文档介绍：Advanced Microeconomics (LECTURE 3:production theory III)
Ye Jianliang
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Cost Minimization
Content:
Definitions
Properties of cost function
WACM
Some forms of cost functions
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One production, cost function is
The optimal solution x(w,q), is the conditional factor demand function.
Question1:calculate the conditional factor demand function of C-D tech. and CES tech.
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Recall the cost minimization condition, let x>0, then set Lagrangian:
We got:
see the fig.
What is?
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of cost function
Proposition1: c(w,q) is homogeneous of degree 1 in w, and non-decreasing in q.
Proposition2: c(w,q) is concave function of w.
Proposition3: x(w,q) is homogeneous of degree 0 in w.
Proposition4:if V(q) is convex, then is x(.) if V(q) is strictly convex, x(.) is single point.
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of cost function
Proposition5: (Shephard’s lemma) if x(w,q) is single point, then
Proposition6: is symmetric negative semi-definite, and
see the fig.
Proposition7: if f(.) is HD1, c(.) and x(.) is HD1 too, if f(.) is concave, c(.) is convex in q.
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Weak Axiom of Cost Minimization (WACM): if xs, xt are in Y, and choice by firm under price ws and wt. then . we can get:
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x2
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x1
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x2
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x1
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x2
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VI
VO
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forms of cost functions
If , corresponding price , and is the limited factor (fixed assets), while is the variable factor. The total cost is
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