Lecture 6
1 Lecture 6
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 Winto a vector of outputs,which are then sold at prices Pfor total revenuesPf(Z)
2 1. Technology • The more tradition approach is to assume: (1) A production correspondence, e.g. or more generally , that maps the vector of inputs Z which cost W into a vector of outputs, which are then sold at prices P for total revenues . f K L ( , ) f Z( )Pf Z( )
In many cases,we think of f()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
3 • In many cases, we think of 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. f Z( )
.(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
4 • (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
(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. 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)
5 • (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. • 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)
pf(Z)=W ∂Z, for each input marginal revenue equals price. And given these first order conditions: Pf(Z)-WZ-0 These two,especially,when combined with demand give you implications about prices
6 • • for each input marginal revenue equals price. • And given these first order conditions: • These two, especially, when combined with demand give you implications about prices. ( ) i f Z P W Z = Pf Z W Z ( ) 0 − =
Now back to the MWG set theoretic approach. ·A production plan is a vector y∈R. This includes both inputs and outputs, and an input is a negative element in this vector. An output is a positive element in this vector. Total profits are p.)
7 • Now back to the MWG set theoretic approach. • A production plan is a vector . • This includes both inputs and outputs, and an input is a negative element in this vector. • An output is a positive element in this vector. • Total profits are . L yR p y
The set of all production plans is y, which is analogous to X,in the consumer chapters. We generally assume that 0Y,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 vectory∈Y wherey>0,for all I andy0 for at least one factor I=k
8 • The set of all production plans is Y, which is analogous to X, in the consumer chapters. • We generally assume that , 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 where , for all l and for at least one factor l=k. 0Y y Y 0 l y 0 k y
·(4)free disposal-if a vector (y,…y,…y）eY where >0,for then all other vectors (y,…yA,y）e y where x<y .(you can always get rid of something). ·(5)irreversibility:ify∈Ythen-y走Y
9 • (4) free disposal– if a vector where , for then all other vectors where • (you can always get rid of something). • (5) irreversibility: if then ( ) 1 , , k L y y y Y 0 k y ( ) 1 , , k L y y y Y k k x y y Y − y Y
These properties are more particular: .(6)Nonincreasing returns to scale. ify∈Y,thenayey for alla∈[0，l-you can always scale down. (7)Nondecreasing returns to scale. Ify∈Y,then ay∈y for all≥l-you can always scale up. ·(⑧)Constant returns to scale Ify∈Y, thenayeY for all a>0-you can always scale up or down. 10
10 • These properties are more particular: • (6) Nonincreasing returns to scale. If , then for all – you can always scale down. • (7) Nondecreasing returns to scale. If , then for all – you can always scale up. • (8) Constant returns to scale If , then for all – you can always scale up or down. y Y y Y 0,1 y Y y Y 1 y Y y Y 0