Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Profit Maximization and the Profit Function
Juan Manuel Puerta
September 30, 2009
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Profits: Difference between the revenues a firm receives and the
cost it incurs
Cost should include all the relevant cost (opportunity cost)
Time scales: since inputs are flows, their prices are also flows
(wages per hour, rental cost of machinery)
We assume firms want to maximize profits.
Let a be a vector of “actions” a firm may take and R(a) and C(a)
the “Revenue” and “Cost” functions respectively.
Firms seek to maxa1 ,a2 ,...,an R(a1 , a2 , ..., an ) − C(a1 , a2 , ..., an )
∗
∗
)
)
= ∂C(a
The optimal set of actions a∗ are such that ∂R(a
∂ai
∂ai
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
From the Marginal Revenue=Marginal Cost there are three
fundamental interpretations
The actions could be: output production, labor hiring and in each
the principle is MR=MC
Also, it is evident that in the long-run all firms should have similar
profits given that they face the same cost and revenue functions
In order to explore these possibilities, we have to break up the
revenue (price and quantity of output) and costs (price and
quantity of inputs).
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
But the prices are going to come as a result of the market
interaction subject to 2 types of constraints:
Technological constraints: (whatever we saw in chapter 1)
Market constraints: Given by the actions of other agents
(monopoly,monopsony etc)
For the time being, we assume the simplest possible behavior,
i.e. firms are price-takers. Price-taking firms are also referred to
as competitive firms
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
The profit maximization problem
Profit Function π(p) = max py, such that y is in Y
Short-run Profit Function π(p,z) = max py, such that y is in
Y(z)
Single-Output Profit Function π(p, w)=max pf (x) − wx
Single-Output Cost Function c(w, y) = min wx such that x is
in V(y).
Single-Output restricted Cost Function c(w, y, z) = min wx
such that (y,-x) is in Y(z).
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Profit Maximizing Behavior
∗)
(x
Single output technology: p ∂f∂x
i
= wi for i = 1, ..., n
Note that these are n equations. Using matrix notation
pDf(x∗ ) = w
∗
∂x∗
where Df(x∗ ) = ( ∂x
∂x1 , ..., ∂xn ) is the gradient of f(x), that is, the
vector of first partial derivatives.
Geometric interpretation of the first order conditions†
Intuitive/Economic interpretation of the first order condition†
Second-order conditions†:
∂ 2 f (x∗ )
≤0
∂x2
∂ 2 f (x∗ )
= ( ∂xi ∂xj ), the
In the two dimensional case:
With multi-inputs: D2 f(x∗ )
negative semidefinite.
Hessian matrix is
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Factor Demand and Supply Functions
For each vector of prices (p, w), profit-maximization would
normally yield a set of optimal x∗
Factor Demand Function: The function that reflects the
optimal choice of inputs given the set of input and output prices
(p, w). This function is denoted x(p, w).
Supply Function: The function that gives the optimal choice of
output given the input prices (p,w). This is simply defined as
y(p, w) = f (x(p, w))
In the previous section (and typically) we will assume that these
functions exist and are well-behaved.
However, there are cases in which problems may arise.
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Problems 1 and 2
The technology cannot be described by a differentiable
production function (e.g. Leontieff)
There might not be an interior solution for the problem†.
This is typically caused by non-negativity constraints (x ≥ 0)
In general, the conditions to handle the boundary solutions in
case of non-negativity contraints (x ≥ 0) are the following:
p ∂f∂x(x)
− wi ≤ 0 if xi = 0
i
p ∂f∂x(x)
− wi = 0 if xi > 0
i
These are the so-called Kuhn-Tucker conditions (see
Alpha-Chiang or appendix Big Varian)
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Problems 1 and 2
The technology cannot be described by a differentiable
production function (e.g. Leontieff)
There might not be an interior solution for the problem†.
This is typically caused by non-negativity constraints (x ≥ 0)
In general, the conditions to handle the boundary solutions in
case of non-negativity contraints (x ≥ 0) are the following:
p ∂f∂x(x)
− wi ≤ 0 if xi = 0
i
p ∂f∂x(x)
− wi = 0 if xi > 0
i
These are the so-called Kuhn-Tucker conditions (see
Alpha-Chiang or appendix Big Varian)
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Problems 3 and 4
There might not be a profit maximizing plan
Example: If the production function is CRS and profits are
positive†
At first sight this may seem surprising but there are some reasons
why this is intuitive: 1) As a firm grows it is likely to fall into
DRS, 2) price-taking behavior becomes unrealistic and 3) other
firms have incentives to imitate the first one
The profit maximizing plan may not be unique (conditional
factor demand set)†.
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Problems 3 and 4
There might not be a profit maximizing plan
Example: If the production function is CRS and profits are
positive†
At first sight this may seem surprising but there are some reasons
why this is intuitive: 1) As a firm grows it is likely to fall into
DRS, 2) price-taking behavior becomes unrealistic and 3) other
firms have incentives to imitate the first one
The profit maximizing plan may not be unique (conditional
factor demand set)†.
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Properties of the Demand and Supply functions
The fact that these functions are the result on optimization puts
some restrictions on the possible outcomes.
The factor demand function is homogenous of degree 0.
Fairly intuitive, if price of output and that of all inputs increase by
a x%, the optimal choice of x does not change†
There are both theoretical and empirical reasons to consider all
the restrictions derived from maximizing behavior
We examine these restrictions in three ways:
1
2
3
Checking FOC
Checking the properties of maximizing demand and supply
functions
Checking the properties of the associated profit and cost
functions.
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Properties of the Demand and Supply functions
The fact that these functions are the result on optimization puts
some restrictions on the possible outcomes.
The factor demand function is homogenous of degree 0.
Fairly intuitive, if price of output and that of all inputs increase by
a x%, the optimal choice of x does not change†
There are both theoretical and empirical reasons to consider all
the restrictions derived from maximizing behavior
We examine these restrictions in three ways:
1
2
3
Checking FOC
Checking the properties of maximizing demand and supply
functions
Checking the properties of the associated profit and cost
functions.
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Properties of the Demand and Supply functions
The fact that these functions are the result on optimization puts
some restrictions on the possible outcomes.
The factor demand function is homogenous of degree 0.
Fairly intuitive, if price of output and that of all inputs increase by
a x%, the optimal choice of x does not change†
There are both theoretical and empirical reasons to consider all
the restrictions derived from maximizing behavior
We examine these restrictions in three ways:
1
2
3
Checking FOC
Checking the properties of maximizing demand and supply
functions
Checking the properties of the associated profit and cost
functions.
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Comparative Statics (single input)
Comparative statics refers to the study of how optimal choices
respond to changes in the economic environment.
In particular, we could answer questions like: how does the
optimal choice of input x1 changes with changes in prices of p or
w1 ?
Using the FOC and the definition of factor demand function we
have:
pf 0 (x(p, w)) − w ≡ 0
(1)
pf 00 (x(p, w)) − w ≤ 0
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Comparative Statics (single input)
Differentiating equation 1 with respect to w we get:
pf 00 (x(p, w)) dx(p,w)
dw − 1 ≡ 0
And assuming f 00 is not equal to 0, there is a regular maximum,
then
dx(p,w)
dw
≡
1
pf 00 (x(p,w))
This equation tells us a couple of interesting facts about factor
demands
1
2
Since f 00 is negative, the factor demand slopes downward
If the production function is very curved then factor demands do
not react much to factor prices.
Geometric intuition for the 1 output and 1 input case †.
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Comparative Statics (multiple inputs)
In the case of two-inputs, the factor demand functions must
satisfy the following first order conditions
∂f (x1 (w1 , w2 ), x2 (w1 , w2 ))
≡ w1
∂x1
(2)
∂f (x1 (w1 , w2 ), x2 (w1 , w2 ))
≡ w2
∂x2
(3)
Differentiating these two equations with respect to w1 we obtain
f11
∂x1
∂x2
+ f12
=1
∂w1
∂w1
(4)
f21
∂x1
∂x2
+ f22
=0
∂w1
∂w1
(5)
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
and similarly, differentiating these two equations with respect to
w2
∂x1
∂x2
f11
+ f12
=0
(6)
∂w2
∂w2
f21
∂x2
∂x1
+ f22
=1
∂w2
∂w2
(7)
writing equations (4)-(7) in matrix form,
∂x1 ∂x1 ! f11 f12
1 0
∂w1
∂w2
=
∂x2
∂x2
f21 f22
0 1
∂w
∂w
1
2
If we assume a technology that has a regular maximum, then the
Hessian is negative definite and, thus, the matrix is non-singular.
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Comparative statics (cont.)
Inverting the Hessian,
∂x1
∂w1
∂x2
∂w1
∂x1
∂w2
∂x2
∂w2
!
=
−1
f11 f12
f21 f22
Factor demand functions (substitution matrix) under (strict)
conditions of optimization implies:
1
2
3
The substitution matrix is the inverse of the Hessian and, thus,
negative definite.
∂xi
Negative Slope: ∂w
≤0
i
Symmetric Effects:
∂xi
∂wj
=
∂xj
∂wi
These derivations were done for the 2-input case, it turns out that
it is straightforward to generalize it to the n-input case using
matrix algebra †.
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Substitution matrix for the n-input case
1
2
3
The first order conditions are Df(x(w)) − w ≡ 0
Differentiation with respect to w yields D2 f(x(w))Dx(w) − I ≡ 0
−1
Which means that Dx(w) = (D2 f(x(w)))
What is the empirical meaning of a negative semi-definite
matrix?
1
2
3
4
Changes in input demand if prices vary a bit: dx = Dx(w)dw0
−1
Use the Hessian formula above: dx = (D2 f(x(w))) dw0
−1
Premultiply by dw, dwdx = dw(D2 f(x(w))) dw0
Negative semi-definiteness means: dwdx ≤ 0
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Properties of the Profit Function
The profit function π(p) results from the profit maximization
problem. i. e. π(p) = max py, such that y is in Y.
This function exhibits a number of properties:
1
2
3
4
Non-increasing in input prices and non-decreasing in output
prices. That is, if p0i ≥ pi for all the outputs and p0i ≤ pi for all the
inputs, π(p0 ) ≥ π(p)
Homogeneous of degree 1 in p: π(tp) = tπ(p) for t ≥ 0
Convex in p. If Let p00 = tp + (1 − t)p0 . Then
π(p00 ) ≤ tπ(p) + (1 − t)π(p0 )
Continuous in p. When π(p) is well-defined and pi ≥ 0 for
i = 1, 2, ..., n
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Proof †
All these properties depend only on the profit maximization
assumption and are not resulting from the monotonicity,
regularity or convexity assumptions we discussed earlier.
The first 2 properties are intuitive, but convexity a little bit so.
Economic intuition in the convexity result †
This predictions are useful because they allow us to check
whether a firm is maximizing profits.
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Supply and demand functions
The net supply function is the net output vector that satisfy profit
maximization, i.e. π(p) = py(p)
It is easy to go from the supply function to the profit function.
What about the other way around?
Hotelling’s Lemma. Let yi (p) be the firm’s net supply function
for good i. Then,
yi (p) =
∂π(p)
∂pi
for i=1,2,...,n
1 input/1 output case. First order conditions imply,
pf 0 (x) − w = 0. By definition, the profit function is
π(p, w) ≡ pf (x(p, w)) − wx(p, w)
Differentiation with respect to w yields,
∂π(p,w)
(x(p,w)) ∂x(p,w)
∂x(p,w)
= p ∂f∂x(p,w)
∂w
∂w − [x(p, w) + w ∂w ]
Motivation
Profit Maximization
Problems
Comparative Statics
∂x(p,w)
∂w
∂π(p,w)
= [p ∂f (x(p,w))
∂w
∂x
The profit function
Supply and demand functions
grouping by
− w] ∂x(p,w)
∂w − x(p, w) = −x(p, w)
The last equality follows from the first order conditions, which
imply p ∂f (x(p,w))
−w=0
∂x
There are 2 effects when prices go up:
1
2
Direct effect: x is more expensive due to the increase in prices.
Indirect effect: Change in x due re-optimization. This effect is
equal to 0 because we are at a profit maximizing point.
Hotelling’s Lemma is an application of a mathematical result:
the envelope theorem
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
The envelope theorem
Assume an arbitrary maximization problem M(a) = maxx f (x, a)
Let x(a) be the value that solves the maximization problem,
M(a) = f (x(a), a).
It is often interesting to check dM(a)/da
The envelope theorem says that
dM(a)
da
=
∂f (x,a)
∂a |x=x(a)
This expression means that the derivative of M with respect to a
is just the derivative of the function with respect to a where x is
fixed at the optimal level x(a).
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
In the 1 input/1 output case, π(p, w) = maxx pf (x) − wx. The a
in the envelope theorem is simply (p,w), which means that
∂π(p, w)
= f (x)|x=x(p,w) = f (x(p, w))
∂p
(8)
∂π(p, w)
= −x|x=x(p,w) = −x(p, w)
∂w
(9)
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Comparative statics with the profit function
What do the properties of the profit function mean for the net
supply functions
1
2
3
π increasing in prices means net supplies positive if it is an output
and negative if it is an input (this is our sign convention!).
π homogeneous of degree 1 (HD1) in p, means that y(p) is HD0
in prices
Convexity of π with respect to p means that the hessian of π
!
!
∂2π
∂2π
∂p21
∂2π
∂p2 ∂p1
∂p1 ∂p2
∂2π
∂p22
=
∂y1
∂p1
∂y2
∂p1
∂y1
∂p2
∂y2
∂p2
is positive semi-definite and symmetric. But the Hessian of π is
simply the substitution matrix.
Motivation
Profit Maximization
Problems
Comparative Statics
The profit function
Supply and demand functions
Quadratic forms and Concavity/Convexity
A function is convex (concave) if and only if its Hessian is
positive (negative) semi-definite
A function is strictly convex (concave) if (but not only if) its
Hessian is positive (negative) definite
A negative (positive) definite Hessian is a sufficient second order
condition for a maximum (minimum). However, a less stringent
necessary SOC are often used: negative (positive) semi-definite.