2019浙江大学高级微观经济学英文PPT课件

• Convexity–
( p (1 ) p') ( p) (1 ) ( p')
for 0,1 .
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• Cost minimization– now assume a fixed vector of outputs and a production function f(Z).
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• Properties of the profit function MWG
• Proposition 5.C.1:
• Suppose that (.)is the profit function
of the production set Y, and that y(.) is the associated supply correspondence.
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• P f (Z ) W
•
Zi
for each input
marginal revenue equals price.
• And given these first order conditions:
Pf (Z ) W Z 0
• These two, especially, when combined with demand give you implications about
and y' Y then y y' Y .
• (10) Convexity if y Y and y' Y then
y (1)y' Y for all 0,1 .
• (11) Y is a convex conde. If for any
y, y' Y and 0, 0, y y' Y .
• (6) Nonincreasing returns to scale.
If y Y , then y Y for all 0,1 – you
can always scale down.
• (7) Nondecreasing returns to scale.
If y Y , then y Y for all 1–ห้องสมุดไป่ตู้you
y1, yk , yL Y where xk yk
• (you can always get rid of something).
• (5) irreversibility: if y Y then y Y
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• These properties are more particular:
• Assume also that Y is closed and satisfies the free disposal property. Then:
• (1) (.)is homogeneous of degree one.
• (2) (.) is convex
• (3) y(.) is homogeneous of deree zero
the same vector that maximizes py
subject to the same constraint for any
positive .
• This also gives us homogeneity of degree zero for the supply correspondence.
• Together, profit maximization and free entry of identical firms gives us the following two sets of conditions (assuming that the production function is continuously differentiable and concave).
Lecture 6
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1. Technology
• The more tradition approach is to assume: (1) A production correspondence, e.g.
f (K, L) or more generally f (Z ), that maps
the vector of inputs Z which cost W into a vector of outputs, which are then sold
• We generally assume that 0 Y, so
thatfirms can shut down. • Generally, we assume that Y is • (1) nonempty (even beyond including 0) • (2) closed, i.e. includes its limit points. • (3) no free lunch– there is no vector y Y
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• (4) If Y is strictly convex, then y(.) is single valued.
• (5) If y(.) is single valued then (.) is
differentiable and ( p) y( p) .
• (6) If y(.) is differentiable then
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• (2) We assume that firms maximize profits, and that they have the option (at least in the long run) to exit, i.e. earn zero profits.
• This is of course a deeply controversial claim.
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• MWG Proposition 5.B.1: The production set Y is additive and satisfies the nonincreasing returns condition if and only if it is a convex cone.
• If it is a convex cone– then the nonincreasing returns condition and additivity immediately follow.
we’re done.
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• Properties of the Cost Function MWG • Proposition 5.C.2: • Suppose that C(W,Q) is the cost function
of a single output technology Y , with production function F(.) and that Z(W,Q) is the associated conditional factor demand correspondence. • Assume also that Y is closed and satisfies the free disposal property. Then: • (1) C(.) is homogeneous of degree one in W • (2) C(.) is nondecreasing in Q
k
• The profit maximization problem is to
maximize p y subject to y Y .
• This yields a supply
correspondence y( p) .
• And a profit function ( p) p y( p) .
• Let k MAX (, ) where k is an integer,
then ky, ky' Y by additivity.
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•
Asclsaoleusinkgy
nonincreasing returns to
Y and ky' Yand we’re
done. k
y '' denote the optimal production
vectors at the three price levels. • Then we know that
( p (1) p') py'' (1) p' y''
• But p' y'' py and p' y'' p' y'and
at prices P for total revenuesPf (Z) .
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• In many cases, we think of f (Z ) as a
function – i.e. only one output– but is doesn’t need to be. • We assume first that firms treat prices as given– i.e. they are price takers– i.e. they don’t have market power.
• An output is a positive element in this vector.
• Total profits are p y .
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• The set of all production plans is Y, which is analogous to X, in the consumer chapters.
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• (3) We make some assumption about the number of firms– perhaps free entry of identical firms, perhaps something else.
• This last assumption gives us a great deal of power– this is the equilibrium assumption in action.
• The correspondence Z (W , Q) is the input
demand that satisfies cost minimization.
• The cost function C(W,Q) W Z(W,Q)
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• Let p'' p (1 ) p' and have y, y' and
• The cost minimization problem is to minimize
W Z subject to f (Z) Q for some fixed
level Q.
• In principle, this quantity constraint could handle a lot of different outputs, but we will certainly focus on the idea of a single output.
can always scale up.
• (8) Constant returns to scale If y Y ,
then y Y for all 0 – you can always
scale up or down.
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• (9) Additivity (also free entry) if y Y
prices.
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• Now back to the MWG set theoretic approach.
• A production plan is a vector y RL.
• This includes both inputs and outputs, and an input is a negative element in this vector.
where yl 0, for all l and yk 0 for at
least one factor l=k.
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• (4) free disposal– if a vector y1, yk , yL Y
where yk 0 , for then all other vectors
Dy( p) D2 ( p) is a symmetric and
positive semidefinite matrix
with Dy( p) p 0 .
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• Homogeneity of degree one in prices occurs because the vector y that maximizes py subject to any constraint is