Transcript Ecnomics D10-1: Lecture 11

Ecnomics D10-1: Lecture 11
Profit maximization and the profit
function
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 profitmaximization 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.
The algebraic approach to the profit
maximization problem
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The problem of the firm is to maxyY p.y
Define the profit function π(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
π(pt) = ptyt = tp0yt + (1-t)p1yt ≤ tp0y0 + (1-t)p1y1 = tπ(p0) + (1-t)π(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
Results using the profit function
• The Derivative Property and Convexity: Dπ=y(p) and D2π
is positive semi-definite
Proof: Let y0 = y(p0) for some p0>>0. Define the function
g(p) = π(p) - p.y0. Clearly, g(p) ≥ 0 and g(p0) = 0. Therefore, g is
minimized at p = p0. If π is differentiable, the associated FONC
imply that Dg(p0) = Dπ(p0) - y(p0) = 0. Similarly, if π is twice
differentiable the SONCs imply that D2g(p0) = D2π(p0) is a
positive semi-definite matrix.
• LAW OF OUTPUT SUPPLY/INPUT DEMAND
– Combining the above results, D2π(p) = Dy(p) is a positive semidefinite matrix. This implies that (yj/pj)≥0: i.e., the physical
quantities of ouputs (inputs) increase (decrease) in own prices.
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)>>0, SONCs require pD2f(z(p,w)) negative semi-definite
• COMPARATIVE STATICS:
Assuming z(p,w)>>0,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)Df[D2f]-1Df ≥ 0)
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
Solving via Cramer' s Rule yields
1
p f12
0 pf22
z1
pf22

 2
w1 pf11 pf12 p (f11f22  f12 f21)
p f21
pf22
pf11 1
pf21 0
z2
 pf21


w1 pf11 pf12 p2 (f11f22  f12 f21)
p f21
pf22