneous move, one shot game? We characterize “optimal” by a situation that involves a dominant strategy. A strategy is dominant if it results in the highest payoff for a player regardless of what the opponent chooses. In Table 10-1, assume player B chooses “Left”, then find the highest payoff for player A over both his/her strategy (UP=10). Similarly, fix player B’s strategy at “Right” and let player A choose the highest payoffs over both his/her strategies (UP=15). Then the dominant (optimal) strategy for payoffs is “UP”. If a player has a dominant strategy he/she will play it. Principle: If a player has a dominant strategy, he/she will play it. In some games a player may not have a dominant strategy (see below). Demonstration 10-1. In Table 10-1 above, does player B have a dominant strategy? (Hint: move row-wise to look for B’s dominant strategy). The answer is No. Note that if player A chooses “UP”, the best choice for player B would be “LEFT” since the payoff 20 is better than the payoff 8 she would earn by choosing “RIGHT”. But if Player A chooses “DOWN” the best choice by player B would be “RIGHT”, since 10 is better than 7 she would realize by choosing “LEFT”. The best choice for B depends on what player A does. Thus, player B does not have a dominant strategy. What should a player do in the absence of dominant strategy? One possibility is to play a secure strategy: a strategy that guarantees the highest payoff given the worst possible scenario (max-min). This situation is not an optimal strategy; it just maximizes the payoff of the “worst case scenario”. 53 Demonstration 10-2 What is the secure strategy for player B in Table 10-1? Answer. If player A’s strategy is fixed at “UP”; then the payoff’s for B are (20 and 8). Choose 8. If A’s strategy is fixed at “DOWN” then the B’s payoff’s are (7 and 10). Choose 7. Then Players B’s worst case scenario is (8, 7). The best worst case for B is then 8 (right). Thus the secure strategy by player B is “RIGHT’. This strategy is a max-min strategy. Shortcomings of Secure Strategy 1. It is a conservative strategy that should be considered only if you have a good reason to be extremely risk-averse. 2. It does not take into consideration the optimal (dominant) strategy of the rival; and thus, it may prevent the player (manager) with the secure strategy from earning a significantly higher payoff. If player B reasons that in such a game player A will choose the dominant strategy and that player will therefore choose “Up”, then player B will earn 20 by choosing “Left” instead of “Right” that brings 8. So if the rival has a dominant strategy, the other player should anticipate that the rival will use it. Nash Equilibrium This equilibrium represents a condition in which each player does the best he/she can, given the decision of the other player. In other words, no player can improve his/her payoff by unilaterally changing his strategy, given the other players’ strategies. No player can improve his/hr payoff without hurting the other player. In Table 10-1, given that player A chooses dominant strategy “UP”, the Nash equilibrium for player B is to take this dominant strategy as given and choose strategy “LEFT” which gives 20 units of payoff’s compared to for “RIGHT”. Similarly, if player B chooses “LEFT” Nash equilibrium for Player A is “UP” which gives 10 units of payoffs. Application of One-Short Games (look for dominant strategies first) An application of simultaneous move, one-shot game is Bertrand duopoly (ZERO PROFITS). Table 10-2 has two players with two possible strategies: to charge high price 54 or low price. The collusion is both charge high price and cheating is one charges the low price. The two number cells are the profits for firm A and firm B. For example, in the cell corresponds for low price for firm B and high price for firm A, the first number (-10) is a loss for firm A, and the second number (50) is the profit of firm B. Table 10-2: A Pricing Game (Bertrand Duopoly) Firm B Firm A Price Low High Low 0,0 50,-10 High -10,50 10,10 In a one shot play of the game, the Nash equilibrium strategies are for each firm to charge low price. Why? Because if firm B charges high price, firm A will make 50 by charging the low price which is better than the 10 it will make by charging a high price. Similarly, if firm B charges the low price, firm A will charge the low price and make zero payoff which is higher than (-10) that firm A will make by charging the high price. This is also a dominant strategy for firm A. Thus firm A will always charge low price regardless of firm B’s decision. The same argument goes for firm B which should charge the low price regardless of what firm A will choose. This is also the dominant strategy for firm B. The outcome of the game is both firms charge the low price and earn zero profit in a Bertrand duopoly. Profits under Nash equilibrium (0, 0) are less than under collusion (10, 10). If the firms collude both would charge the High price and make 10 profits for each of them. This makes the Nash equilibrium inferior to the collusion. This result is called a dilemma. But collusion is illegal and if the firms colluded secretly, one firm may cheat by charging the low price and make the other firm’s customers switch to it. In this case the firm that did not cheat will suffer from a loss (-10). The manager of this firm that did not cheat either has to reveal to the shareholders that he colluded but did not cheat and consequently suffered a loss. This will bring him to jail. The second alternative is to explain nothing for making a loss and in this case he will be fired. Then this manger will cheat in one shot games. The situation can be different under repeated games. 55 Demonstration 10-4: Advertising and Dominant Strategies Firms advertise in order to entice customers from other competing companies. Suppose there are two firms: A and B, and two strategies: to advertise or not to advertise as illustrated in Table 10-3. Table 10-3: An Advertising Game Firm B Firm A Strategy Advertise Do not Advertise Advertise $4,$4 $20,$1 Do not $1,$20 $10,$10 Advertise The profit maximizing strategies for both firms are to “advertise” to cancel each other advertising out. These to-advertise strategies are dominant strategies for both firms. For each player “TO ADVERTISE” brings more money than the “DO NOT ADVERTISE”, regardless of what the strategy of the other player, because of cheating. Thus if both “advertise” each will make $4. Note that if both collude and agreed to “Do not advertise” each will make more money ($10). But collusion does not work in one-shot games. If one cheats and “advertises” it will make $20 and the one that “did not advertise” will make $1. In one-shot game, the game is over right after it is played and there is no chance for punishment. So collusion (10, 10) does not work. Here advertising brings more money. The advice is to advertise. Coordination Games In the previous games, the firms were competing in the sense, what one firm’s gains are at the expense of the other firm. In coordination games, firms find it more profitable to coordinate their actions and do like wise. An example of coordination games is two producers of electric appliances. Each firm has a choice of producing one of two types of outlets: 120 volt, two prong outlets; and 90 volt, four prong outlets. If those firms coordinate and produce likewise, 56 then in this case consumers do not have to spend more money on wiring their houses with different outlets and will have more money to buy appliances. If they do not, then that will make consumers spend less to buy the appliances because in this case they have to spend more money in wiring their houses. Let us assume that the two firms’ profits are given by the matrix in Table 10-4. Table 10-4: A Coordination Game Firm B Firm A Strategy 120 volt 90 volt 120 volt $100,$100 $0,$0 90 volt $0,$0 $100,$100 Given firm B’s strategy of producing 120 volt or 90 volt outlets, firm A would maximize profit by choosing to profit by matching B’s chosen strategy. In this case firm A’s profit would be 100 compared to zero profit by not coordinating. Similarly, given firm A’s strategy of choosing either 120 volt or 90 volt, firm B would maximize profit by matching A’s chosen strategy. In this case, the firms must match each other. In this coordinating game, there are two Nash equilibriums: an equilibrium of $100 profit for both firms choosing 120 volt outlet, and another equilibrium with $100 profit for both choosing 90 volt outlets. In this case each firm should guess what the other firm is going to do. If the firm has no clue of the other firm’s choice, then this firm will have a very tough decision. If the firms cannot talk and coordinate, then the government can set up standards requiring all firms to operate on, for example, 120 volt. In this case, there are no incentives to cheat. Coordination games are different from competing (advertising) game in Table 10-3. INFINITELY REPEATED GAMES In the simultaneous move, one shot games, collusion is not very likely because games are played only once and punishment is too late. There is today but no tomorrow, if one firm cheats in such unrepeated games the profits from cheating exceed those from collusion. However, in reality, firms compete every week, every year over and over again 57 and forever. Thus, the games are repeated. In the case when games are repeated infinitely it is possible under certain conditions that collusion will stick (i.e., be the solution). Theory When a repeated game is played, players receive payoffs during each repetition of the game. Payoff received today has a higher time-value than payoff that will be received tomorrow. The future payoffs must be discounted and we must compare the present values of the future payoffs to today’s value of the current payoff. In case of cheating, we have to compare the value of current one-time profit from cheating plus present value of Nash payoffs (no more cheating after this and we will have the present value of the Nash payoff for each game after that) with the present value of the stream of profits from cooperation or collusion over infinite time. Present Value (PV) PVfirm = π 0 + π1 / (1+i) + π2 / ( 1+i)2 + -----+ πT / ( 1+i)T where π 0 is profit today, π 1 profit a year from now, π T is from T years from now, and i is the interest rate or discount rate and [1/ (1+i)] is the discount factor and also the term of the series. If the period is not infinite and profit is constant (π I = π ), then the above series can be written as: PVfirm = π / ( 1+i)0 + π / ( 1+i) + π / ( 1+i)2 + -----+ π / ( 1+i)T for T repeated games.. PVfirm = π[1 + 1/ (1+i) +1/ (1+i)2 + 1/ (1+i)3 +…+ 1 / ( 1+i)inf] = π * t =0 t= inf 1 / ( 1+i)t for infinitely repeated games. If the period is infinite and profit πi is constant, then the series is expressed as 1 PVfirm = t 0 (1 i) t 1 The series (1 i) t 0 t converges or has a limit because it’s typical term or element (1/1+i) is a fraction. The converging limit of this series is1/[1- ELEMENT OF SERES] = 1/[1-1/(1+i)] = (1+ i) /i. Substitute this limit into PVfirm equation above, we have PVfirm = [(1+i)/i] *π This is the term which we will use for cheating to compare the profit from cheating today (plus PV of Nash payoffs if non zero in the future) with the present value, PVfirm ,of 58 streams of current and future profits from collusion or cooperation over the life span of the firm. Table: Infinitely Repeated Games Firm B Firm A Price Low High Low 10, 10 70,-40 High -40,70 50,50 PVfirm (cheat) = Cheating Payoff at current period + PV(Nash Payoffs) PVfirm (cheat) = 70 /( 1+i)0 + 10 / (1+i) + 10/ (1+i)2 + -----+ 10 / (1+i)Inf where 70 is one time payoff from cheating. Break 70 into 60 and 10 because 10 is Nash and is needed for convergence for a series with a constant which 10 in this case) PVfirm (cheat)= 60 + 10(1+i)0 + 10 / (1+i) + 10/ (1+i)2 + … + 10 / (1+i)Inf 1 = 60 + 10[ ] t 0 (1 i) t = 60 + 10(1+i)/i where (1+i)/i is the limit and 10 is Nash payoff. PVfirm (Collusion) = 50 + 50/ (1+i) + 50/ (1+i)2 + … + 50 / (1+i)Inf 1 = 50[ ] = 50(1+i)/i t 0 (1 i) t You can plug the value for i in both PV equations and then compare. Supporting Collusion with Trigger Strategies As mentioned above, collusion in infinitely repeated games is possible under certain conditions. Firms enter into a collusion agreement based on past plays of the firms. If a firm deviates and cheats then there will be a deviation from past plays. In this case other firms will use trigger strategies that are intended to punish the deviation. The punishment means that other firm will punish the cheater by doing exactly what he did, they would 59 lower the price (they go to Nash if exists). If every firm relies on trigger strategies collusion will last if current profit from cheating (plus discounted non zero future Nash profits if exist) < PV of future streams of profits under collusion. as explained in the PV equations above. Table 10-8 is an example for that with Nash payoffs are zeros (Bertrand). . Table 10-8: A Pricing Game That is Repeated (Bertrand) Firm B Firm A Price Low High Low 0,0 50,-40 High -40,50 10,10 If both firms collude in this repeated game, then the stream of future profits for each firm is 10. If one player cheats, while the other one sticks to the collusion agreement, the cheater will make 50, while the non cheater would make -40. If the collusion breaks down and both firms recourse to low prices (or Nash) then each will make zero profits in the future for this Bertrand oligopoly. The PV Approach: Suppose firm A cheats, while firm B does not. The game is over after one play. Then PVCheatFirm A = current $ cheating payoff + disct’d Nash1 + disct’d Nash1 +… = $50 + 0 + 0 +--- = $50 (Note: in this example Nash exists and equals zero). If firm A does not cheat and “cooperates” in this repeated game, the present value is PVCoopFirm A = 10 + 10/ ( 1+i) + 10 / ( 1+i)2 +10 / ( 1+i)3 + …. = 10(1+i) / i 60 where i is the interest rate. Thus, there is no incentive for Firm A to cheat if: PVCheatFirm A ≤ PVCoopFirm A PVCheatFirm A = 50 + 0 + 0+ … ≤ 10 (1+i) / i = PVCoopFirm A 50 ≤ 10 (1+i)/i or 50/10 ≤ (1+i)/I (divide both sides by 5) 5 ≤ (1+i)/i (multiply both sides by i) 5i ≤ 1+ i 5i -i ≤ 1 4i ≤ 1 (subtract one from both sides) Or i ≤ ¼ (no cheating). Solve for i. (In this case i = 25%) If i < 25%, Firm A will lose more in present value by cheating than it will gain by cooperating. The solution is COOPERATION. In general, the lower the interest rate is, the more likely that conclusion will persist, and vice versa. More generally, we can write the principle for sustaining collusion in terms of one shot- game payoffs without using preset values as follows. (ΠCheat - ΠCoop) / (ΠCoop – ΠN ) ≤ 1/i ( no cheating) Or (ΠCheat - ΠCoop) ≤ 1/i (ΠCoop – ΠN) where ΠCheat is the maximum one-shot payoff if the player cheats, ΠCoop is the one-shot cooperative or collusive payoff and ΠN is the one-shot Nash equilibrium. If any player cheats, the trigger strategy is to punish the player by choosing the Nash one-shot equilibrium strategy forever after. Apply this condition to example 10-8 when i = 10%. (50 – 10)/(10 - 0) < 1/0.1 = 10 for no cheating 4 < 10. (collusion is more profitable). Each firm can earn a payoff of 10. 61 Intuitively, the above condition for sustaining collusive or cooperative outcomes is that “provided the one-time gain from breaking the collusive agreement (cheating) is less than the present value of what would be given up by cheating (cooperation), players find it to their interest to live up to the agreement” The lower the interest rate, the more likely that conclusion will persist, and vice versa. Demonstration 10-6: The Principle to the Sustainability Approach Use the information in Table 10-8 and apply the above principle to the sustainability of collusive agreements when i = 40% (higher than before). Check if cooperation or collusion will persist over cheating. (50 – 10)/(10 - 0) < ? 1/0.4 = 2.5 for no cheating (cooperation) 40/10 > 2.5 (cheating and no collusion because interest rate is very high)) The other PV approach requires that we check if PVCheatFirm A ≤ PVCoopFirm A 50 + 0 + … ? 10 + 10/(1.4) + 10/(1.4)2 + 10/(1.4)3 + …. where 0.40 = i (very high). 50 ? 10*(1+i) / i 50? 10*[1.4 / 0.4] 50 > 35 (cheat no collusion). Since the matrix in table 10-8 is symmetric each firm has the incentive to cheat. (You can also use the principle of collusion sustainability stated above here) Factors Affecting Collusion in Pricing Games (increases in monitoring costs reduce incentives to collude). 1. Number of Firms (Remember the Heinz case study) Collusion is easier when there are fewer firms rather than many. If there are five firms in the market, each firm must be monitored four times by its rivals. Total number of monitoring in the industry is 5*4 = 20 total firms monitored. The cost of monitoring reduces the gains to colluding. 62 2. Firm Size It is easier for a large firm with 20 outlets to monitor a small firm with one outlet. The large firm must monitor 1 store, but the small firm must monitor 20 stores). 3. History of the Market Firms may not meet to collude but they can reach an understanding of the way the game has been played over time. Thus, the firms reach a tacit coordination. So they accomplish collusion indirectly by learning from past experience. 4. Punishment Mechanism Punishing a rival has a cost. If a firm posts a single price to all its current and potential customers then if it punishes its rival by lowering the price it must lower it on all the customers including those of the rival. This results in high punishment cost for the firm. But if this firm charges different prices to different customers, it can just lower the prices for the rival’s customers. In this case the cost of punishment for the firm is lower. FINITELY REPEATED GAMES There are two types of finitely repeated games: one type that has a known final end period; and the second that has an uncertain or unknown final or end period Finite Games with Uncertain end of period. These games, although finite, is similar to infinite games. The firms know that their product has a finite lifespan and some day they will become obsolete but they do not know when. Thus, the end of the finite period is uncertain. Suppose both firms played the game today. They know that the probability that the game will end tomorrow is Ө and that it will continue until tomorrow is (1- Ө). Note that 0< Ө <1. For the next two days, the probability it will end is Ө * Ө = Ө 2 and the probability that it will continue for two days is (1- Ө)*(1- Ө) = (1- Ө)2. The probability it will end in three days is Ө3 and will continue for three days is (1- Ө)3 and so on. Let us apply this type of games to Table 10- 10. 63 Table 10-10: A Pricing Game that is Finitely Repeated Firm B Firm A Price Low High Low 0,0 50,-40 High -40,50 10,10 If there is collusion and both firms adopt high price strategies, the payoff for each of them is 10. This will continue until the product terminates and the game ends. Assume interest rate is zero (no discounting) for simplicity. Then (PV of) payoff from cooperation for firm A is: PV П firm A Coop = 10 + (1-Ө)10 + (1- Ө)2 10 + (1- Ө)310 ….=10/ Ө The term of seres is (1- Ө) and the limit of the series is 1/(1- term of series)] =[1/(1- (1- Ө)) = [1/ Ө]) (Footnote: this equation is similar to the equation of collusion for the infinitely repeated games with (1- Ө) are replacing [1/(1+ i)] as the term of series. In both games they receive the same benefits. We assume i =0). Note that if the games will end or terminate tomorrow and that Ө =1 then the pay from collusion is 10 (Nash is zero in this example). This is a one-shot game. If the firm cheats, then the relationship between the payoff from cheating and that from collusion which assumes that the game will continue is: PV П firm A Cheat = 50 + 0 + …> PV П firm A Coop = 10/1 = 10. (that is greater than the payoff from collusion in a one shot game). In this case there is no incentive for firms to collude. Since the matrix in Table 10-10 is symmetric (compare the off-diagonal cells) firm B will have the same thing and there will be no cooperation. On the other hand, if Ө is a small fraction such that 64 PV П firm A Coop = > PV П firm A Cheat = 50 (which is the cheating payoff) + PV (Nash for future payoffs) 10/ Ө > 50 + 0 + 0 + … 0r 10/ Ө > 50, (where Ө = 10/50 = 0.20). then the firms will cooperate and collude. More precisely, if Ө < 0.20 (i.e., 1/5) then the firms will cooperate and collude. We can conclude by saying that the lower Ө and the higher (1- Ө), the more likely the firms will cooperate and collude. Demonstration 10-7: Billboard Advertising Game Suppose two cigarette manufacturers repeatedly played the following simultaneous –move billboard advertising game as illustrated in Table 10-11. Table 10-11: A Billboard Advertising Game Firm B Firm A Strategy Advertise Do not Advertise Advertise 0,0 20,-1 Do not -1,20 10,10 Advertise In this table, if both companies cooperate and “DO NOT ADVERTISE” (collusion) each will earn $10, while if they both “ADVERTISE” (Nash) each will make zero. If one advertise and the other does not (cheating), the one that advertised makes $20 and the one that does not make -$1. Assume there is a 10% chance that the government will ban (end) cigarette sales in any given year, can the firms “collude” by agreeing not to advertise? Note that 1-Ө = 0.9. If firm A cooperates and doesn’t cheat it can expect to earn: 65 PV П firm A Coop = 10 + (1-Ө)10 + (1- Ө)2 10 + (1- Ө)310 ….=10/ Ө PV П firm A Coop = 10 + (.9)10 + (.9)2 10 + (.9)310 +…. = 10/.10 = $100. Since $20 < $100 the firm has no incentive to cheat (that is the solution is collusion). The incentives for firm B are the same. Thus, firms can collude by using this type of trigger strategy which involves punishing the cheating firm by charging a lower price until the game ends. Repeated Games with a Known Final Period: The End-of-Period Problem Suppose a game is repeated some known number of times with strategies and payoffs as supposed in Table 10-12. Table 10-12: A pricing Game Firm B Firm A Price Low High Low 0,0 50,-40 High -40,50 10,10 Let us assume for simplicity the game is repeated twice (two one-shot games) and the players know the game will end in period two. This means after the game is played twice there is no tomorrow (at the end of the second period). At that time there are no trigger strategies and no punishments even if player A cheats. The two-shot game is really played as a one-shot game twice. Player A kept charging the high price. In this case since there is no tomorrow. Player A can charge a low price in the second period and player B cannot punish him/her. In fact player A would be happy if player B continues charging the high price in the second. In this case player A if charges the low price it will earn 50. But player B knows that player A will charge the low price and thus B will do likewise. This means this two-shot game will end in the first period and will not go to the second or end period in this example. Nash equilibrium in this two-shot game is to charge low price in each period. The game is played as two one-shot games and each player will earn zero profit in each of the two periods. 66 In that collusion will not work even if the game is played three, four, 1000 times. This type of “backward unraveling” continues until the players realize no effective punishment can be used during any period. The key reason is that each player knows that promises of cooperation will be broken any time because the period has an end and then there is no tomorrow. So the solution is low prices with zero profits. Demonstration 10-8 Suppose firms A and B will play the game in Table 10-12 twice. Assume that firm A’s strategy is to charge high price each period provided that firm B (the opponent never charged a low price in any previous period. Assume interest rate = 0. 1. How much will firm B earn? 2. How much firm A earn. Answer: Since firm A will also charge a high price each period, the opponent firm B will be able to trick firm A in the second period because in this period the game will end. Firm A will stick to its strategy for the first and second periods because it will not discover B’s cheating until the second period, and at that time it will be too late to punish firm B. Then firm B will charge a high price in the first period and earn 10 and charge a low price and earn 50 in the second period for a total of 60 (this is better than cooperating and charging higher price in each period for a total profit of 10 + 10 in the two periods). Correspondingly, Firm A will earn 10 in the first period and make a loss of 40 in the second period, for a total loss of 30 in the two periods. Since each player knows when the game will end and trigger strategies will not enhance profits. Applications of the End-of-Period Problem End of period problem arises when workers know precisely when a repeated game will end. In the final period, there is no tomorrow and there is no way to punish a player for doing something wrong in the last period. Here is an implication of the end-of-period problem for managerial decisions. Resignations and Quits Workers weigh the benefits from shirking with the cost of being fired. If the benefits are less than the costs, workers will find it in their interest to work hard. If the worker 67 announces today that he/she will quit tomorrow than there is no reason for the worker to work hard because the threat of being (the trigger strategy) fired has no bite. What can the manager do to overcome the end-of-period problem? He can fire the worker today but legally this may not be feasible. Moreover, there is a more fundamental reason why the manger should not adopt this policy. To avoid being fired on announcement, workers will not announce their plans of quitting until the end of the day and in this case they get to work longer than if they announce their plans. Consequently, the manager will not solve the end-of-period problem but instead he/she will be continuously be surprised by worker resignations. A good strategy is to give the workers some rewards for good work that extend beyond the termination of employment with the firm. In this case the worker will not take advantage of the end-of-period problem. But if the worker takes advantage of the end-of- the period problem the manager, being well connected, can punish the worker by informing potential employers about it. Multistage Games These games differ from the class of simultaneous games one-shot infinitely repeated games in the sense that timing is very important for multistage games. In particular, multistage games permit players to make sequential rather than simultaneous decisions. Theory In order to understand how multistage games differ from one shot and infinitely repeated games. We need to introduce the extensive form of a game. An extensive- form game summarizes who these players are, the information sets available to those players at each stage, the strategies available to the players, the order of moves and the payoffd from the alternative strategies. Fig. 10-1 depicts the extensive form of a game assume that there are two players: A and B; and that player A is the first mover and player B is the second mover. Each player has two strategies: Up and Down. The numbers at the end of branches in this figure are the players payoffs since player A is the first mover the first number is that players payoff and the second number is player B’s payoff. 68 (Fig. 10) Up (10, 15) B Up (5, 5) Down A Up (0, 0) Down B Down (6, 20) 69 In Fig. 10-1, player A moves first, and once this player moves, it’s player B’s turn. If player A chooses Up and player B makes the same Up move, then the payoff for A and B, respectively, are (10,15). But if player B moves in the other direction and chooses the Down strategy then their respective payoffs are (5, 5). As in simultaneous- move games, each player’s payoff depends on both player’s actions. This is the similarity between these types of games. For example, if the first move of player A is Down and player B chooses Up then player A’s payoff is (0), but if B chooses Down player A’s payoff is (6). There is important difference between the sequential and simultaneous types of games. Since player A is the first mover in this case, this player cannot make decisions based on player B’s moves, but player B gets to make decision after player A. Thus, there is no conditional “if” in player A’s strategy. Let’s see how strategies work in this game. Suppose the strategies are: player B chooses Down if player A chooses Down. What is the best strategy for A? The best strategy for A is Down because in this case A will make 6, which is better than 5. Given that player A chooses Down, does player B have an incentive to change his strategy? The answer is NO. Choosing Down instead of Up, B earns 20 instead of 0. Since neither player has an incentive to change his/her strategies then there is a Nash equilibrium associated with those strategies. Player A: Down; Player B: Down if player A chooses Up, and Down if player A chooses Down. (player B threatens to play Down all the time). The payoff: (6, 20) Is this a reasonable game? Why doesn’t A choose Up and make 10 instead of choosing Down and making 6? The answer is in the way B’s strategy is formulated. If A chooses Up, B threatens to choose Down all the time. In this case A will make 5 instead of 6. Should A believe B’s threats? If B chooses Down it will make 5. What to make out of all this? There are two Nash equilibria in this game. Nash Equilibrium: As explained above when B threatens to play Down all the time. Nash Equilibrium: When A finds that B’s threats are not credible. 70 Player A: Up Player B: Up if player A chooses Up and Down if player A chooses Down. Player B will have to chooses Up if A chooses Up. In this case, the neither player has an incentive to change his/her mind. The second Nash equilibrium is more reasonable because B’s threats are not credible in the sense that A can choose Up and this will force B to choose Up and NOT Down because it will have a lower payoff (5 instead of 15) if it follows upon its threat to choose Down. 71 Chapter 11 Pricing Strategies for Firms with Market Power In this chapter we deal with pricing strategies of firms that have some market power: firms in monopoly, oligopoly and monopolistic competition. As we learned in chapter 8, firms in perfect competition are price takers and they don’t have a pricing strategy of their own. This chapter goes as far as providing practical advice on implementing pricing strategies for those firms with market power, typically using information that is readily available to managers, including publicly available information such as the price elasticity of demand. The optimal pricing strategies for firms with market power vary depending on the underlying market structure and the instruments (e.g., advertising) available. To account for that, this chapter presents more sophisticated pricing strategies that enable a manger to extract greater profits from the consumers. BASIC PRICING STRATEGIES We will first look at the very basic pricing strategy which relies on single or uniform pricing. This strategy uses the profit-maximizing rule: MR=MC to derive the optimal price. This rule is then mathematically manipulated to provide a rule of thumb that makes use of the markup to arrive at the price. Review of the Basic Rule of Profit Maximization Firms with market power can restrict output to charge a higher price; thus they have a downward-sloping demand curve. In this case the price is different from marginal revenue. The profit-maximizing rule for firms with market power is given by MR = MC. This rule is first solved for the equilibrium output which in turn is substituted in the inverse demand equation to solve for the optimal or equilibrium price as was illustrated in chapter 8. Managers of large firms may have research department that have economists who can estimate demand and cost functions and apply this rule and to solve for optimal price and output 72 Demonstration 11-1 Suppose the inverse demand equation is given by P = 10 -2Q (downward sloping demand = market power) and the cost function is C(Q) = 2Q. Determine the profit-maximizing output and price. Answer: Recall MR has twice the slope of the price in this case. Then MR =10 – 4Q. Set MR = MC 10-4Q* = 2 Solve for Q*. Then Q* = 2 units. Plug Q* into the inverse demand equation P* = 10 -2Q* = $6. A Simple Pricing Rule for Monopoly and Monopolistic Competition Some small firms such as retail clothing stores do not hire economists to estimate their demand and cost functions. They can, however, rely on publicly available information such as information on price elasticity of demand (see chapter 7 for estimates of price elasticity for different industries). We can derive a rule of thumb from the profit- maximization rule and estimate the price with minimal or crude information and still be consistent with profit-maximization. Formula: Marginal Revenue for a firm with Market Power (Monopoly and Monopolistic Competition): MR = P[(1+Ef)/Ef] where Ef = %∆Q/%∆p = (∆Q/∆P)*P/Q where Ef is the firm’s own direct price elasticity of demand. Substitute this in the profit- maximization rule P[(1+Ef)/Ef] = MC 73 Solve for the price: P = [Ef /(1+Ef)]MC or P = (K)MC where K = Ef /(1+Ef) can be viewed as the profit maximization (optimal factor) markup factor. Example: The clothing store’s best estimate of elasticity is -4.1 and this is known. Thus, the optimal markup is K = -4.1/(1- 4.1) = 1.32. Then the optimal price P = (K)MC = 1.32*MC (That is, 1.32 times marginal cost). The manger should note two things about this price elasticity: First, the more elastic the price is, the lower the markup factor and the price (if Ef = -infinity, then K= 1 and P = MC as is the case in perfect competition); the lower MC is, lower the price. Demonstration 11-2 Suppose the manger of a convenience store competes in a monopolistically competitive market and buys Soda at a price of $1.25 per liter. Chapter 7 reports that the price elasticity of demand for the typical grocery is -3.8. The manger of this convenience store believes that demand is slightly more elastic than -3.8. Let the price elasticity of the convenience store is -4. What is the profit maximizing price for this store? P = [-4/(1-4)]MC = 1.3 MC A Simple Pricing Rule for Cournot Oligopoly Strategic interaction is an important issue in Cournot oligopoly. Each firm maximizes profit taking into account of the output of the rival firms in the industry. It believes that the output of the rivals will stay constant. The maximization rule is the same as in the monopoly case, 74 MR = MC. But under Cournot monopoly, MR depends on the firm’s output and on the rivals’ output as well. Each oligopolistic firm uses this rule to derive its interaction functions in which its own output depends on the rivals’ outputs. Then the interaction functions are used to determine the profit-maximizing outputs (Q1*, Q2*) Fortunately and similar to monopoly, a simple markup pricing rule can be used in Cournot oligopoly when the oligopolistic firms have identical cost structures and producing similar products. Suppose the industry consists of N firms with each firm having identical cost structures and produces similar products. In this case we can use the markup pricing rule for monopoly and monopolistic competition to derive a pricing formula for a firm in a Cournot Oligopoly. First, it can be shown that if products are similar then Ef = N*EM where Ef is the price elasticity of demand for the typical firm, EM is the industry’s price elasticity of demand and N is the number of firms in the industry. Recall that the markup pricing rule under monopoly and monopolistic competition is given by P = [Ef /(1+Ef)]MC where MC is the individual firm’s marginal cost. Upon substitution for Ef from above, the profit maximizing price for a firm under Cournot is given by: P = [NEM /(1+NEM)]*MC (rule of thumb pricing under Cournot) Demonstration 11-3 Suppose a Cournot industry has three firms, with market elasticity Em equal -2 and the individual firm’s MC is $50. What is the firm’s profit maximizing price under Cournot oligopoly P = {(3)(-2)/[1+(3)(-2)] }*$50 = $60 75 STRATGIES THAT YIELD EVEN GREATER PROFITS These are strategies that can be implemented under monopoly, monopolistic competition and oligopoly by which the manager can earn a profit greater that it can get using the single pricing rule (MR = MC) whether directly or through a pricing formula. These strategies which include: price discrimination, two–part pricing, block pricing and commodity bundling, are appropriate for firms with various cost structures and degrees of market interdependence. Extracting Surplus from Consumers All the above four strategies aim at extracting consumer surplus and turn it into profit for the producers. I. Price Discrimination Price discrimination is the practice of charging different prices to different consumers for the same good or service sold. There are three types of discrimination; each requires that the manager have different types of information about consumers. First- degree price discrimination (perfect price discrimination) This type of prices discrimination amounts to charging each customer the maximum price it is willing and able to pay. This price is called the reservation price. Definition: Reservation Price: The maximum price the customer is willing to pay (e.g. P1 and P2 ), which is greater than or equal to the actual price. P P1 P2 Actual P D Q 76 If monopoly single pricing strategy is used and the monopoly price is P*M, then consumer surplus (CS) in the graph below is the yellow triangle above the P*M- line and below the D-curve. CS MC P*M M PC Q*M MR If 1st degree price discrimination is practiced then: Consumer surplus (rectangle area) = 0, (because the price is the maximum price the consumer is willing to pay). Fig. 11-1a below shows the firms’ total (operating) profit (CS + PS) when the firm charges the maximum price. It is the area below the demand curve and above the MC curve up to Q*M. Note that the area below the MC curve and below the price line P*M up to the quantity Q*M is the producer surplus (PS). First-degree price discrimination is also called perfect price discrimination because it requires identifying the reservation price for each consumer under alternative quantities. This is not possible in the real world. 77 Fig. 11-1 First and Second Degree Price Discrimination Second Degree Price Discrimination (discrimination based on quantity) This type of price discrimination leaves the consumer with some consumer surplus. Thus relative to the first degree price discrimination, the total profit under the second degree is lower. This discrimination practice is based on giving discount for buying extra quantities of the good. In Fig. 11-1b, the firm charges the consumers $8 a unit for the first two units. In this case it extracts [1/2*(8-5)*2= $3] of the consumer surplus which would have gone to the consumers under single pricing. It also extracts some more by charging $5 per unit of on the units from 2 to 4. This is an additional extraction of CS. The firms cannot extract all consumer surpluses; some consumer surplus will be left to the consumers under the 2nd degree-price discrimination. Example: Electric companies: it works by charging different prices for different quantities or blocks of the same good or service (KWH). This is the case of natural monopoly (economies of scale) where both AV and MC curves are declining all the way. 78 Natural Monopoly: MR = MC Breakeven: P = AC or TR = TC P1 PM* P2 Break even P3 EM AC MC MR D Q1 QM* Q2 Q3 1st block 2nd block 3rd block Graph: Natural monopoly with second-degree price discrimination. Fig. 11-1(b) above shows how much of the consumer surplus is extracted by the firm when the second-degree practice is used. Third-Degree Price Discrimination Customers are divided into few groups with a separate demand curves or elasticities for each group. This is the most prevalent form of price discrimination. Example: Airline fares: Airline passenger tickets are divided into groups 1st class fare, regular unrestricted economy fare, and restricted economy fare. How are customers divided into groups? Some characteristic is used to divide consumers’ into distinct groups: willingness to pay, Identity can be readily established (ID ….etc) What price to charge each group? Given whatever total output is produced, this total output is allocated among the groups based on the profit maximization rule 1. 79 1. MR1 = MR2 = --- = MRN That is, prices should be designed as a result of equating MRS and read off their corresponding demand curves. If for example MR1 > MR2 output should be shifted from group 2 to group 1 (because the first group is adding more to total revenue), this will lower P1 and increase P2 until that MR1 = MR2 2. Determination of total output (Q*) is by equating MRT = MCT Where MRT is the horizontal sum of all groups MRi , i = 1,…, n. That is, fix MRi at a certain level then add up the corresponding quantities Q1, Q2,. ..,Qn. Then repeat this process by fixing MRi at a different level and so on. You will get MRT. Then equate MR1 = MR2 = --- = MRN = MCT to divide the total output among the n customer groups. Where MCT is the marginal cost of total output. If MRi > MCT for all groups i, then profit will increase by increasing total output and lowering prices. MRi < MCT then profit will increase by decreasing total output and increasing prices. This continues until MRi = MCT for all groups i = 1,…., n. Suppose there are two groups Group 1 Group 2 Total output Q1 Q2 QT = Q1 + Q2 P1 P2 Total cost function C = C (QT) TR1 = P1Q1 TR2 = P2 Q2 π = P1Q1 + P2Q2 – C(QT) (profit) 80 Q1 will increase until incremental profit ∆π / ∆Q1= 0 ∆π /∆Q1 = ∆ (P1Q1) / ∆Q1 – ∆C / ∆Q1 = 0 which means MR1 – MC = 0 this implies that MR1 = MC Similarly Q2 will increase until incremental profit ∆π / ∆Q2 = 0 MR2 = MC Putting these relationships together MR1 = MR2 = MC (which is the condition allocating total output Q* among the two groups). This is the condition for profit maximization under third degree monopoly. Monopolists practicing this price discrimination may find it easier to think in terms of the relative prices that should be charged to each group and to relate these prices to elasticity. Recall MR1 = P1 + P1(1 / EP1D1) = P1(1+1/EP1D1) Recall MR2 = P2 + P2(1 / EP2D2) = P2(1+1/EP2D2) Note that Ep11 /(1+Ep1 D1) = ( 1 +1/EP1D1) This can be rewritten as P1[(1+E1)/E1] = MC P2[(1+E2)/E2] = MC Therefore from 1st profit max ruler under 3rd price discrimination: MR1 = MR2 P1(1+1/EP1D1) = P2(1+1/EP2D2) 81 P1 = [1+(1/EP2D2)] P2 [1+(1/EP1D1)] The higher price will go to the consumers with the lower elasticity. Example: EP1D1 = - 2 (lower elasticity) EP2D2 = - 4 (higher elasticity). P1 / P2 = (1-1/4) / (1-1/2) = 1.5 Or P1 = 1.5P2 Demonstration 11-4 Local monopoly is near campus. Let MC =$6 per pizza. During the day only students eat there, while at night faculty members eat. If student’s elasticity of demand is -4 and of faculty is -2, what should be the pricing policy be to maximize profit? Answer: The faculty has more elastic demand P1[(1+E1)/E1] = MC P2[(1+E2)/E2] = MC Let L =lunch or day pizza, and D = Dinner pizza. PL[(1-4)/-4] = $6 PD [(1-2)/-2] = $6 Then PL =$8 (more elastic )and PD =$12 (less elastic) II. Two-Tier (Part) Pricing With two-part pricing, the firm charges a fixed fee for the right to purchase its goods, plus a per-unit charge for each unit purchased. This pricing policy is commonly used by athletic and night clubs. As is the case with price discrimination, the purpose of this 82 policy is to enhance the seller’s profit by extracting consumer surplus from consumers. Similar to the first-degree price discrimination, this two-part pricing strategy allows firms to extract the entire consumer surplus. To address this pricing strategy, we first present the case of profit maximization by a firm with market power (say monopoly) and estimate its profit based on using a single pricing policy. Then we use the two-part pricing policy and estimate the profit for this policy. In this example, we will show how the two-part pricing gives higher profit. Fig. 11-2: Comparison of Standard Monopoly Pricing and Two-Part Pricing 83 Fig.11-2(a) gives the profit maximization for a firm with market power using single pricing which based on the rule: MR =MC. Suppose that the demand curve is given by Q = 10- P. Then the inverse demand is given by P = 10 –Q and, thus, MR = 10 – 2Q. Suppose that the total cost function is given by: C(Q) = 2Q, which implies that MC = 2 (in this case MC = AC and constant). The firm’s equilibrium output and price based on single pricing are determined by 10 -2Q = 2. Then Q* = 8/2= 4 units and P* = $6. Total profit = (P – MC)*MC = (6 – 2)*4 = $16 Consumer surplus = (1/2)*(10 -6)* 4 = $8 Now let us use the two-part pricing strategy. Suppose the demand function in Fig. 11-2 (a) be for a single consumer. The firm can use the following two-part pricing strategy: the fixed initiation fee for the right to purchase units $32 and that the price per unit is $2. This situation is depicted in Fig. 11-2(b).With a price of $2 per unit, the consumer will purchase Q = 10 – P = 10 -2 = 8 units. The consumer surplus with 8 units is CS = (1/2)*(10 - 2)*8 = $32. To implement this pricing strategy, the firm can charge a fixed initiation fee (whether as membership fee or an entrance fee) of $32. This fee will extract the entire consumer surplus. Note that at $ 2/ unit, revenue will equal cost (net of fixed cost). That is, (Variable) Profit = (P – MC)*Q = (2-2)*8 = $0. 84 But the firm receives $32 as a fixed payment which is greater than the $18 profit which receives by charging a single price Demonstration 11-5 Suppose the total demand for golf services is Q = 20 – P and MC =$1. The total demand function is based on individual demands of 10 golfers. What is the optimal two part pricing strategy for this golf services firm? How much profit will the firm earn? Answer: The optimal per unit charge is marginal cost. At this price, 20-1 = 19 rounds of golf will be played each month. The total consumer surplus received by all 10 golfers at this price is thus: ½[(20-1)19] = $180.50 Since this is the total consumer surplus enjoyed by all 10 consumers, the optimal fixed fee is the consumer surplus enjoyed by an individual golfer ($180.50/10 = $18.05 per month). Thus, the optimal two part pricing strategy is for the firm to charge a monthly fee to each golfer of $18.05, plus greens fee of $1 per round. The total profits of the firm thus are $180.05 per month, minus the firm’s fixed costs. III. Block Pricing Here the seller packs units of the same product and sells them as one package. The consumer is faced with buying either the whole package or none of it. An example of this practice is selling eight rolls of toilet paper or 12–pack of soda. The seller will assign a value to the package that covers the cost as well as the consumer surplus. Example: Suppose an individual consumer’s demand is given by Q = 10 – P The inverse demand is expressed as P = 10 – Q Let the cost be C(Q) = 2Q. Then P = MC 10 – Q = $2 85 Q = 8 units. In this case, the firm will sell eight units. (see Fig. 11 – 3; Block pricing). The cost of buying the eight consumer is $16 and the CS = ½ (10-2)*8 = $32 Total value of the eight units = 16+32 = $48 Fig. 11-3: Block pricing Then the profit maximizing price for the package of eight units = $48 Demonstration 11-6 Suppose a consumer’s (inverse) demand for gum produced by a firm with market power is given by P = 0.2 – 0.04 Q And the marginal cost is zero. What price should the firm charge for a package containing five pieces of gum? Answer 86 When Q = 5, P = 0.2 – 0.04 * (5) = 0 When Q = 0, P = $0.2 . The linear demand is graphed in Fig. 11-4 (optimal Block Total Pricing with zero marginal cost) Value of the five units = C5 = ½ ($0.2 - $0) * 5 = $ 0.50 The firm extracts all consumer surplus and charges a price if $0.50 for a package of five pieces. IV. Commodity Bundling Travel bundle may include “airfare, hotel, car rental, meals”. A computer bundle may include “computer, printer, scanner, software …”. This pricing practice is different from block pricing because under bundle pricing the goods or the services are not the same, while they are identical under price discrimination because under bundling the sellers know that for different consumers, price the components of the bundle differently but cannot identify them into groups. Because of this lack of information the profit under bundling is usually less than under price discrimination. Suppose the manager of a computer firm knows there are two groups of consumers who value its computers and monitors differently. Table 11-1 shows the maximum prices the two groups would pay for a computer and a monitor. Table 11-1: Commodity Bundling Consumer Valuation of Computer Valuation of Monitor 1 $2000 $200 2 $1,500 $300 The manager does not know the identity of those two groups, and thus cannot practice price discrimination. Suppose the cost is constant and equals to zero to simplify matters. The manager can separately sell one computer and total profit equals TR – TC = 2,000 – 0 = $2,000 If it sells it at $1,500, then 87 TP = 3,000 – 0 =3,000 Moreover, it can also sell monitors separately. At $300 it can sell one. At $200, it can sell two and then total profit equals = $3,000 + 2 * $200 = $3,400 If the manager bundles the computers and the monitors and sell them at $1,800 a bundle then Total profits = 2 * $1,800 = 3,600 which $200 more than selling the computers and the monitors separately. Thus commodity bundling can hence profit. Demonstration 11-7 Suppose there are three purchasers of a new car that has the following valuations of two options: air conditioner and power brakes. Consumer Air Conditioner Power brakes 1 $1000 $500 2 $800 $300 3 $100 $800 Suppose the costs are zero 1. If the manager knows the valuations and consumer identities what is the optimal pricing strategy? Profit from consumer 1 = 1,000 + 500 = 1,500 Profit from consumer 2 = 800 + 300 = 1,100 Profit from consumer 3= 100 + 800 = 900 Total Profit = $3,500 2. Suppose the manager does not know the identities of the buyers. Hoe much will the firm make if the manager sells brakes and air conditioners for $800 each but offers a special options, package (power brakes and an air conditioner) for $1,100. Consumer 1 and 2 will buy the bundle Profit = 2 * $1,100 Consumer 2 will buy power brakes at $800 Total Profit = $3,000 88 Chapter 12: The Economics of Information In the previous chapters it was assumed that consumers and firms operate in an environment of perfect information and certainty whether in terms of prices, output, income etc. But the real world is far from that. In this chapter we will examine consumers and firms’ behavior under imperfect information and uncertainty. We demonstrate means (piecing, advertising etc.) by which managers can cope with uncertainty. Mean and Variance. Under uncertainty the variable is random and has possible outcome and their respective probability of occurrence. All this information about a variable can collapse into a single number which is the mean or the expected value. Mean Example: Offshore oil exploration company Event : Oil exploration. Possible outcomes: success or failure of finding oil. Payoffs: Price of the stock of this company in cases of success and failure Outcomes Probabilities Payoffs Success 1/2 $ 40 / Share Failure 1/2 $ 20 / Share Expected value = Payoff 1* q1 + Payoff 2 * q2 = (40)*1/2 + (20)*1/2 = $30. In general, suppose event = X X1 = Payoff 1; X2 = Payoff 2; ….. ; Xn = Payoff N Expected values : E(X) = q1*X1 + q2*X2 + …… + qn*Xn Variability (Risk): 89 The mean or the risk is not enough to convey all the information about a random variables. Some variables may have the same mean but the outcomes are deviated far from their mean . In other words the two variables have different spread or risk. Risk is measured by variance or the standard deviation. If the event has two possible outcomes (X1, X2) then the variance can be written as σ2 = Pr1 * (X1 – E (X) )2 + Pr2 * (X2 – E(X) )2 where E(X) = q1*X1 + q2*X2 is the expected value or weighted average for X1 and X2. Example: Suppose the two events have the same expected income (E (X)) but different risks. These events are two different sales jobs. Job A : a commission job with two possible outcomes. Job B : a salaried job with two possible outcomes. EVENT OUTCOME 1 OUTCOME 2 Income q1 Income q1 Job A $ 2,000 0.5 $ 1,000 0.5 Job B $ 1,510 0.99 $ 510 0.01 In job B, $510 is a severance pay if the company that offers this job goes out of business. Then the expected incomes for these jobs are: E (XA) = 0.5*XA1 + 0.5*XA2 = 0.5* (2000) + 0.5*(1000) = $1,500. E (XB) = 0.99*XB1 + 0.01*XB2 = 0.99*(1510) + 0.01*(510)= $1,500. The Variances are σ2A = q1*(X1A – E(XA) )2 + q2*(X2A – (XA) )2 = = 0.5*( 2000 – 1500)2 + 0.5*( 1000 – 1500 )2 = $250,000. The Standard deviation is σA = 500. σ2B = q1*( X1B – E (XB) )2 + q2*( X2B – E(XB) )2 = = 0.99*(1510 – 1500) 2 + 0.01* (510 – 1500)2 = $9,900. The standard deviation for this job is σB = $ 99.5. Thus, job A is much riskier than job B. 90 Demonstration 12-2 The manager of XYZ company is introducing a new product that will yield $1,000 in profits if the economy does not go into recession. If the recession occurs, then the company will lose $4,000. If economists project that there is a 10% chance the economy will go into a recession how risky is the introduction of the new product. Answer E(Profit) = 0.9*(1,000) + 0.1*(-4,000) = $500 σ2 = 0.9*(1,000 – 500)2 + 0.1*(-4,000 -500)2 σ2 = 0.9*(500)2 + 0.1*(-4,500)2 σ2 = 2,250,000 σ2 = √2,250,000 = $1,500 Uncertainty and Consumer Behavior We will see how the presence of uncertainty affects both consumers and managers. Risk Aversion People may have different tastes for the same set of risky prospects, and thus they exhibit different preferences for these prospects. Suppose F represents the uncertain prospects associated with buying 100 shares of stock F, and G is the uncertain prospect of buying 100 shares of stock G. Because attitude and preferences among consumers differ, a risk averse person prefers a sure amount of $M to a risky prospect with an expected value of $M. A risk-loving individual prefers a risky prospect with an expected value of $M to the same amount of $M. A risk neutral individual is indifferent between a risky prospect with an expected value of $M and a sure amount of $M. 91 Managerial Decisions with Risk Averse Consumers Here are some examples of how risk aversion affects managerial decisions. Product Quality The analysis of risk can be used to show how uncertainty about product quality affect consumer’s behavior and how managers can deal with it. There is risk associated with buying new products. If risk averse individual is faced with the new product Y and the regular product X and views these two products to be of the same quality, he will buy the regular product X. The manager has two primary tactics to induce the risk averse consumers to buy the new product. 1. The manager may lower the price of the new products. For example, he can give free samples (where the price is zero). 2. The manager can use comparative advertising to convince the consumer that the new product is of better quality than the regular brand. If consumers are convinced they may buy the new product. Chain Stores Risk aversion may explain that it is in a firm’s best interest to be part of a chain store instead of remaining independent. The type and quality of products offered by national chains are certain. Example, imagine a family driving through a small town and looking for a restaurant to eat. In this town there are two restaurants to eat: a local diner and a national hamburger chain. The family is uncertain about quality of the food of the diner, but it is more sure about the food of the national chain. It would choose to eat at the national chain. The same applies to retailing outlets, gas stations, etc. Insurance 92 People buy insurance on their automobiles and homes. They give a small amount of money to avoid loosing a huge sum if a catastrophic event occurs. Here buying insurance represents choosing the “sure thing” over the risky prospect of a catastrophic event. Uncertainty and the Firm As uncertainty affects consumer behavior and managers must account for that uncertainty also effects the managers’ input/output decisions. Risk Aversion (and the firm) Just as consumers have different preferences regarding different risky prospects, so does the manager of the firm. 1. A manager who is risk neutral is interested in maximizing expected profits. The variance of profits does not have an effect on his/her decision. 2. A manager is risk averse if he/she prefers the project that has a lower risk with a lower expected value to the project that has higher risk and expected value. When a manager faces a decision to choose among risky projects, it is important to evaluate the risks and expected returns of the projects and then to document this evaluation. Risky projects may have bad outcomes and that could get the managers fired. The manager is not likely to get fired if he/she provide evidence that based on the available information the decision was sound. A convenient way to do this is to use mean variance analysis as shown below. Demonstration 12-3 Suppose a risk averse manager is considering two options: expanding the market for bologna and expanding the market for caviar. Suppose that there is a 90% chance of an economic boom and 10% of a recession. Suppose also, there is a risk free alternative (say, a treasury bill). The manager can have a joint project that 93 combines bologna and caviar. The four projects and their payoffs during boom and recession are given below. What should the manager do? Why? Project Boom (90%) Recession (10%) Mean Standard Deviation Bologna -$10,000 $12,000 -$7,800 6,600 Caviar 20,000 -8,000 17,200 8,400 Joint 10,000 4,000 9,400 1,800 T-Bill 3,000 3,000 3,000 0 Answer The manager should not invest in T-bill because the lowest payoff for joint project is greater than 3,000 which is the payoff for T-bills. Moreover, risk averse and risk neutral managers will not invest in a project with negative expected payoff. This will eliminate the bologna project. This will leave the manager with two projects: the caviar and joint projects. Which of these two projects, which have different expected values and risk, would the manager invest? It all depends on his/her preferences toward risk. The payoffs associated with the joint project above reveal the importance of diversification. By investing in multiple projects the manager, can reduce the systematic risk. Diversification also reveals why shareholders are risk neutral. They want managers to maximize the value of the firm without a regard to risk. This is because shareholders diversify in different stocks. We know that diversification diversifies systematic risk away. Producer Search Producers search for low prices of inputs when there is uncertainty regarding input prices, firms employ optimal search strategies. Demonstration 12-4 94 Profit Maximization Under Uncertainty Under certainty profit is π = PQ – C (Q) Then first profit maximization rule is MR = MC And under perfect competition this rule is: P = MC which is solved for Q*. Under uncertainty, demand is uncertain and thus total revenue is uncertain. The firm maximizes expected profit. E π = EP*Q – C(Q) where EP = q1 * p1 + q2 * p2 +….. is the expected price and qi is the expected price and qi is the ith probability. Then first-profit maximization rule is EMR = M. Where EMR is the expected marginal revenue under perfect competition, this rule is EP = MC Which can be solved for Q0? Demonstration 12-5 Suppose the perfectly competitive firm Appleway must determine how much to produce before the actual price is unknown. The firm knows the expected price. There is a 10% probability that the market price is $2 and 70% that the market price $1 when the apple juice hits. Then the expected price is EP = 0.1 * ($2) + 0.7 * (1) = 0.6 + 0.70 = $1.30 Suppose the cost function is given by C = 200 + 0.0005 Q2 How much should this firm produce to maximize expected profit? What are the expected profits of this firm? Answer Set EP = MC Then $1.30 = 0.001Q Q* = 1,300 gallons 95 Expected Profit = EP*Q – 200 – 0.0005Q2 = (1.30) (1,300) – 200 – 0.0005 (1,300)2 = 1,690 – 200 – 0.0005 (1,300)2 = $645 Uncertainty and the Market The presence of uncertainty may have a strong impact on the ability of the markets to efficiently allocate resources because it creates problems with the market. Asymmetric Information A situation that exists when some people in the market has more information than other. The people with the least information may choose not to participate in the market. For example, if a person has a box and she knows it has $10. These people who do not have this information will not accept to buy the box from her because she will not sell the box at a loss. In the stock market when some traders have insider information and others do not, there is a asymmetric information. In extreme cases asymmetric information can lead to the destruction of stock markets if asymmetric information continues to exist. Asymmetric information between consumers and the firm can affect the firm’s profit. Suppose the firm invests heavily and produces a superior product. If the consumer does not have this information, it will not buy this superior product. Asymmetric information may also affect many managerial decisions including hiring workers (workers know this abilities better then managers do), issuing credit to consumers (consumers know their credit abilities). This is why companies spend a lot of money checking on individuals and their backgrounds. There are two specific manifestations (types) of asymmetric information: adverse selection and moral hazard. Those two concepts are difficult to distinguish. Adverse selection arises when an individual has hidden characteristics (characteristics that she knows but unknown by the other proxy in an economic transaction). For example, the job applicant knows his own abilities, but the employee does not. The workers abilities thus reflect a hidden characteristic. 96 Moral hazard generally takes place when one proxy takes hidden actions (actions that it knows another party cannot observe). For example, if a manager cannot monitor (observe hidden action) then the workers effort represent a hidden action. Adverse Selection A situation where individuals have hidden characteristics and in a selection process results in a pool of individuals with economically undesirable characteristics. Example1: An industry with firms that allow 5 days of paid sick leave. One firm decides to allow 10 days. If the workers have hidden characteristics (the firm cannot distinguish between healthy and unhealthy workers). This increase in the monthly sick leave will mostly attract unhealthy workers. Healthy workers are not interested in this policy. Example2: A pool of poor drivers may have adverse selection. This pool includes two types of poor drivers: those who have bad driving habits, and those who had a string of bad luck. If the insurance companies increase the insurance premium on this pool, only drivers with bad habits would accept to pay the higher premium but those who had bad luck won’t accept to pay the higher premium. Then the insurance company will end up with the bad drivers and in this selection there is adverse impact. The insurance company should not increase the premium but should refuse to insure the bad drivers. There are insurance companies who specialize in bad drivers and they ask them to pay a high premium. Moral Hazard A situation where one party to a contract takes a hidden action that will benefit him/her at the expense of another party is called a moral hazard. Example 1: The principal agent problem. In this case the principal (the owner) offers the agent (the manager) a contract (a salary + benefits) to do certain tasks. Since the manager will receive the salary, and his/her behavior is unobservable by the owner, he/she has incentives to work less (hidden actions). The reduced effort may result in reducing profit. One way to mitigate this moral hazard by the owner is to monitor the behavior of the manager (taking away the hidden action). Another way is to compensate the manager based on his performance. 97 Example 2: Health insurance: Insurance companies are vulnerable to the moral hazard problem. The probability of a loss depends on the hidden efforts expended by the insured to avoid the loss. This moral hazard exists. When individuals are fully insured they have a reduced incentive to put forth effort to avoid a loss. Signaling and Screening Managers and other market participants can use signaling and screening to mitigate some of the problems that arise when one party to a transaction has hidden characteristics. Signaling is an attempt by an informed party to send an observable indicator of his/her hidden characteristic to an uninformed party. Thus signal must be observable. Example of “observable indicators” in the product markets are that companies send signals such as money back guarantees, free trial, labeling that indicates the product has won a “special award” or the manufacturer has been in business since say, 1933. In the labor markets, the signal takes the form that the job applicant graduated from a certain prestigious school. If the productivity of the job seeker is unobservable that will lead to lower salaries for both the productive and unproductive workers. In this case productive workers should find ways to provide information to the manager that reveals that they are indeed productive. How can productive workers send the right signals to the manager that they are productive? Talk is cheap. Unproductive workers should not easily mimic the signal. Screening The uninformed party can use screening to reduce the effects of hidden characteristics. Example: In this job market, the manager can use a self-selection device to distinguish between peoples’ skills. Example: Two people with different characteristics are applying for a job in a company. One applicant is an administrator and the other is a salesman. The manager can use a self- selection device to fit the two workers to the right job. The device may stipulate that the manager’s job will pay $20,000 and the salesman job pays 10% of total sales. The second worker who identifies his characteristic to be a salesman he will ask for the salesman job. He knows he is a salesman and can generate a million dollars in sales. Then this salesman 98 compensation is $100,000 (10%*1Mn), which is higher than the $20,000 job. The manager knows that his ability does not fit the salesman job. He will go for the administrative job.- 99