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Chapter 4 - Consumer and Firm Behavior The Work-Leisure

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Chapter 4 - Consumer and Firm  Behavior The Work-Leisure Powered By Docstoc
					            Consumer and Firm Behavior:
             The Work-Leisure Decision
              and Profit Maximization
                          Chapter 4




                    ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                       Professor C. K. Yip              1
            Static vs Dynamic Decision Making
• In this and the next chapters, we are considering static decision making,
  i.e., planning over a single period.
• From chapter 6 to 9, we are going to discuss dynamic decision making,
  i.e., planning over more one period.
• Chapter 4 first recalls what you’ve learnt in the last semester: the micro
  behavior of a representative consumer and a representative firm.
• Chapter 5 then assembles these in a macro model in order to address
  some important macro issues. The role of government is also introduced.




                          ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                             Professor C. K. Yip                 2
Objectives of the Representative Consumer &
          the Representative Firm
• Representative Consumer: To maximize utility subject to budget (and
  time) constraint by allocating time between work and leisure;
• Representative Firm: To maximize profits subject to technological
  constraint by deciding how much labor to be hired.




                          ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                             Professor C. K. Yip               3
                   Assumptions of the Model
1) Two Goods:
    – Consumption good, which is an aggregation of all consumer goods in the
      economy.
    – Leisure, which is any time spent other than working in the market.
      e.g. Recreational activities, sleep and household work.
2) One Consumer:
    – All consumers are identical in terms of preferences, ability, time constraint
      and budget constraint. Then, the economy will behave as if there were only
      one consumer, one that we refer to as the representative consumer.




                             ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                                Professor C. K. Yip                      4
                   Assumptions of the Model
3) Price-Taking:
    – The representative consumer is a price-taker, i.e., he takes all market prices
      as given, and acts as if his actions had no effect on those prices.
4) No Money:
    – The economy we’re considering is a barter economy, i.e., all trade involves
      barter exchanges of goods for goods in the absence of money.




                             ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                                Professor C. K. Yip                        5
            The Representative Consumer’s
                Optimization Problem
• Objective: to make himself as well off as possible given the constraints
  he faces.
• Two Ingredients in this problem:
    – Consumer’s preferences
    – Consumer’s budget constraint




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7/12/2013                              Professor C. K. Yip                   6
                          Preferences
• The preferences of the representative consumer is captured by the
  utility function,
                                      U(C , l)
  where C is the quantity of consumption,
           l is the quantity of leisure
• Any particular pair of consumption and leisure (C , l) is called a
  consumption bundle.
• For each consumption bundle, the utility function U assigns a real
  number so that different bundles can be ranked.




                        ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                           Professor C. K. Yip              7
                                      Preferences
• Consider two distinct bundles (C1 , l1) and (C2 , l2)
    – (C1 , l1) is strictly preferred to (C2 , l2) if
                                 U(C1 , l1) > U(C2 , l2)
    – Consumer is indifferent between the two bundles if
                                         U(C1 , l1) = U(C2 , l2)
• Assumptions on Preferences:
  1) More is always preferred to less
    – A consumer always prefers a consumption bundle that contains more
      consumption, more leisure, or both.




                                    ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                                       Professor C. K. Yip              8
                                Preferences
   2) Consumer prefers a more diversified consumption bundle.
    – If the consumer is indifferent between (C1 , l1) and (C2 , l2), then some
      mixture of the two will be preferable to either one.
    – Example: Consider a new bundle (C3 , l3), where C3 = lC1 + (1 – l)C2,
      l3 = ll1 + (1 – l)l2 and l lies between 0 and 1 (a fraction), then
                            U(C3 , l3) > U(C1 , l1) = U(C2 , l2)
   3) Consumption and leisure are normal goods.
    – A good is normal (inferior) for a consumer if the quantity of the good that
      he/she purchases increases (decreases) when income increases.




                              ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                                 Professor C. K. Yip                     9
      Graphical Representation of Preferences
• An indifference curve connects a set of points, with these points
  representing consumption bundles among which the consumer is
  indifferent.
• A family of indifference curves is called indifference map.




                          ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                             Professor C. K. Yip              10
            Properties of Indifference Curves
• Consider a consumption bundle
  B. Since a consumer prefers                       Consumption
  more to less, any bundle that is
  indifferent to B must lie within
  quadrant II and IV.                                             IV        I

• Implication: An indifference curve                                    B
  slopes downward.
                                                                  III       II




                                                                                 Leisure




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7/12/2013                              Professor C. K. Yip                                 11
            Properties of Indifference Curves
• Consider any two bundles A and
  B, since a consumer prefers a                       Consumption
  more diversified bundle C to
  either A or B, the set of bundles                                   A
  that are indifferent to A and B
  must lie below the straight line                                        C
  AB.
• Implication: An indifference curve is                                       B
  convex, that is bowed-in toward the
  origin.
                                                                                  Leisure




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      Graphical Representation of Preferences

                 Consumption, C




                                       A



                   C1                  B

                                                D         I2
                   C2                               I1



                                  l1       l2            Leisure, l




                   ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                      Professor C. K. Yip                    13
                Marginal Rate of Substitution
• Marginal rate of substitution of
  leisure for consumption (MRSl,C) is                      Consumption, C
  the rate at which the consumer is                                                   Slope = MRSl,C
  just willing to substitute leisure for
  consumption good.                                                               A
                                                              C1
• It is also minus the slope of the
  indifference curve.
                                                              C2                  D          B
• Convexity of indifference curve is                                                                    I2

  equivalent to                                                                                  I1

                                                                             l1         l2
    – Diminishing marginal rate of substitution.                                                      Leisure, l

      (compared slope at A and slope at
      B)


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7/12/2013                                   Professor C. K. Yip                                                    14
             Marginal Rate of Substitution
• The MRS at A is larger (in terms of
  absolute magnitude) than the MRS                   Consumption, C
  at B.                                                                         Slope = MRSl,C
• As we increase l and reduce C, i.e.
  moving from A to B along I1, the                                          A
                                                        C1
  consumer needs to be
  compensated more in terms of l to
                                                        C2                  D          B
  give up another unit of C.                                                                      I2

• The consumer requires this extra                                                         I1

  consumption because of a                                             l1         l2            Leisure, l
  preference for diversity.


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              Marginal Rate of Substitution
Mathematical Derivations:
• Suppose indifference curve I1                       Consumption, C
  represents the utility level ,                                                 Slope = MRSl,C


                                                                             A
• Totally differentiate this with                        C1
  respect to C and l gives
                                                         C2                  D          B
                                                                                                   I2
                                                                                            I1

                                                                        l1         l2            Leisure, l




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      Constraints faced by The Representative
                     Consumer
• Two constraints:
    – Time constraint for l
    – Budget constraint for C
• The time constraint for the consumer is given by
                                      s
                                 l+N =h
  where h is the total number of hours available (e.g., 24 hours a day), l is
                         s
  the leisure time and N is the time spent working (or labor supply).




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                         Budget Constraint
• Sources of income:
                          s
  1) Real wage income, wN
    – w is the real wage, i.e., the price of one unit of labor time in terms of
      consumption goods (the numeraire).
  2) Real dividend income, p
    – Since the firms are owned by the representative consumer, any profits made
      by firms are distributed to him as dividends.




                               ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                                  Professor C. K. Yip                    18
                    Budget Constraint
• Taxation T: A lump-sum tax, i.e. a tax that does not depend on the
  actions of the economic agent who is being taxed.
                                          s
• Real Disposable Income = wN + p – T
• The consumer first receives income and pays taxes in terms of
  consumption goods, and then decides on how much to consume out of
  the disposable income.
• All disposable income is consumed, i.e.
                                            s
                                C = wN + p – T
                                   = w(h – l) + p – T




                        ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                           Professor C. K. Yip              19
                        Budget Constraint
• Reasons:
    – Since the consumer only lives for one period, there is no incentive to save
      anything.
    – Since more is preferred to less, any wastage is not optimal.
• The consumer’s budget constraint can be written as
                           C + wl = wh + p – T
• RHS = Total implicit real disposable income
• LHS = Implicit real expenditure on consumption goods and leisure
• Note: w can also be interpreted as the market price, or the opportunity
  cost, of leisure time.



                             ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                                Professor C. K. Yip                    20
       Graphical Representation of the Budget
                    Constraint
• Budget constraint:
       C = –wl + (wh + p – T)                          Consumption, C

  thus slope = –w.                            wh + p – T
                                                            A

• The vertical intercept, wh + p – T,
  is the maximum consumption that                                      C = –wl + wh + p – T
  can be achieved when the
  consumer consumes no leisure.
• Case 1: p < T
                                                                                      B

                                                                            h + (p – T)/w     h
                                                                                                  Leisure, l




                          ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                             Professor C. K. Yip                                                  21
       Graphical Representation of the Budget
                    Constraint
• Case 2: p > T
                                                   Consumption, C
    – The consumer can still enjoy
      C = p – T > 0 even if he                     A

      chooses not to work.
    – When C = 0, l = h + (p – T)/w,                                    C = –wl + wh + p – T
      but it is not feasible as the
      maximum time can only be h
                                                p–T                                        B
    – When l = h, C = p – T

                                                                                        D
                                                                                       h
                                                                                           Leisure, l




                           ECO 2021 Intermediate Macroeconomic Theory
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       Graphical Representation of the Budget
                    Constraint
• The budget constraint tells us what
                                                  Consumption, C
  consumption bundles are feasible
  to consume given the market real                A

  wage (w), dividend income (p) and
  taxes (T).                                                            Not Feasible

• The consumption bundles within
  the shaded regions and on the
                                                                 Feasible                  B
  budget constraint, are feasible.
• Thus the shaded region together                                                      D
  with the budget constraint is called                                                 h
  the feasible set.                                                                        Leisure, l




                          ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                             Professor C. K. Yip                                               23
                   Consumer Optimization
• The representative consumer is assumed to be rational, i.e. he always
  chooses the best feasible consumption bundle, or the optimal
  consumption bundle.
• “Best” in the sense that it lies on the highest possible indifference curve.
• “Feasible” in the sense that it lies within the feasible set.




                             ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                                Professor C. K. Yip                     24
                       Graphical Solution
• Suppose p > T.
• Claim: H is the optimal consumption                 Consumption, C

  bundle.
                                                      A
Reasons:
                                                                F
• Any bundle inside the budget                                          H
  constraint is not optimal
                                                                J                    I2
  (compare J to F).                                                         E
                                                                                I1
                                                   p–T                                   B
• B is preferred to any point on
  BD.
                                                                                     D
• For any point on AB, the
                                                                                     h
  consumer can always improve by                                                          Leisure, l
  moving closer to H.

                           ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                              Professor C. K. Yip                                             25
                    Mathematical Solution
• The consumer tries to solve the following constrained optimization problem
                               max U(C , l)
                                      C,l

  subject to                 C = w(h – l) + p – T
  and                         C ³ 0, h ³ l ³ 0.
• Lagrangian
                    L = U(C , l) + l[w(h – l) + p – T – C]
   where l is the Lagrangian multiplier.




                            ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                               Professor C. K. Yip               26
                   Mathematical Solution
• We assume that an interior solution can be obtained. This means
  choosing C = 0, l = h or l = 0 are not optimal (so that we can ignore the
  last two constraints).
• Formally, we can impose the restrictions:
                        Uc(0 , l) = ∞ and Ul(C , 0) = ∞
  For any C and l, to guarantee an interior solution.
• First-order (Necessary) conditions (FOCs):
   – Obtained by differentiating the Lagrangian with respect to C, l and l.
      (Recall: Lagrangian equation L = U(C , l) + l[w(h – l) + p – T – C]
                                  Uc(C , l) = l,
                                 Ul(C , l) = lw,
                          w(h – l) + p – T – C = 0.

                          ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                             Professor C. K. Yip                27
                   Mathematical Solution
• From the FOCs, we obtain
                                                     Consumption, C


                                                     A

• At H, where an indifference curve
                                                               F
  is just tangent to the budget                                        H
  constraint, the above equality holds.                        J                    I2
• If MRS > w (e.g. at F), the                                              E
                                                                               I1
                                                  p–T                                   B
  consumer would be better off by
  increasing l and reducing C, thus
                                                                                    D
  moving closer to H.
                                                                                    h
                                                                                         Leisure, l




                          ECO 2021 Intermediate Macroeconomic Theory
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                      Comparative Statics
• To determine how C and l changes when any of p, T and w changes.
• Recall the FOC of the consumer’s problem,



  which can be written as
                      Ul(C , l) – wUc(C , l) = 0.                 (1)
• From the budget constraint,
                     w(h – l) + p – T – C = 0. (2)
• The two form a system of equations in terms of C and l (endogenous
  variables).


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                      Comparative Statics
• Totally differentiate the two equations
              –dC – wdl + (h – l)dw + dp – dT = 0                        from (2)
            [Ucl – wUcc]dC + [Ull – wUcl]dl – Ucdw = 0                      from (1)
In matrix form,



                  A
• Determinant of the bordered Hessian matrix A is
                         Ñ = –Ull + 2wUcl – w2Ucc
• Strict quasiconcavity of U è Ñ > 0.

                            ECO 2021 Intermediate Macroeconomic Theory
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               1) Changes in p and/or T
• Using Cramer’s Rule, we get




• The assumption that consumption and leisure are normal goods is
  equivalent to the conditions –Ull + wUcl > 0 and Ucl – wUcc > 0.




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                1) Changes in p and/or T
                  Graphical Illustration
• Consider a net increase in p – T.                         Consumption, C

• Since w (slope) remains the same,                     F
  the budget constraint makes a
  parallel shift (from AB to FJ).                      A

• Since disposable income ­while
                            ,                          C2                                  K

  prices remain the same, there is                     C1
                                                                                  H
                                                                                                            I2
                                                                                                        J
  only a pure income effect on the                                                             I1
  consumer’s choices.
                                                                                                        B
• The new optimal consumption
  bundle is K, where both C and l ­                                                                 D
  (normal goods).                                                            l1       l2
                                                                                                    h
                                                                                                        Leisure, l



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                  1) Changes in p and/or T
                    Graphical Illustration
Remark:                                                        Consumption, C

• The increase in consumption                              F

   (C2 – C1) is less than the increase
  in nonwage income (distance                             A

  AF).                                                    C2                                  K

• Since the consumer is working                           C1
                                                                                     H
                                                                                                       J
                                                                                                             I2

  less (leisure ­ wage income ¯.
                ),                                                                                I1

• This will offset part of the                                                                         B

  consumption increase.
                                                                                                       D
                                                                                l1       l2
                                                                                                       h
                                                                                                           Leisure, l



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                         2) Changes in w
• Using Cramer’s rule,




• C is normal good è –Ull + wUcl > 0, together with Ñ > 0 and Uc > 0 è




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                       2) Changes in w
• However, we cannot determine the effect of a change in w on l.
• Reason: It depends on the relative magnitude of the opposing income
  and substitution effects.
• Substitution effect: w­ Opportunity cost of leisure ­
                          è
                             (l becomes more expensive relative to C)
                            è Demand for leisure ¯


• Income effect: w ­ Wage income ­
                   è
                    è Demand for leisure ­
                                         (normal good)



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                     2) Changes in w
                   Graphical Illustration
• Suppose p > T and w ­      .                              Consumption, C
• The budget constraint shifts from
                                                                   I1 I
  ABD to EBD (with a steeper                            E
                                                                        2


  slope).
                                                        J
• This shows a special case in which                   A
                                                                                  H
  leisure remains unaffected.                          C2
                                                                             O
• Pure substitution effect: Movement                   C1
                                                                             F
  from F to O (on the same                                                                B
  indifferent curve).                                                                     K

• Pure income effect: Movement from O
  to H.                                                                               D
• Both income and substitution                                               l1
                                                                                      h
  effects act to ­ C.                                                                     Leisure, l



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                      2) Changes in w
                    Graphical Illustration
• Labor supply curve which specifies
  how much labor the consumer wishes                  Real Wage, w
  to supply given any real wage.                                             s
                                                                         N
• Algebraically, the labor supply curve is
              s
            N (w) = h – l(w),
  where l(w) is the demand function for
  leisure.
• Substitution effect > Income effect
  è Upward sloping labor supply curve
• Net ­ (p – T)
        in                                                              Employment, N

  è Upward shift in labor supply curve

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  Example: C and l are perfect complements
• Suppose the consumer’s utility                          Consumption, C
  function can be represented by
        U(C , l) = min{C , al}.                                                         C = al
           (Leontief Function)
                                                         A
  where a is a positive constant.                                                           I2
                                                                               E
• Note that more is not always                                                     I1
                                                                           F
  preferred to less. The consumer can                                                   B
  be better off only if he receives
  more of both goods.
• Thus, it is always optimal to choose                                              D
                                                                                   h
                 C = al.                                                                Leisure, l



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  Example: C and l are perfect complements
• Combining C = al and the budget constraint gives



• In this case,



• This is because with perfect complements, there are no substitution
  effects. Thus leisure ­ real wages ­
                        as           .




                          ECO 2021 Intermediate Macroeconomic Theory
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                 The Representative Firm
• The firm owns productive capital and hires labor to produce
  consumption goods.
• Production technology is captured by the production function, which
  describes the technological possibilities for converting factor inputs
                         d
  (capital K and labor N ) into outputs Y.
                                             d
                               Y = zF(K , N )
  where z is total factor productivity.
                      d
• z ­ both K and N will be more productive.
     è




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            Assumptions on Production Function
• Production function exhibits constant returns to scale (or homogenous of
  degree one).
                                            d
    – For any x > 0, xY = zF(xK , xN ).
    – If all factor inputs are changed by a factor x, then output changes by the
      same factor x.
    – In this case, a perfectly competitive economy with numerous small firms
      will behave in exactly the same way as one with a single representative
      firm (same level of efficiency).
                                              d              d
    – Increasing return to scale: zF(xK , xN ) > xzF(K , N ).
                                                d              d
    – Decreasing return to scale: zF(xK , xN ) < xzF(K , N ).




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            Assumptions on Production Function
• Positive marginal product of capital
  (MPK) and marginal product of labor
                                                        Output, Y
  (MPN).
                                                                      Slope = MPK
    – MPK (MPN) is the additional output                                                           *
                                                                                           F(K , N )
      that can be produced with one
      additional unit of capital (labor),                                         A
      holding constant the quantities of labor
      (capital).
                                      *
    – Fix the quantity of labor at N , then
                   *
      the MPK at K is the slope of the
      production function at point A.
                                                                              *
                                                                          K           Capital Input, K




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            Assumptions on Production Function
    – Algebraically, we assume that,
              d                    d
      FK(K , N ) > 0 and FNd(K , N ) > 0.




                                                        Marginal Product of Labor, MPN
    – Conceptually, this simply means:
      more inputs yield more output.
• Diminishing Marginal Product
    – The declining MPK and MPN is
      equivalent to the concavity of the
      production function.
                                                                                           MPN
    – Algebraically, this means
                               d
                   FKK(K , N ) < 0,
                                 d                                                                        d
            and FNdNd(K , N ) < 0.                                                       Labor Input, N

    – Implicitly, we assume that
      F(. , .) is twice differentiable.
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            Assumptions on Production Function
• MPN ­ K ­
      as
    – Algebraically, this means




                                                         Marginal Product of Labor, MPN
    – Increase in the quantity of
      machinery and equipment
      enhances the productivity of the                                                                       2
                                                                                                       MPN
      workers.
                                                                                                   1
                                                                                             MPN
• F(. , .) is quasiconcave.
                                                                                                             d
                                                                                          Labor Input, N




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            Cobb-Douglas Production Function
• Probably the most commonly used form of production function which
  satisfies all the above properties
                                        a   d b
                                 Y = zK (N )
  where 0 < a, b < 1.
• a + b = 1 è Constant return to scale.
  a + b > (<) 1 è Increasing (decreasing) return to scale.
• If there are profit-maximizing price-taking firms and a + b = 1, then a
  will be the share that capital receives of national income.




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      Changes in Total Factor Productivity (z)
• Changes in z is critical to our     Output, Y
  understanding of the causes of                                                    *
                                                                               Z2F(K , N )
                                                                                             d

  economic growth and business cycles                                               *
                                                                               Z1F(K , N )
                                                                                             d

  (real business cycles theory).
• Effects of z ­ :
  1)dOutput ­ given values of K and
                for                                                                 Labour Input, N
  N.
                                      Marginal Product of Labor, MP
  2) MPN ­ given value of K.
                                                                           N
             for
• Factors that would affect z:
    –   Technological innovation
    –   Weather
    –   Government regulations                                                              MPN
                                                                                                  2

                                                                                        1
    –   Price of energy                                                          MPN
                                                                                                             d
                                                                                            Labor Input, N
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              Profit Maximization Problem
• Assume that capital K is fixed. Then the firm’s problem is to choose a
                d
  quantity of N in order to maximize its profits.
• The representative firm is assumed to behave competitively, i.e. taking
  the real wage w as given.
                                              d
• The problem can be stated as (choosing N )
                                          d       d
                         max p = zF(K , N ) – wN
• Similar to the consumer’s problem, we assume
                     FNd(K , 0) = ∞ and FNd(K , ∞) = 0
   to ensure interior solution in the firm’s profit maximization problem.




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              Profit Maximization Problem
• The optimal condition (FOC) is
                    d      d
         z[∂F(K , N )/∂N ] = w                        Revenue, Variable Costs            d
                                                                                    wN
• This states that it is optimal for the                                                     d
                                                                                     zF(K , N )
  firm to hire workers up to a level in
  which the MPN equals the real wage.
                                                                   A
• Graphically, the optimal quantity of
  labor N* is at A, where the slope of
  total revenue function is equal to                               B

  the slope of the total variable cost
  function.                                                             *
                                                                    N           Labor Input, N
                                                                                                 d

• The maximized profits p* is given
  by the distance AB.

                           ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                              Professor C. K. Yip                                   48
             Profit Maximization Problem
• The FOC of the profit-
  maximization can also be
                                                    Real Wage, w
  interpreted as the firm’s demand
  curve for labor, for given values
  of z and K .
• The optimal condition (FOC) is                                          MPN or Labor Demand Curve

         MPN(K , N) = w.
• Diminishing MPN implies w and
  N are inversely related.
                                                                                                       d
                                                                       Quantity of Labor Demanded, N




                          ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                             Professor C. K. Yip                                                  49
                   Comparative Statics
• Recall the FOC of the firm’s problem
                                         d
                               zFNd(K , N ) = w
• Totally differentiate this gives
                             d
                 zFNdNddN – dw + FNddz + zFKNddK = 0.
  Thus, we obtain




                        ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                           Professor C. K. Yip              50
                        Quasiconcavity
• A function f(x) is quasiconcave if
                   f(x1) ³ f(x2) è f[ax1 + (1 – a)x2] ³ f(x2)
  for any 1 ³ a ³ 0.
• f is strictly quasiconcave if
                   f(x1) ³ f(x2) è f[ax1 + (1 – a)x2] > f(x2)
  for any 1 > a > 0.
• Consider a strictly quasiconcave utility function U(C , l).
  Suppose x1 = (C1 , l1), x2 = (C2 , l2), then
            U(x1) = U(x2) è U[ax1 + (1 – a)x2] > U(x1) = U(x2)
  for any 1 > a > 0.
  Thus the indifference curves are strictly convex.


                         ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                            Professor C. K. Yip              51
                         Quasiconcavity
• Strict quasiconcavity also implies that the bordered Hessian matrix of
  the utility function is negative definite, i.e.,




                         –Ull + 2wUcl – w2Ucc > 0




                          ECO 2021 Intermediate Macroeconomic Theory
7/12/2013                             Professor C. K. Yip                  52

				
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