Week 10: Theory of the Firm (Jehle and Reny, Chapter 3)economics-course.weebly.com/uploads/2/5/7/2/25725158/...Week 10: Theory of the Firm (Jehle and Reny, Chapter 3) Tsun-Feng Chiang*

Week 10: Theory of the Firm (Jehle and Reny, Chapter3)

Tsun-Feng Chiang*

*School of Economics, Henan University, Kaifeng, China

November 22, 2015

First Last (shortinst) Short title November 22, 2015 1 / 34

3.1 Primitive Notions

The second important actor on the microeconomic stage is the individualfirm. A firm is an entity created by individuals for some purpose. Thisentity will typically acquire inputs and combine them to produce output.Inputs are purchased on input markets and these expenditures are thefirm’s costs. Output is sold on product markets and the firm earns revenuefrom those sales.Similar to utility maximization, a firm is assumed to pursue profitmaximization.The owners of a firm are consumers, the more profit the firmcan make, the greater will be its owner’s command over goods andservices.Of course, profit maximization may not be the only motive behind firmbehavior. Sales, market shares, or even prestige maximization are alsopossibilities. However, in the long run, it is not sustainable for a firmwithout profit. So we keep the assumption of profit maximization.


3.2 Production

Production is the process of transforming inputs into outputs. The firmhas a production possibility set, Y ⊂ Rm, where each vectory = (y1, · · · , ym) ∈ Y is a production plan whose components indicatethe amounts of the various inputs and outputs (where m are the numbersof output and input factors). A common convention is to write element ofy ∈ Y so that yi < 0 if resource i is used up in the production plan, andyi > 0 if resource i is produced in the production plan.For simplicity, we consider firms producing only a single product frommany inputs. For that, it is more convenient to describe the firm’stechnology in terms of a production function. When there is only oneoutput produced by many inputs, we shall denote the amount of output byy , and the amount of input i by xi , so that with n inputs, the entire vectorof inputs is denoted by x = (x1, · · · , xn). Of course, the input vector aswell as the amount of output must be nonnegative.


The production function, f , is therefore a mapping from Rn+ into R+.

When we write y = f (x), we mean that y units of output can be producedusing the input vector x. We make the further assumption on theproduction function f .

Assumption 3.1 Properties of the Production Function

The production function, f : Rn+ → R+, is continuous, strictly increasing,

and strictly quasiconcave on Rn+, and f (0) = 0.

Continuity of f ensures that small changes in the vector of inputs lead tosmall change in the amount of output produced. We require f to bestrictly increasing to ensure that employing strictly more of every inputresults in strictly more output. The strictly quasiconcave says that anyconvex combination of two input vectors can produce at least as much asone of the original two. The last condition states that a positive amountoutput requires positive amount of some of the inputs.


When the production function is differentiable, its partial derivative,∂f (x)/∂xi , is called the marginal product of input i and gives the rate atwhich output changes per additional unit of input i employed. Theassumption of strictly increasing production function implies∂f (x)/∂xi > 0 for all i .For any fixed level of output, y , the set of input vectors producing y unitsof output is called the y -level isoquant, a level set of f . We denote thisset by Q(y),

Q(y) ≡ {x ≥ 0|f (x) = y}.

An analog to the marginal rate of substitution in consumer theory is themarginal rate of technical substitution (MRTS) in production theory.This measures the rate at which one input can be substituted for anotherwithout changing the amount of output produced. Formally, the MRTSbetween inputs i and j when the current input vector is x, denotedMRTSij(x), is defined as the ratio of marginal products,


MRTSij(x) =∂f (x)/∂xi∂f (x)/∂xj

This show the rate at which input i can be substituted for input j whenholding output constant. In the two-input case, as depicted in Figure 3.1,MRTS12(x1) is the absolute value of the slope of the isoquant through x1

at the point x1.

Figure 3.1. The marginal rate of technical substitution


In general, the MRTS between any two inputs depends on the amount ifall inputs employed (see x in the formula). However, if it is independent ofall inputs, then we call the production is separable.

The MRTS measures how easy an input can be substituted for anotherinput at a point in the isoquant, sometimes we are more interested in howeasy an input can be substituted for another input in the whole isoquant.This measure is called the elasticity of substitution. Intuitively, ”If theMRTSij does not change at all for change in xj/xi , we might say thatsubstitution is easy, because the ratio of the marginal productivities of thetwo inputs does not change as the input mix changes. Alternatively, if theMRTSij changes rapidly for small change in xj/xi , we would say thatsubstitution is difficult, because minor variations in the input mix will havea substantial effect on the inputs’ relative productivities. ”(Nicholson,2004). Formally,

Definition 3.2 The Elasticity of Substitution

For a production function f (x), the elasticity of substitution betweeninputs i and j at the point x is defined as


Definition 3.2 (Continued)

inputs i and j at the point x is defined as

σij =percent change in xj/xi

percent change in MRTSij=

d ln(xj/xi )

d ln(fi (x)/fj(x))=

d(xj/xi )

xj/xi· fi (x)/fj(x)

d(fi (x)/fj(x)),

where fi and fj are the marginal products of input i and j .

When the production function is quasiconcave, the elasticity ofsubstitution can never be negative, so σij ≥ 0. See Figure 3.2. (the nextslide) as an example. In (a), the isoquant is linear so d MRTS12 = 0, orσ12 is infinite. The inputs 1 and 2 are perfect substitutable. In (c), thetwo inputs are productive only in fixed proportions with one another,d MRTS12 →∞, or σ12 = 0. In (b), the production function is anintermediate case where σ is neither zero nor infinite, and the isoquant areneither straight lines nor right angles.


Figure 3.2.

(a) σ is infinite and there is perfect substitutability between inputs. (b) σis finite but larger than zero, indicating less than perfect substitutability.(c) σ is zero and there is no substitutability between inputs.


Example 3.1

Consider the CES production function y = (xρ1 + xρ2 )1/ρ for 0 6= ρ < 1. Tocalculate the elasticity of substitution, we first note thatln(x2/x1) = ln(x2)− ln(x1), so that when we take the total differential inthe numerator of σ, we get

d ln

(x2

x1

)= d ln(x2)− d ln(x1) =

1

x2dx2 −

1

x1dx1 = −

(1

x1dx1 −

1

x2dx2

). (E .1)

Calculating the partials of f , forming the required ratio, and taking logs,we have

ln

(f1f2

)= ln

(xρ−1

1

xρ−12

)= (ρ− 1)[ln(x1)− ln(x2)].

Taking the total differential gives the denominator of σ:

d ln

(f1f2

)= (ρ− 1)

(1

x1dx1 −

1

x2dx2

). (E .2)


Example 3.1 (Continued)

Divide (E .1) by (E .2), we get

σ =1

1− ρ

which is a constant: hence, the initials CES, which stand for constantelasticity of substitution. The closer ρ is to unity, the larger is σ; when ρis equal to 1, σ is infinite and the production function is linear.

Other popular production functions can be derived from specific CESforms. Such as Cobb-Douglas production function,

y =∏n

i=1 xαii , where

∑ni=1 αi = 1.

with σij = 1. And the Leotief form,

y = min{a1x1, · · · , anxn}.

with σij = 0.


All CES production functions (including the cases of Cobb-Douglas andLeontief) are members of the class of linear homogeneous productionfunctions. One of the important properties of linear homogeneousproduction functions is they are always concave.

Theorem 3.1 (Shephard) Linear Homogeneous Production Functionsare Concave

Let f (x) be a production function satisfying Assumption 3.1 and supposeit is homogeneous of degree one. Then f (x) is a concave function of x.

We frequently want to know how output responds as the amounts ofdifferent inputs, or returns, are varied. For instance, in the short run, theperiod of time in which one input is fixed, output can be varied only bychanging the amounts of some inputs but not others. Figure 3.3.(see thenext slide) shows how output behaves as we move through the isoquantmap along the horizon at x̄2, keeping x2 constant and varying the amountof x1. In the long run, each input is flexible


to change. Returns to scale refer to how output responds when all inputsare varied in the same proportion, so they exist in the context of long run.In Figure 3.3., returns to scale have to do with how output behaves as wemove through the isoquant map along a ray such as OA, where the levelsof x1 and x2 are changed simultaneously, always staying in the proportionx2/x1 = α.

Figure 3.3. Returns to scale and varying proportions


The common measures of returns includes the marginal productMPi (x) ≡ fi (x), and the average product, APi (x) ≡ f (x)/xi , of eachinput. The output elasticity of input i , measuring the percentageresponse of output to a 1 percent change in input i , is given by

µi (x) =df (x)/f (x)

dxi/xi=

df (x)/dxif (x)/xi

=MPi (x)

APi (x)

Each of these is a local measure, defined at a point, i.e. at differentpoints, the values differ. The scale properties of the technology may bedefined either locally or globally. A production function is said to haveglobally constant, increasing, or decreasing returns to scale according tothe following definitions.

Definition 3.3 (Global) Returns to Scale

A production function f (x) has the property of (globally):1. Constant returns to scale if f (tx) = tf (x) for all t > 0 and all x.2. Increasing returns to scale if f (tx) > tf (x) for all t > 1 and all x.3. Decreasing returns to scale if f (tx) < tf (x) for all t > 1 and all x.


Many technologies exhibit increasing, constant, and decreasing returnsover only certain range of output. It is therefore useful to have a localmeasure of returns to scale. One such measure, defined at a point, tells usthe instantaneous percentage change in output that occurs with a 1percent increase in all inputs. The measure is known as the elasticity ofscale or the elasticity of output, and is defined as follows.

Definition 3.4 (Local) Returns to Scale

The elasticity of scale at the point x is defined as

µ(x) ≡ limt→1d ln[f (tx)]

d ln(t)=

∑ni=1 fi (x)xif (x)

.

Returns to scale are locally constant, increasing, or decreasing as µ(x) isequal to, greater, than, or less than one. The elasticity of scale and theoutput elasticities of the inputs are related as follows:

µ(x) =∑n

i=1 µi (x)


Example 3.2

Let’s examine a production function with local returns to scale:

y = k(1 + x−α1 x−β2 )−1

where α > 0, β > 0, and k is an upper bound on the level of output, sothat 0 ≤ y < k. Calculating the output elasticities for each output, weobtain

µ1(x) = α(1 + x−α1 x−β2 )−1x−α1 x−β2 ,

µ2(x) = β(1 + x−α1 x−β2 )−1x−α1 x−β2 ,

Adding these two gives the elasticity of scale:

µ(x) = (α + β)(1 + x−α1 x−β2 )−1x−α1 x−β2 = (α + β)(1− yk )

which also varies with x1 and x2, or output y . If (α + β) > 1, theproduction function could exhibit increasing and decreasing to scale forlow and high levels of output, respectively.


3.3 Cost

For a given level of output, there are various input vectors that canachieve it. The firm must decide which of the possible production plans(input vectors) it will use. If the object of the firm is to maximize profits,it will necessarily choose the least costly, or cost-minimizing, productionplan for every level of output.We will assume throughout that firms are perfectly competitive on theirinput markets and that therefore they face fixed input prices. Letw = (w1, · · · ,wn) ≥ 0 be a vector of prevailing market prices at which thefirm can buy inputs x = (x1, · · · , xn). Because the firm is a profitmaximizer, it will choose to produce some level of output while using thatinput vector requiring the smallest money outlay. One can speak thereforeof the cost of output y - it will be the cost at prices w of the least costlyvector of inputs capable of producing y .


Definition 3.5 The Cost Function

The cost function, defined for all input prices w� 0 and all output levelsy ∈ f (Rn

+) is the minimum-value function,

c(w, y) ≡ minx∈Rn+w · x s.t. f (x) ≥ y .

If x(w, y) solves the cost-minimization problem, then

c(w, y) = w · x(w, y).

Because f is strictly increasing, the constraint will always be binding at asolution. Consequently, the cost minimization problem is equivalent to

minx∈Rn+w · x s.t. y=f (x).

Let x∗ denote a solution to the minimization problem and f isdifferentiable at x∗. By Lagrange’s theorem, there is a λ∗ ∈ R such that


wi = λ∗∂f (x∗)

∂xi, i = 1, · · · , n.

Because wi > 0, i = 1, · · · , n, we may divide the preceding ith equation bythe jth to obtain

∂f (x∗)/xi∂f (x∗)/xj

=wi

wj.

Thus, cost minimization implies that the marginal rate of technicalsubstitution between any two inputs is equal to the ratio of their prices.

From the first-order condition, it is clear the solution depends on theparameters w and y , i.e. x∗(w, y). Given Assumption 3.1 and assumew� 0, the solution is unique. Figure 3.4. (see the next slide) illustratesthe solution to the cost-minimization problem with a two-input case. Thesolution is point where the y -level isoquant Q(y) and an isocost line if theform w · x = α, α > 0 tangent. If x1(w, y) and x2(w, y) are solutions, thenthe minimum cost is c(w, y) = w1x1(w, y) + w2x2(w, y).


Additionally, x∗(w, y) is referred to as the firm’s conditional inputdemand, because it is conditional on the arbitrary level of output y . So atthis point, cost minimization may or may not be profit maximizing.

Figure 3.4. The solution to the firm’s cost-minimization problem. In the figure,

α < α′.


Example 3.3

There is a cost minimization problem with the CES production function,

minx1≥0,x2≥0w1x1 + w2x2 s.t. (xρ1 + xρ2 )1ρ ≥ y

Assuming the interior solution exists, the first-order Lagrangian conditionsreduce to the two conditions

w1

w2=

(x1

x2

)ρ−1

y = (xρ1 + xρ2 )1ρ

Solve these two equations for x1 and x2,

x1 = yw1/(ρ−1)1 (w

ρ/(ρ−1)1 + w

ρ/(ρ−1)2 )−1/ρ,

x2 = yw1/(ρ−1)2 (w

ρ/(ρ−1)1 + w

ρ/(ρ−1)2 )−1/ρ.



To obtain the cost function, just substitute x1 and x2 into the objectivefunction for the minimization problem.

c(w, y) = w1x1(w, y) + w2x2(w, y) = y(wρ/(ρ−1)1 + w

ρ/(ρ−1)2 )ρ−1/ρ.

The cost function and the expenditure function in consumer theory isidentical, so they share the same properties. We state them here withoutproofs for their proofs are identical to the those given for the expenditurefunction.

Theorem 3.2 Properties of the Cost Function

If f is continuous and strictly increasing, then c(w, y) is

Zero when y = 0.

Continuous on its domain.

For all w� 0, strictly increasing and unbounded above in y .


Theorem 3.2 (Continued)

Increasing in w.

Homogeneous of degree one in w.

Concave in w.Moreover, if f is strictly quasiconcave we have

Shephard’s lemma: c(w, y) is differentiable in w at (w0, y0) wheneverw0 � 0, and

∂c(w0, y0)

∂wi= xi (w

0, y0), i = 1, · · · , n.

Example 3.4

Consider a cost function with the Cobb-Douglas form, y = xα1 xβ2 , where

α + β = 1. We can obtain the cost function c(w, y) = Awα1 w

β2 y where

A = β−βα−α.



By Shephard’s lemma,

x1(w, y) =∂c(w, y)

∂w1= αAwα−1

1 wβ2 y

x2(w, y) =∂c(w, y)

∂w2= βAwα

1 wβ−12 y

As solutions to the firm’s cost-minimization problem, the conditional inputdemand functions possess certain general properties. These are analogousto the properties of Hicksian demands, so once again it is not necessary torepeat the proof.

Theorem 3.3 Properties of Conditional Input Demands

Suppose the production function satisfies Assumption 3.1 and that theassociated cost function is twice continuously differentiable. Then


Theorem 3.3 (Continued)

1 x(w, y) is homogeneous of degree zero in w.

2 The substitution matrix, defined and denoted

σ∗(w, y) ≡

∂x1(w,y)∂w1

· · · ∂x1(w,y)∂wn

.... . .

...∂xn(w,y)∂w1

· · · ∂xn(w,y)∂wn

is symmetric and negative semidefinite. In particular, the negativesemidefiniteness property implies that ∂xi (w, y)/∂wi ≤ 0 for all i .

Definition: Homothetic Function

A real-valued function h on D ⊂ Rn is called homothetic if it can bewritten in the form, h(x) = g(f (x)), where g : R→ R is strictly increasingand f : D → R is homogeneous of degree 1.


The cost and conditional input demands associated with homotheticproduction technologies have some special properties collected in whatfollows.

Theorem 3.4 Cost and Conditional Input Demands When Productionis Homothetic

When the production function satisfies Assumption 3.1 and is homothetic,

the cost function is multiplicatively separable in input prices andoutput and can be written c(w, y) = h(y)c(w, 1), where h′(y) > 0and c(w, 1) is the unit cost function, or the cost of 1 unit of output;

the conditional input demands are multiplicatively separable in inputprices and output and can be written x(w, y) = h(y)x(w, 1), whereh′(y) > 0 and x(w, 1) is the conditional input demand for 1 unit ofoutput.


The general form of the cost function that we have been considering untilnow is most properly viewed as giving the firm’s long-run cost because thefirm may freely choose the amount of every input it uses. In the short run,the firm is ”stuck” with fixed amounts of certain inputs. We should expectits cost in the short run to differ from its costs in the long run. Thefollowings define the short-run cost function.

Definition 3.6 The Short-Run, or Restricted, Cost Function

Let the production function be f (z), where z ≡ (x, x̄). Suppose that x is asubvector of variable inputs and x̄ is a subvector of fixed inputs. Let w andw̄ be the associated input prices for the variable and fixed inputs,respectively. The short-run, or restricted, total cost function is defined as

sc(w, w̄, y ; x̄) ≡ minx w · x + w̄ · x̄ s.t. f (x, x̄) ≥ y .

If x(w, w̄, y ; x̄) solves this minimization problem, then


Definition 3.6 (Continued)

sc(w, w̄, y ; x̄) = w · x(w, w̄, y ; x̄)︸︷︷︸total variable cost

+ w̄ · x̄︸︷︷︸total fixed cost

It should be clear that for a given level of output, long-run costs, wherethe firm is free to choose all inputs optimally, can never be greater thanshort-run costs, where the firm may choose some but not all inputsoptimally. This point is illustrated in Figure 3.5. (see the next slide) wherex2 is fixed in x̄2 in the short run.For the level of output y1, the optimal choice of x2 is x̄ ′2 < x̄2 and thecorresponding x1 (the input combination B) where the isoqant tangent tothe long-run cost c(y1). But in the short run, x2 is fixed in x̄2, to minimizethe cost given y1, the firm has to use the input combination A. And thecost is sc(y1) > c(y1). For the case where the level of output y3 isproduced, in the short run the level of input x̄2 is smaller than the long-runoptimal level x̄ ′′2 , so sc(y3) > c(y3). The only exception is the case wherethe short-run constraint of x̄2 is exactly equivalent to the


long-run choice, like the case that sc(y2) = c(y2) in the center of Figure3.5.

Figure 3.5. sc(w, w̄, y ; x̄) ≥ c(w, w̄, y) for all output levels y .


To explore the relationship between the long-run and short-run costfunctions a bit further, let x̄(y) denote the optimal choice of the fixedinput to minimize short-run cost of output y at the given input prices.Then we’ve argue that

c(w, w̄, y) ≡ sc(w, w̄, y ; x̄(y)) (eq.1)

must hold for any y . Differentiate this equation with respect to y ,

dc(w, w̄, y)

dy=∂sc(w, w̄, y ; x̄(y))

∂y+∑i

∂sc(w, w̄, y ; x̄(y))

∂x̄i

∂x̄i (y)

∂y(eq.2)

But since x̄(y) is the optimal choice of the fixed input at the level y ,

∂sc(w, w̄, y ; x̄(y))

∂x̄i≡ 0

for all fixed input i . Therefore, (eq.2) becomes


dc(w, w̄, y)

dy=∂sc(w, w̄, y ; x̄(y))

∂y(eq.3)

The short-run cost-minimization problem involves more constraints on thefirm than the long-run problem, so we know that sc(w, w̄, y ; x̄)≥ c(w, w̄, y) for all levels of the fixed input. (eq.1) shows that for somelevels of output y∗, sc(w, w̄, y∗; x̄) is equal to c(w, w̄, y∗). (eq.3) furthershows that when sc(w, w̄, y∗; x̄) = c(w, w̄, y∗), the short-run and long-runcost coincides at a point where their slopes are equal in the cost-outputplane, illustrated in Figure 3.6. (see the next slide) where at y1, y2 andy3, the short-run cost is equal to the long-run cost.


Figure 3.6. Long-run total cost is the envelope of short-run total cost.


3.4 Duality in Production

Given the obvious structural similarity between the firm’scost-minimization problem and the individual’s expenditure minimizationproblem, there is also a duality between production and cost just as thereis between utility and expenditure. It is possible to recover the productionfunction through the conditional input demand functions. And if theconditional input demand function x(w, y) is homogeneous of degree zeroin w, and its substitution matrix is negative semidefinite and symmetric,then we can ensure it is consistent with the cost-minimization behavior.


2nd Midterm Exam

Date/Time: Monday Nov. 30th, 2015 / 9:00 am -11:30 am

Location: The meeting room on the 2nd floor, School of Economics

Coverage: Consumer Theory (Week 6- Week 9)

Rules: Same as the 1st midterm


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Week 10: Theory of the Firm (Jehle and Reny, Chapter 3)economics-course.weebly.com/uploads/2/5/7/2/25725158/...Week 10: Theory of the Firm (Jehle and Reny, Chapter 3) Tsun-Feng Chiang*