Ecnomics D101: Lecture 11

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Ecnomics D10-1 Lecture 11

• Profit maximization and the profit function

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Profit maximization by the price-taking
competitive firm

• The firm is assumed to choose feasible
  input/output vectors to maximize the excess of
  revenues over expenditures under the assumption
  that it takes market prices as given.
• There are 3 equivalent approaches to the
  profit-maximization problem (and associated
  comparative statics)
• The algebraic approach using netput notation
• The dual approach using the properties of the
  profit function.
• The Neoclassical (calculus) approach using FONCs
  and the Implicit Function Theorem.

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The algebraic approach to the profit maximization
problem

• The problem of the firm is to maxy?Y p.y
• Define the profit function p(p) as the value
  function
• Let y(p) argmaxy?Y p.y denote the solution set
• CONVEXITY
• Let y0?y(p0), y1?y(p1), and yt?y(pt), with pt
  tp0 (1-t)p1
• p(pt) ptyt tp0yt (1-t)p1yt tp0y0
  (1-t)p1y1 tp(p0) (1-t)p(p1)
• LAW OF OUTPUT SUPPLY/INPUT DEMAND
• ?p?y (p1-p0)(y1-y0) (p1y1-p1y0) (p0y0-p0y1)
  0
• Implies all own price effects are nonnegative
  i.e.,
• ?yi/?pi 0

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Results using the profit function

• The Derivative Property and Convexity Dpy(p)
  and D2p is positive semi-definite
• Proof Let y0 y(p0) for some p0gtgt0. Define
  the function g(p) p(p) - p.y0. Clearly, g(p)
  0 and g(p0) 0. Therefore, g is minimized at
  p p0. If p is differentiable, the associated
  FONC imply that Dg(p0) Dp(p0) - y(p0) 0.
  Similarly, if p is twice differentiable the
  SONCs imply that D2g(p0) D2p(p0) is a positive
  semi-definite matrix.
• LAW OF OUTPUT SUPPLY/INPUT DEMAND
• Combining the above results, D2p(p) Dy(p) is a
  positive semi-definite matrix. This implies that
  (?yj/?pj)0 i.e., the physical quantities of
  ouputs (inputs) increase (decrease) in own prices.

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The Neoclassical approach to profit maximization
the single output case

• Problem maxz pf(z)-w.z
• Solution z(p,w) argmaxz pf(z)-w.z
• Assume f is twice continuously differentiable.
• FONCs pDf(z(p,w))-w 0 z(p,w) 0
  (pDf(z(p,w))-w).z 0
• For z(p,w)gtgt0, SONCs require pD2f(z(p,w))
  negative semi-definite
• COMPARATIVE STATICS
• Assuming z(p,w)gtgt0,differentiate the FONCs to
  obtain
• pD2fDwz I or Dwz (1/p)D2f-1 when the
  Hessian matrix of f is nonsingular. In that
  case, Dwz is negative semi-definite.
• (Also, Df pD2fDpz 0 or Dpz -(1/p)D2f-1Df
  so that?q/?p DfDpz -(1/p)DfD2f-1Df 0)
  

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The Neoclassical approach single output, two
input example

• Max pf(z1,z2) - w1z1 - w2z2
• Let (z1(p,w1,w2),z2(p,w1,w2)) argmax
  pf(z1,z2)-w1z1 - w2z2
• FONCs for interior solution
• pf1(z1(p,w1,w2),z2(p,w1,w2)) - w1 0
• pf2(z1(p,w1,w2),z2(p,w1,w2)) - w2 0
• Differentiating with respect to, e.g. w1, yields
• pf11(?z1/?w1) pf12(?z2/?w1) 1
• pf21(?z1/?w1) pf22(?z2/?w1) 0