Lecture 4
1
Summary of Lecture 1: Technology
2
Summary of Lecture 2: Pro°t maximization
2.1
The pro°t maximization problem
2.2
Implications of pro°t maximization
2.3
The pro°t function
2.4
Cost minimization
3
Summary of Lecture 3: The cost function and
duality
3.1
De°nitions
Short and longterm total costs, average costs (SAC and LAC), and marginal
cost functions (SMC and LMC).
3.2
Properties of the cost function
c
(
°
)
is increasing in
y
and nondecreasing in
w
;
c
(
°
)
is homogeneous of degree 1
in
w
;
c
(
°
)
is concave in
w
; continuous in
w
; Shepard°s lemma.
3.3
Implications
3.4
Duality
3.4.1
Mathematical introduction
3.4.2
Duality in Production

We can recover information on a ±rm°s technology ² as described by the
input requirement set V(y) ²using the intersection of halfspaces built by
means of the cost function
c
(
°
)
, so that the cost function summarizes the
economically relevant information about the technology;

A di/erentiable function satisfying the properties for cost functions above is
indeed a cost function for some technology;

Functions satisfying the properties of the conditional demand functions (ho
mogeneity of degree 0 in prices and symmetric and negative semide±nite
matrix of partial derivatives with respect to prices) can be show to be a
conditional demand function for some technology.
1
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3.4.3
Geometry of duality
The isocost curve is the set of input prices that allows us to produce a given
output at the same cost. The slope of the isocost curve is simply, i.e.it equals
the ratio of the factor demands. The slope of the isoquants is given by TRS
which must equal the ratio of the input prices in optimum.
A very curved isoquant means that large changes in factor prices lead to small
changes in input choices. Thus the ratio of factor demands will remain relatively
unchanged which means that the isocost curve is quite ³at. Conversely, if the
technology is linear (linear isoquant) we will only use the best input.
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 Spring '10
 gilo
 Economics, Convex function, Utility maximization problem, Hicksian demand function

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