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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


Profits: Difference between the revenues a firm receives and thecost 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)The optimal set of actions a∗ are such that ∂R(a∗)

∂ai= ∂C(a∗)

∂ai


From the Marginal Revenue=Marginal Cost there are threefundamental interpretations

The actions could be: output production, labor hiring and in eachthe principle is MR=MCAlso, it is evident that in the long-run all firms should have similarprofits given that they face the same cost and revenue functions

In order to explore these possibilities, we have to break up therevenue (price and quantity of output) and costs (price andquantity of inputs).


But the prices are going to come as a result of the marketinteraction 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 toas competitive firms


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 inY(z)

Single-Output Profit Function π(p,w)=max pf (x)− wxSingle-Output Cost Function c(w, y) = min wx such that x isin V(y).

Single-Output restricted Cost Function c(w, y, z) = min wxsuch that (y,-x) is in Y(z).


Profit Maximizing Behavior

Single output technology: p∂f (x∗)∂xi

= wi for i = 1, ..., n

Note that these are n equations. Using matrix notationpDf(x∗) = wwhere Df(x∗) = (∂x∗

∂x1, ..., ∂x∗

∂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†:

In the two dimensional case: ∂2f (x∗)∂x2 ≤ 0

With multi-inputs: D2f(x∗) = (∂2f (x∗)∂xi∂xj

), the Hessian matrix isnegative semidefinite.


Factor Demand and Supply Functions

For each vector of prices (p,w), profit-maximization wouldnormally yield a set of optimal x∗

Factor Demand Function: The function that reflects theoptimal 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 ofoutput given the input prices (p,w). This is simply defined asy(p,w) = f (x(p,w))In the previous section (and typically) we will assume that thesefunctions exist and are well-behaved.

However, there are cases in which problems may arise.


Problems 1 and 2

The technology cannot be described by a differentiableproduction 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 incase of non-negativity contraints (x ≥ 0) are the following:p∂f (x)∂xi− wi ≤ 0 if xi = 0

p∂f (x)∂xi− wi = 0 if xi > 0

These are the so-called Kuhn-Tucker conditions (seeAlpha-Chiang or appendix Big Varian)


Problems 3 and 4

There might not be a profit maximizing planExample: If the production function is CRS and profits arepositive†At first sight this may seem surprising but there are some reasonswhy this is intuitive: 1) As a firm grows it is likely to fall intoDRS, 2) price-taking behavior becomes unrealistic and 3) otherfirms have incentives to imitate the first one

The profit maximizing plan may not be unique (conditionalfactor demand set)†.


Properties of the Demand and Supply functions

The fact that these functions are the result on optimization putssome 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 bya x%, the optimal choice of x does not change†

There are both theoretical and empirical reasons to consider allthe restrictions derived from maximizing behaviorWe examine these restrictions in three ways:

1 Checking FOC2 Checking the properties of maximizing demand and supply

functions3 Checking the properties of the associated profit and cost

functions.


Comparative Statics (single input)

Comparative statics refers to the study of how optimal choicesrespond to changes in the economic environment.

In particular, we could answer questions like: how does theoptimal choice of input x1 changes with changes in prices of p orw1?

Using the FOC and the definition of factor demand function wehave:

pf ′(x(p,w))− w ≡ 0 (1)

pf ′′(x(p,w))− w ≤ 0


Comparative Statics (single input)

Differentiating equation 1 with respect to w we get:pf ′′(x(p,w))dx(p,w)

dw − 1 ≡ 0

And assuming f ′′ is not equal to 0, there is a regular maximum,then

dx(p,w)dw ≡ 1

pf ′′(x(p,w))

This equation tells us a couple of interesting facts about factordemands

1 Since f ′′ is negative, the factor demand slopes downward2 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 †.


Comparative Statics (multiple inputs)

In the case of two-inputs, the factor demand functions mustsatisfy the following first order conditions

∂f (x1(w1,w2), x2(w1,w2))∂x1

≡ w1 (2)

∂f (x1(w1,w2), x2(w1,w2))∂x2

≡ w2 (3)

Differentiating these two equations with respect to w1 we obtain

f11∂x1

∂w1+ f12

∂x2

∂w1= 1 (4)

f21∂x1

∂w1+ f22

∂x2

∂w1= 0 (5)


and similarly, differentiating these two equations with respect tow2

f11∂x1

∂w2+ f12

∂x2

∂w2= 0 (6)

f21∂x1

∂w2+ f22

∂x2

∂w2= 1 (7)

writing equations (4)-(7) in matrix form,(f11 f12f21 f22

)( ∂x1∂w1

∂x1∂w2

∂x2∂w1

∂x2∂w2

)=(

1 00 1

)If we assume a technology that has a regular maximum, then theHessian is negative definite and, thus, the matrix is non-singular.


Comparative statics (cont.)

Inverting the Hessian,(∂x1∂w1

∂x1∂w2

∂x2∂w1

∂x2∂w2

)=(

f11 f12f21 f22

)−1

Factor demand functions (substitution matrix) under (strict)conditions of optimization implies:

1 The substitution matrix is the inverse of the Hessian and, thus,negative definite.

2 Negative Slope: ∂xi∂wi≤ 0

3 Symmetric Effects: ∂xi∂wj

= ∂xj

∂wi

These derivations were done for the 2-input case, it turns out thatit is straightforward to generalize it to the n-input case usingmatrix algebra †.


Substitution matrix for the n-input case1 The first order conditions are Df(x(w))− w ≡ 02 Differentiation with respect to w yields D2f(x(w))Dx(w)− I ≡ 03 Which means that Dx(w) = (D2f(x(w)))−1

What is the empirical meaning of a negative semi-definitematrix?

1 Changes in input demand if prices vary a bit: dx = Dx(w)dw′

2 Use the Hessian formula above: dx = (D2f(x(w)))−1dw′

3 Premultiply by dw, dwdx = dw(D2f(x(w)))−1dw′4 Negative semi-definiteness means: dwdx ≤ 0


Properties of the Profit Function

The profit function π(p) results from the profit maximizationproblem. i. e. π(p) = max py, such that y is in Y.This function exhibits a number of properties:

1 Non-increasing in input prices and non-decreasing in outputprices. That is, if p′i ≥ pi for all the outputs and p′i ≤ pi for all theinputs, π(p′) ≥ π(p)

2 Homogeneous of degree 1 in p: π(tp) = tπ(p) for t ≥ 03 Convex in p. If Let p′′ = tp + (1− t)p′. Thenπ(p′′) ≤ tπ(p) + (1− t)π(p′)

4 Continuous in p. When π(p) is well-defined and pi ≥ 0 fori = 1, 2, ..., n


Proof †All these properties depend only on the profit maximizationassumption 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 checkwhether a firm is maximizing profits.


Supply and demand functions

The net supply function is the net output vector that satisfy profitmaximization, 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 functionfor good i. Then,

yi(p) = ∂π(p)∂pi

for i=1,2,...,n

1 input/1 output case. First order conditions imply,pf ′(x)− w = 0. By definition, the profit function isπ(p,w) ≡ pf (x(p,w))− wx(p,w)Differentiation with respect to w yields,∂π(p,w)∂w = p∂f (x(p,w))

∂x(p,w)∂x(p,w)∂w − [x(p,w) + w∂x(p,w)

∂w ]


grouping by ∂x(p,w)∂w

∂π(p,w)∂w = [p∂f (x(p,w))

∂x − w]∂x(p,w)∂w − x(p,w) = −x(p,w)

The last equality follows from the first order conditions, whichimply p∂f (x(p,w))

∂x − w = 0There are 2 effects when prices go up:

1 Direct effect: x is more expensive due to the increase in prices.2 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


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 thatdM(a)

da = ∂f (x,a)∂a |x=x(a)

This expression means that the derivative of M with respect to ais just the derivative of the function with respect to a where x isfixed at the optimal level x(a).


In the 1 input/1 output case, π(p,w) = maxx pf (x)− wx. The ain the envelope theorem is simply (p,w), which means that

∂π(p,w)∂p

= f (x)|x=x(p,w) = f (x(p,w)) (8)

∂π(p,w)∂w

= −x|x=x(p,w) = −x(p,w) (9)


Comparative statics with the profit function

What do the properties of the profit function mean for the netsupply functions

1 π increasing in prices means net supplies positive if it is an outputand negative if it is an input (this is our sign convention!).

2 π homogeneous of degree 1 (HD1) in p, means that y(p) is HD0in prices

3 Convexity of π with respect to p means that the hessian of π(∂2π∂p2

1

∂2π∂p1∂p2

∂2π∂p2∂p1

∂2π∂p2

2

)=

(∂y1∂p1

∂y1∂p2

∂y2∂p1

∂y2∂p2

)is positive semi-definite and symmetric. But the Hessian of π issimply the substitution matrix.


Quadratic forms and Concavity/Convexity

A function is convex (concave) if and only if its Hessian ispositive (negative) semi-definite

A function is strictly convex (concave) if (but not only if) itsHessian is positive (negative) definite

A negative (positive) definite Hessian is a sufficient second ordercondition for a maximum (minimum). However, a less stringentnecessary SOC are often used: negative (positive) semi-definite.

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