2019浙江大学高级微观经济学英文PPT课件

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.
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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.
• 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
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.
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.
• 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.
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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
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
• 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.
• 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).
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
• 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).
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) .
• (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– you
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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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• 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.
Lecture 6
1
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 outpБайду номын сангаасts, which are then sold
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
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
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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
• 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
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
• 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
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 .
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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.