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Chapter 8 and 9 Cost Theory and Applications 1. Relevant Costs- costs which vary over alternatives of a decision 2. Sunk costs- costs incurred regardless of alternative action. Also, cost of purchased resources with no opportunity value 3. Incremental cost- change in cost with a change in activity level · Proper measure (long run v. short run) must be geared to the duration of the planning horizon. · long run- all inputs are variable (i.e., the flow of resources per time period can be changed). · short run- the flow of one or more resources are fixed per period of time. 4. Alternative use or opportunity cost basis for valuation 4. Traceable v. Nontraceable costs- Two types of nontraceable costs a. joint costs- costs incurred in the production of two or more types of output which are produced in fixed proportion. 1. passengers above deck and freight below deck on an aircraft. Direct aircraft flying expenses are joint between passenger services above deck and freight services below deck. 2. Marginal costs may not be precisely determined b. common costs- costs incurred in the production of two or more types of output in which the outputs can be varied seperately. 1. Refined petroleum products from crude oil 2. Marginal cost can be precisely determined 5. Implicit costs Cost Functions 1. See derivation of cost function in Production Theory notes Example- If Q = K.5L.5 and PK = PL = 1 Long Run Total Cost = 2 Q If K = 1, short run total costs = 1 + 1 Q2 If K = 4, short run total costs = 4 + 1/4 Q2 Appendix Q = a Lb1 K b2 C = CL L + CK K C = CL[b1/(b1+b2)] CK [b1/(b1+b2)] [(b2/b1)[b1/(b1+b2)]+(b2/b1)[-b2/(b1+b2)]] (Q/a )[1/(b1+b2)] COST FUNCTION FOR COBB-DOUGLAS PRODUCTION FUNCTION: Q = X.5 Y.5|PX=PY=1 $/PERIOD SHORT RUN 40 SHORT RUN TOTAL COST 35 |Y=4 TOTALCOST |Y = 1 LONG RUN 30 TOTAL COST 25 20 15 10 5 0 0 5 10 15 20 QUANTITY / PERIOD MARGINAL, AVERAGE TOTAL AND AVERAGE VARIABLE COST FUNCTIONS 9 8 SMC|Y=1 SAVC|Y=1 7 6 5 SAC|Y=1 4 SAC|Y=4 3 2 LMC=LAC 1 0 AFC|Y=1 0 2 4 6 8 10 12 2. Long and Short Run Cost Concepts- a. long run: all inputs can be varied Returns to Scale v. long run costs b. Short run: certain inputs are fixed per time period Average fixed cost = AFC = total fixed cost/Q Average variable cost = SAVC = total variable cost/Q Average total cost = SAC = total cost/Q Incremental cost = marginal cost = SMC = d total cost/dQ 3. Cost Elasticity = dC/dQ Q/C = Marginal cost/ average cost a. if cost elasticity < 1, economies of scale in LR economies of utilization in SR b. if cost elasticity > 1, diseconomies of scale in LR diseconomies of utilization in SR c. If long run cost elasticity = short run cost elasticity, firm has efficient size plant for that output d. Railroad Example: Long Run Short Run Small Railroads .70 .67 Large Railroads .99 .77 Small roads have too little output Large roads have excess capacity Unit Cost vs. Cost Elasticity $/Q SRAC1 8 7 6 SRMC2 SRMC1 5 SRAC2 4 LRAC 3 2 1 0 0 5 10 15 20 QUANTITY Factors Producing Scale Economies • Specialization of Labor • Technological factors • Quantity discounts • Lower cost of capital • Principle of massed reserves • Principle of multiples Breakeven Analysis: Used to examine the profitability of new product lines Assume a linear total cost function and a linear total revenue curve (completely elastic demand curve): Total Revenues = Total Costs P Q = F + V Q implies Solving for Q: Qbreakeven = F/(P-V) = (Fixed Cost)/(Unit Profit Contribution) Example: P(Price) = $2 per unit, F(Fixed Cost) = $40,000, and V (Variable Cost per unit) = $1.20 per unit Qbreakeven = F/(P-V) = 40,000/(2-1.2)= 50,000 Breakeven Analysis $ of Revenue and Cost 400 Total Revenue 350 300 Total Cost 250 Profit 200 150 100 50 Output 0 0 20 40 60 80 100 120 140 160 180 200 QBE Operating Leverage- extent to which fixed production facilities are used in the operation to lower cost and increase risk 1. Degree of Operating Leverage (DOL) = the percentage change in profits with a percentage change in output A measure of risk of a more capital intensive production process 2. DOL = total profit contribution/total profits = Q (P - V) / [Q (P - V) - F] 3. Example: P = 2, F = 40,000, V = 1.2, then at Q = 100,000 DOL = 100,000(2-1.2)/[100,000 (2-1.2) - 40,000] = 80,000/40,000 = 2 Degree of Operating Leverage 400 Total Revenue 350 300 Total Cost 250 Profit DOL = 2 200 150 100 50 0 0 20 40 60 80 100 120 140 160 180 200 QBE Degree of Operating Leverage Comparison of Two Cost Functions at Q = 100 400 Total Revenue 350 DOL = 2 300 Total Cost= 250 Profit 40 + 1.2 Q 200 Total Cost= 150 100 + .6 Q 100 DOL= (2-.6)100/40 50 = 3.5 0 0 20 40 60 80 100 120 140 160 180 200 QBE Empirical Cost Estimati Approaches A. Accounting B. Statistical C. Engineering D. Survey or Survivor Tech A. Accounting: may involve the following 1. separate fixed and variable cost components 2. assignment of variable portion to output measures, input measures, quality measures, etc. 3. obtain unit costs by dividing the cost assigned to any category by the number of units 4. To estimate the cost for a particular product or service, multiple the unit costs by their respective number of units output, input, etc Accounting Costing All Costs 100 variable costs Fixed Costs Costs Unrelated to cost study 80 10 10 Output #1 Output #2 Output #3 Units of #1 10 Units of #2 5 Units of #3 3 Costs #1 40 Costs #2 25 Costs #3 15 Cost per unit of #1 Costs per unit of #2 Costs Per unit of #3 40/10=4 25/5=5 15/3=5 Cost = F + v1 X1 + v2 X2 + v3 X3 10 4 5 5 Statistical A. Long Run (Planning) v. Short Run (Operating) Cross sectional data v. Time series data Cross sectional: data gathered on a number of individuals at approximately the same point in time Time Series: data gathered on a single individual at different points in time Long Run and Short Run Costs 30 25 Chrysler Ford 20 15 199 10 1994 4 5 0 0 5 10 15 20 B. Requirements 1. Output Matching - example of deferred maintenance in RR 2. Uniform production with a time period 3. no technological change- might add a time variable to the regression equation 4. no changes in factor prices or inflation 1. deflate by a price index 2. reconstruct costs based on future prices and historic input and output levels 3. include factor prices in the cost function Output Matching Railroad Maintenance of Way and Structures ($000) 12 10 8 6 4 2 0 0 2 4 6 8 10 12 Gross Ton Miles (000,000) Nonuniform Production If 1/2 month at Q=2 and 1/2 month at Q=8, 30 Theoretical Cost = 10 25 Actual Cost = 16 20 15 10 5 0 0 2 4 6 8 With Technical Change We can add a time term to the Cost Function ln Cost = b0 + b1 ln Q + b2 t + e Q = output t = time b2 = the percentage change in cost per year Adjust for Factor Prices • Deflate by a price index – Cost / CPI = b0 + b1 Q • Include factor prices in the Cost Function – ln Cost = b0 + b1 ln Q + b2 ln PL + b3 ln PF + b4 ln PK + e • Reconstitute Costs based on future prices Reconstituted Costs Q= L.5 K.5 PL PK Labor Cap. Out. Cost Rev. 10 10 10 10 10 200 210 12 12 8 8 8 192 168 20 10 8 16 11.3 320 240 18 9 5 10 7.1 180 150 15 10 6 12 8.5 210 180 12 9 9 12 10.4 216 216 Recorded and Revised Costs 350 Recorded Costs vs. Output 300 250 Reconstituted Cost vs. Output 200 150 100 50 0 0 2 4 6 8 10 12 Biases with Cross Sectional Data 16 A cross section of plants but each one is of a 14 different vintage. So the decrease in costs may 12 be due to the technology of the plant rather than to economies of scale. 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 11 12 C. Common Functional Forms for statistical estimation 1. linear TC = a + b Q 2. quadratic TC = a + b Q + c Q2 3. Cubic TC = a + b Q + c Q2 + d Q3 4. log linear log TC = a + b log Q 5. log quadratic log TC = a + b log Q + c (log Q)2 6. Translog log TC = a + b log Q + c (log Q)2 + S di log wi + S S eij log wi log wj + S fi log wi log Q where wi is the price of factor i (labor, capital, etc.) D. Long Run Cost estimation: use cross- sectional data Empirical results of earlier studies- L shaped cost functions L Shaped Average Cost Curve $/Q 12 10 Minimum Efficient Size 8 6 4 2 0 Q 0 1 2 3 4 5 6 7 8 9 10 11 12 Survivor Technique Example- steel ingots open hearth Bessimer process Firm Size Percent of Industry Capacity Number of Firms % ind cap 1930 1938 1951 1930 1938 1951 Very small: < .5% 7 6 5 39 29 22 Small: .5-2.5% 19 13 14 18 13 13 Medium: 2.5-25% 35 45 46 6 7 7 Large: > 25% 39 36 35 1 1 1 Engineering Technique Example: Oil Pipeline Throughput = f(diameter of pipe, horsepower of engines driving fluids, number of pumping stations) T2.735 = H D4.735/.01046 or T = k H.37D1.73 where T = throughput H = horsepower D = diameter of pipe Oil Pipeline Long-Run and Short-Run Average Costs 5 4 Long Run Ave. Cost 3 Short Run Diam. = 10 $/Q 2 Short Run Diam. = 20 1 0 11 13 15 17 19 21 1 3 5 7 9 Q Breakeven Analysis Exam Problems 1. The Ajax Company estimates its fixed cost at $500,000 and its average variable cost at $2.00 per unit. Ajax sells its product at a price of $4.00 per unit. a. What is Ajax's current break even output level? Q = 500000/(4-2) b. What is the firm's cost elasticity at Q = 500,000? CE = 2 (500,000)/[2 (500,000) + 500,000] c. At Q = 500,000, what is the firm's degree of operating leverage? Interpret it. DOL=(4-2)(500,000)/[(4-2)(500,000)-500,000] d. What price would yield an average profit contribution of 40 percent? .4=(P - 2)/P or P=2/.6 2. The Baker Company estimates its fixed cost at $1,000,000 and its average variable cost at $4.00 per unit. The firm's goal is to sell 500,000 units. a. What is Baker's break even price? Profits = (P-4)500,000 - 1000000 = 0 P=6 b. What is its cost elasticity at Q = 500,000? Cost Elas. = 4(500,000)/[1,000,000+4(500,000)] = 0.67 c. What price must Baker charge if its average profit contribution is to be 60 percent? (P-4)/P = .6 P = 10 d. At Q = 500,000 and price = $8, what is Baker's degree of operating leverage? Interpret it. DOL = (8-4)500,000/((8-4)500,000-1000000) =2