Producer Theory
1Technology
1.Production is a process of transforming inputs into outputs.The
fundamental problem…rms must contend with in this process is the technological feasibility.The state of technology determines and restricts what is possible in combining inputs to produce out-put.
2.We…rst concentrate on the ca in which the…rm produces only
one output.The ca of multiple outputs will be dealt with later.
When there is only one output,we will u q2R+to denote the …rm’s output and x2R n+to denote the…rm’s inputs.
3.We de…ne the production function as:
q=f(x):
The production function captures the technology of production.It tells us how much input x is needed to
produce a a…xed amount of output q.
4.Assumption3.1.The production f:R n+!R+,is continuous,
strictly increasing,and strictly quasiconcave on R n+,and f(0)=0.
5.The marginal product of x i,
MP i=@f @x i
;
tell us how many extra units of output an extra unit of x i produces. Unlike marginal utility in consumer theory,marginal product is objective and measurable.
6.The strict quasiconcavity assumption means that any convex com-
bination of two input vectors can produce at least as much output as one of the original two.This would be the ca if we have diminishing marginal product.
7.For any…xed level of output q,the t of input vectors producing
q is called q-level isoquant.This plays the same role as indi¤erence curve in consumer theory.An isoquant traces out all the combina-tions of inputs that allow that…rm to produce the same quantity of output.
8.The substitutability between any pair of inputs x i and x j is the
marginal rate of technical substitution(MRTS).MRTS measures the amount of one input i the…rm would require in exchange for using a little less of another input j in order to just be able to produce the same output as before.A q-level isoquant is de…ned as:
f(x)=q:
Given x ij,let x j(x i)be the amount of x j required to keep out-put constant.If we di¤erentiate the isoquant with respect to x i, holding x ij constant,we get
@f @x j dx j
dx i
+
@f
@x i
=0:
Rearranging the terms gives the slope
MRT S ij(x)= dx j
dx i
=
@f
@x i
@f
j
:
Since f is strictly quasiconcave,MRTS is diminishing.
9.Elasticity of substitution:for a production function q=f(K;L);
the elasticity of substitution ;measures the proportionate change in K=L relative to the proportionate change in the MRTS along the isoquant.That is,
=
% K
L
% MRT S L;K
=
dK=L
dMRT S
MRT S
K=L
=
d ln(K=L)
d ln MRT S
:
The shape of the isoquants indicates the degree of substitutability.
10.De…nition:we classify the returns to scale of a production function,
f(K;L),as follows:E¤ect of Output(m>1)Returns to Scale f(mx)=mf(x)Constant
f(mx)<mf(x)Decreasing
f(mx)>mf(x)Increasing
11.Constant returns to scale means if you multiply all inputs by factor
m>1,the output is incread by factor m.Increasing returns to scale means if you multiply all inputs by a factor m,the out-put increas by a factor more than m.That is,you can get proportionally more as you expand.Decreasing returns to scale is just the opposite to increasing returns.A production function can have increasing returns over some range and decreasing returns over another.In fact,a lot of times there is some optimal scale in between:it is ine¢cient to be too small and ine¢cient to be too big.
12.Locally,the elasticity of scale at point x is de…ned by
(x) lim
m!1d ln[f(mx)]
d ln(m)
=P n i=1f i(x)x i
f(x)
:
1.1Special production functions
1.Linear production function:perfect substitutes
2.Cobb-Douglas production functions
q=AL K ;
where A; ; >0.
3.Leontief production functions
q=min K; L :
4.CES production function
q=A h L 1 +(1 )K 1 i 1
where A>0; 2(0;1); >0. is the elasticity of substitution.
(a) =+1,linear production function.
(b) =1,Cobb-Douglas production function.
(c) =0,Leontief production function.
5.Homothetical production function:homothetical production pro-
duces a linear expansion path starting from the origin,optimal input ratio at various output level is constant given…xed input prices.
2Cost Minimization
2.1Long-run Cost minimization
1.Some cost de…nition
(a)Opportunity cost:the value of a resource in its best alterna-
tive u.
(b)Sunk costs are tho unrecoverable costs that have already
been incurred and the resources have no alternative u.
2.Short-run vs.long-run cost minimization
(a)Long-run:free to vary quantities of all its inputs as much as
it desires.
(b)Short-run:unable to adjust the quantities of some of its in-
puts.
3.Long-run cost-minimization.Let w denote input prices.The cost
minimization problem is de…ned as:
min
x
wx
s:t:f(x) q:
L=wx+ (q f(x)):
The…rst-order conditions:
@L
@x i
=w i f x i=0;i=1;:::;n
@L
@
=q f(x)=0;
From the…rst n equations,we get for all i
w i
f x
i
= .
4.Recall f x
i is the extra output the…rm can make from one extra
unit input i,so1
x i is the amount of input i required to produce
one unit of output,and w
f x
i is the cost for producing one extra
unit of output using input i.A cost-minimizing…rm choos an input combination such that the cost fo
r producing one extra unit of output is the same no matter what input mixes the…rm choo to increa output.
5.Example 1:Cobb-Douglas production function
q =50L 1=2K 1=2:
MRT S L;K =K L =w r =)K (r;w )=w r
L:Plugging K (r;w )into the production function
q =50L 1=2 w r L 1=2=)L =q 50 r w
1=2:Similarly,K =q 50 w r 1=2:6.Comparative statics of change in output
(a)The cost minimizing input combinations,as q 0varies,trace
out the expansion path .
(b)If the cost minimizing quantities of labor and capital ri as
output ris,labor and capital are normal inputs .
(c)If the cost minimizing quantity of an input decreas as the
…rm produces more output,the input is called an inferior input .
7.Properties of Cost Function
If f is continuous and strictly increasing,then the cost function
c (w;q ) min x wx s:t:f (x ) q is
(a)Zero when q =0.
(b)Continuous on its domain.
(c)For all w >>0,strictly increasing and unbounded above in
q .
(d)Increasing in w:
(e)Homogeneous of degree one in w:
(f)Concave in w .
(g)Shephard’s lemma:c (w;q )is di¤erentiable in w at (w 0;q 0)
whenever w 0>>0;
and @c (w 0;q 0)@w i
=x i w 0;q 0 ;i =1;:::;n:
