Week 11: Theory of the Firm (Malinvaud, Chapter 3) / The...

Week 11: Theory of the Firm (Malinvaud, Chapter 3)/ The Competitive Firm and Perfect Competiton

(Jehle and Reny, Chapter 3.5 - 4.1)

Tsun-Feng Chiang*

*School of Economics, Henan University, Kaifeng, China

December 6, 2015

Microeconomic Theory Week 11: Theory of the Firm (Malinvaud, Chapter 3) / The Competitive Firm and Perfect Competiton (Jehle and Reny, Chapter 3.5 - 4.1)December 6, 2015 1 / 26

3.5 The Competitive Firm

In this section, we examine behavior when the firm is both a perfectcompetitor on input markets and a perfect competitor on its outputmarket. On the other words, the firm faces fixed output and inputprices prevailing both markets. Such a firm which takes prices as givenis called a price taker. The firm could not reduce the prices paid toinputs or it would hire no input; it has no incentive to raise prices eitherbecause it does not satisfy cost-minimization. Similarly, the firm couldnot sell its output for a price higher than the market price, because noconsumer want to buy expensive price when cheaper one is available;it has no incentive to sell output for less because it does not satisfyprofit-maximization.


Profit Maximization

As we had assumed in the beginning of this chapter, a firm’s object isto maximize its profits. Profit is the difference between revenue fromselling output and the cost of acquiring the factors necessary toproduce it. The competitive firm can sell each unit of output at themarket price, p. Its revenues are therefore a simple function of output,R(y) = py . To produce the level of output y , the firm can hire input x inthe feasible production plan. if w is the vector of factor prices, the costof using x to produce y is simply w · x. Now the profit-maximizationproblem of the firm can be expressed as

max(x,y)≥0 py −w · x s.t . f (x) ≥ y ,

where f (x) is a production function satisfying Assumption 3.1. Thesolution to this problem tell us how much output the firm will sell andhow much of which inputs it will buy.


Because the production function is strictly increasing, we can rewritethe constraint as f (x) = y . Replace this for y in the objective function,the maximization problem becomes

max(x≥0) pf (x)−w · x.

Assume this profit-maximization problem has an interior solution at theinput vector x∗ � 0. Then the first-order conditions require that thegradient of the maximand be zero because there are no constraints.That is,

p∂f (x∗)∂xi

= wi , for every i = 1, · · · ,n.

The term on the left-hand side, the output price times the marginalproductivity of input i , is called the marginal revenue of product ofinput i . At the optimum, it must equal the cost per unit of input i ,namely, wi . Using any two first-order conditions, we receive

∂f (x∗)/∂xi

∂f (x∗)/∂xj=

wi

wj, for all i , j ,


or the MRTS between any two inputs is equal to the ratio of theirprices. This is precisely the same as the condition for cost-minimizinginput choice. It therefore confirms our earlier intuition that profitmaximization requires cost minimization in production.We can think maximizing profits a two-step procedure. In the first step,calculate the cost function c(w, y) for each possible level of output y .Then in the second step, solve the maximization problem by choosinga specific level of y :

maxy≥0 py − c(w, y).

If y∗ is the optimal output, it therefore satisfies the first-order condition,

p − dc(w, y∗)dy

= 0,

or output is chosen so that price equals marginal cost. Second-orderconditions require that marginal cost be nondecreasing at theoptimum, or that d2c(y∗)/dy2 = d MC(y∗)/dy ≥ 0. Output choice isillustrated in Figure 3.7. (see the next slide)


Figure 3.7. Output Choice for the competitive firm. Profits are maximized aty∗, where price equals marginal cost, and marginal cost is nondecreasing.


When the solutions to the maximization problem exist, the optimalchoice of output, y∗ ≡ y(p,w), is called the firm’s output supplyfunction, and the optimal choice of inputs, x∗ ≡ x(p,w), gives thevector of firm input demand functions. Unlike the conditional inputdemands that depend partly on output, these input demands achievethe ultimate objective of the firm; they maximize the firm’s profit. Theprofit function, defined in what follows, is a useful tool for studyingthese supply and demand functions.

Definition 3.7 The Profit FunctionThe firm’s profit function depends only on input and output prices andis defined as the maximum-value function,

π(p,w) ≡ max(x,y)≥0 py −w · x s.t. f (x) ≥ y .

When the profit function is well-defined, it possesses several usefulproperties.


Theorem 3.7 Properties of the Profit FunctionIf f satisfies Assumption 3.1, then for p � 0 and w� 0, the profitfunction π(p,w), where well-defined, is continuous and

1 Increasing in p.2 Decreasing in w.3 Homogeneous of degree one in (p,w).4 Convex in (p,w).5 Differentiable in (p,w)� 0. Moreover, (Hotelling’s lemma),

∂π(p,w)

∂p= y(p,w), and

−∂π(p,w)

∂wi= xi(p,w), i = 1,2, · · · ,n.

Proof:Here we’ll just give a quick proof of the fourth property, convexity.Let y1 and x1 maximize profits at p1 and w1, and y2 and x2 maximizeprofits at p2 and w2. Defined pt ≡ tp1 + (1− t)p2, and


Theorem 3.7 (Continued)

wt ≡ tw1 + (1− t)w2 for 1 ≤ t ≤ 1, and let y t and xt maximize profitsat pt and wt . Then

π(p1,w1) = p1y1 −w1 · x1 ≥ p1y t −w1 · xt

π(p2,w2) = p2y2 −w2 · x2 ≥ p2y t −w2 · xt

Time the first and second inequalities by t and (1− t), then sum bothup,

tπ(p1,w1) + (1− t)π(p2,w2) ≥ (tp1 + (1− t)p2)y t + (tw1 + (1− t)w2)xt

= π(pt ,wt )

This proves convexity.

Note that by Hotelling’s lemma, output supply and input demands canbe obtained directly by simple differentiation. From the properties ofthe profit function, we can derive some important properties of theoutput supply and input demand functions.


Theorem 3.8 Properties of Output Supply and Input DemandFunctionsLet π(p,w) be a twice continuously differentiable profit function forsome competitive firm. Then, for all p > 0 and w� 0 where it is welldefined:

Homogeneity of degree zero:y(tp, tw) = y(p,w) for all t > 0,

xi(tp, tw) = xi(p,w) for all t > 0 and i = 1, · · · ,n.Own-price effects:

∂y(p,w)

∂p≥ 0,

∂xi(p,w)

∂wi≤ 0 for all i = 1, · · · ,n.

The substitution matrix


Theorem 3.8 (Continued)∂y(p,w)∂p

∂y(p,w)∂w1

· · · ∂y(p,w)∂wn

−∂x1(p,w)∂p

−∂x1(p,w)∂w1

· · · −∂x1(p,w)∂wn

......

. . ....

−∂xn(p,w)∂p

−∂xn(p,w)∂w1

· · · −∂xn(p,w)∂wn

is symmetric and positive semidefinite.Proof:First we prove the property of homogeneity of zero. Because the profitfunctions is homogeneous of degree one in (p,w). By Hotelling’slemma and Theorem A2.6, the derivative of the profit function respectto p and w, or y(tp, tw) and xi(tp, tw) = xi(p,w), are homogeneous ofdegree zero.To prove the second property, take the second derivatives of the profitfunction with respect to p and wi for i = 1, · · · ,n and apply the propertythat the profit function is convex in (p,w). Then this property is proved.


Theorem 3.8 (Continued)Then we prove the third property. By Young’s theorem, it is clear thesubstitution matrix is symmetric. Any by the property that the profitfunction is convex in (p,w), the matrix (the Hessian matrix of the profitfunction) is positive semidefinite.

Example 3.5

Let the production function be the CES form, y = (xρ1 + xρ2 )β/ρ whereβ < 1 and 0 6= ρ < 1. Calculate the input demand functions, outputsupply function and profit function.

The objective function for the profit maximization problem is

π(p,w) = p(xρ1 + xρ2 )β/ρ − (w1x1 + w2x2)

The first-order conditions are


Example 3.5 (Continued)

−w1 + ρβ(xρ1 + xρ2 )(β−ρ)/ρxρ−11 = 0,

−w2 + ρβ(xρ1 + xρ2 )(β−ρ)/ρxρ−12 = 0,

(xρ1 + xρ2 )β/ρ − y = 0.

Solve the equation system, we obtain

xi = y1/β(wρ/(ρ−1)1 + wρ/(ρ−1)

2 )−1/ρw1/(ρ−1)i , i = 1,2

Substitute these two equations into the first first-order condition givesthe output supply function,

y = (pβ)−β/(β−1)(wρ/(ρ−1)1 + wρ/(ρ−1)

2 )β(ρ−1)/ρ(β−1)

Put this equation back to the previous two equations, we can obtainthe input demand functions

xi = w1/(ρ−1)i (pβ)−1/(β−1)(wρ/(ρ−1)

1 + wρ/(ρ−1)2 )(ρ−1)/ρ(β−1) i = 1,2


Example 3.5 (Continued)Substitute the input demand functions into the objective function,

π(p,w) = p−1/(β−1)(w r1 + w r

2)β/r(β−1)β−β/(β−1)(1− β).

where r ≡ ρ/(ρ− 1).

1The profit function we have defined so far is the long-run profitfunction, because we have supposed the firm is free to choose itsoutput and all input levels as it see fits. As we did for the cost function,we can construct a short-run or restricted profit function to describefirm behavior when some of its inputs are variable and some are fixed.

Theorem 3.9 The Short-Run, or Restricted, Profit FunctionLet the production function be f (x, x), where x is a subvector ofvariable inputs and x of fixed inputs. Let w and w be the associatedinput prices for variable and fixed inputs,


Theorem 3.9 (Continued)respectively. The short-run, or restricted, profit function is defined as

π(p,w, w, x) ≡ maxy ,x py −w · x− w · x s.t . f (x, x) ≥ y .

The solutions y(p,w, w, x) and x(p,w, w, x) are called the short-run, orrestricted, output supply and input demand functions, respectively.For all p > 0 and w� 0, π(p,w, w, x) where well-defined is continuousin p and w, increasing in p, decreasing in w, and convex in (p,w). Ifπ(p,w, w, x) is twice differentiable, y(p,w, w, x) and x(p,w, w, x)possess all three properties listed in Theorem 3.8 with respect tooutput and variable prices.

1For one last perspective on the firm’s short-run behavior, let’s abstractfrom input demand behavior and focus on output supply. The short-runprofit maximization problem is to find a y such that the revenue minusthe short-run cost would achieve the maximum


π(p,w, w, x) = maxy py − sc(y ,w, w, x).

The first-order condition tell us that for optimal output y∗ > 0,

p =d sc(y∗)

d y

which says price equals short-run marginal cost is the short-runcondition for the firm to produce y∗ > 0.However, it is not the only condition. Suppose in the short run priceequals marginal cost at some y1 > 0. Now we express the short-runtotal cost sc as the sum of the total variable cost tvc(y) = w · x and thetotal fixed cost, tfc = w · x. Given the level of output y1 > 0, short-runprofits can be expressed as

π1 ≡ py1 − tvc(y1)− tfc.

If π1 is negative. Is it still best for the firm to produce y1 even though itis making a loss? Let’s compare two cases where y1 > 0 is producedor nothing is produced


When nothing is produced, the firm still have to pay tfc in the short-run,and its (negative) profit π0 is

π0 ≡ −tfc.

Compare this with π1 which is also negative, if π1 is larger than π0, or

π1 > π0 ⇒ py1 − tvc(y1)− tfc > −tfc ⇒ p >tvc(y1)

y1 ≡ avc(y1)

where avc(y1) is the average variable cost. This says the firm hadlarger profit (smaller loss) by producing y1 > 0. Otherwise, it shutsdown. In summary, in the short-run, if the firm produces a positiveamount of output, then it will produce an amount of output where priceequals marginal cost, and price is not below the average variable costat that level of output.


4.1 Perfect Competition

In perfectly competitive markets, buyers and sellers are sufficientlylarge in number to ensure than no single one of them, alone, has thepower to determine market price. Buyers and sellers are price takers,and each decides on a self-interested course of action in view ofindividual circumstances and objectives. In this section we can seehow buyers and sellers’ self-interest leads to the market equilibrium.

The demand side of a market is made up of all potential buyers of thegood, each with their own preferences, consumption set, and income.Let I ≡ {1, · · · , I} index the set of individual buyers and qi(p,p, y i) bei ’s nonnegative demand for good q as a function of its own price, p,income, y i , and prices, p, for all other goods. Market demand for q issimply the sum of all buyers’ individual demands

qd (p) ≡∑i∈I

qi(p,p, y i).


qd (p) gives the total amount of q demanded by all buyers in themarket. Each buyer’s demand for q depends on the price of q and theprices of all other goods, p, as well. Besides, whereas a single buyer’sdemand depends on the level of her own income, market demanddepends both on the aggregate level of income and income distributionamong buyers.The supply side of the market is made up of all potential sellers if q. Inthe short-run, the number of sellers are fixed and limited to those firmsthat "already exist" and are in some sense able to be up and runningsimply by acquiring the necessary variable inputs. If we letJ ≡ {1, · · · , J} index those firms, the short-run market supplyfunction is the sum of individual firm short-run supply functionsqj(p,w):

qs(p) ≡∑j∈J

q j (p,w).

Market demand and market supply together determine the price andtotal quantity traded. We say that a competitive market is in short-runequilibrium at price p∗ when qd (p∗) = qs(p∗).


Example 4.1Consider a competitive industry composed of J identical firms. Firmsproduce output according to the Cobb-Douglas technology,q = xαk1−α, 0 < α < 1 and where x is some variable inputs and k isthe fixed input. At price p for the product, wx and wk for the variableinputs, the is output supply of any firm j is

qj = pα/1−αwα/α−1x αα/1−αk

If α = 1/2. wx = 4, wk = 1, and k = 1, the output supply for firm jreduces to qj = p/8. The market supply function with J = 48 firms willbe

qs = 48(p/8) = 6p

Let the market demand be qd = 294/p. In the equilibrium whereqd = qs, we obtain the equilibrium price and quantity as p∗ = 7 andq∗ = 42, respectively. Because there are 48 firms, each firm produces



qj = 42/48 = 7/8. And we can calculate in the short-run, the profit foreach firm is πj = 2.0625 > 0.

Figure 4.1. Short-run equilibrium in a single market.


In Example 4.1, a competitive firm earns positive profits. In the longrun, new firms may decide to begin producing the good in question.When supply of the good increases, the price declines, so do theprofits. As long as the profits are positive, new firms keep entering intothe market. On the other hand, if a firm keeps earning negative profitsin the long run when it can just all inputs, it leaves the market. Thenthe price of the good increases until the profits are nonnegative. Insummary, in the long run, a firm is free to exit or enter a market untilthe profit is zero. Thus we have the following two conditionscharacterize the long-run competitive market,

qd (p) =∑J

j=1 qj(p),

πj(p) = 0, j = 1, · · · , J.

The first condition simply says the market must clear. The second sayslong-run profits for all firms in the industry must be zero so that no firmwishes to enter or exist. In the short-run, the number of firms is given.


In the long run, the number of firms as well as the equilibrium price aredetermined by the two conditions.

Example 4.2There is a linear market demand

p = 39− 0.009q.

An identical firm has the profit function

πj(p) = p2 − 2p − 399.

By Hotelling’s lemma, the output supply function for this firm is

y j =dπ(p)

dp= 2p − 2.

Let p be the equilibrium price and J be the equilibrium number of firms.By the two conditions of the long run equilibrium, we have



(1000/9)(39− p) = J(2p − 2).p2 − 2p − 399 = 0.

Solving the equation system gives p = 21 and J = 50.

Figure 4.2. Long-run equilibrium in a competitive market.


Example 4.3Following Example 4.1 but consider what is different in the long run.In the run long, firms may enter in response to positive profits andincumbent firms are free to choose the level of input k optimally.Market price will be driven to a level where maximum firm profits arezero. Therefore, we can rewrite the profit function as

π(p, k) = k(p2/16− 1) = 0

For all k > 0 if and only if p = 4. The market-clearing condition with Jfirms, each operating some level of input k , requires thatqd (p) = qs(p), or

2944

=48

J k ⇒ 147 = J k

The number of firms is determined by the size of k .


Announcement

Final ExamDate (Scheduled): Monday, December 28th, 2015Time: 9:00 am ∼ 11:30 amLocation: The large meeting room on the second floor, School ofEconomicsCoverage: Firm Theory and Partial Equilibrium (Week 10 - Week12)Others: The regulations of this exam are the same as they werefor the two midterms.


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