Optimization problems in economics usually involve maximizing or minimizing functions which depend not only on endogenous variables one can choose, but also on one or more exogenous parameters like prices, tax rates, income levels, etc. Although these parameters are held constant during the optimization, they vary according to the economic situation.
For example, we may calculate a firm’s profit-maximizing input and output quantities while treating the prices it faces as parameters. But then we may want to know how the optimal quantities respond to changes in those prices, or in whatever other exogenous parameters affect the problem we are considering.
Consider first the following simple problem. A functionf depends on a single variable x as well as on a single parameterr. We wish to maximize or minimizef (x, r)w.r.t.x while keepingrconstant:
max(min)x f (x, r)
The value ofxthat maximizes (minimizes)f will usually depend onr, so we denote it by x∗(r). Insertingx∗(r)intof (x, r), we obtain
f∗(r)=f (x∗(r), r) (thevalue function)
What happens to the value function asrchanges? Assuming thatf∗(r)is differentiable, the chain rule yields
df∗(r)
dr =f1(x∗(r), r)dx∗(r)
dr +f2(x∗(r), r)
Iff (x, r)has an extreme point at an interior pointx∗(r)in the domain of variation forx, thenf1(x∗(r), r)=0. It follows that
df∗(r)
dr =f2(x∗(r), r) (1)
Note that whenr is changed, thenf∗(r)changes for two reasons. First, a change in r changes the value off∗ directly because r is the second variable inf (x, r). Second, a change in r changes the value of the function x∗(r), and hence f (x∗(r), r)is changed indirectly. Formula (1) shows that the total effect is simply found by computing the partial derivative off (x∗(r), r)w.r.t.r, ignoring entirely the indirect effect of the dependence of x∗onr. At first sight, this seems very surprising. On further reflection, however, you may realize that the first-order condition forx∗(r)to maximizef (x, r)w.r.t.ximplies that any small change inx, whether or not it is induced by a small change inr, must have a negligible effect on the value off (x∗, r).
E X A M P L E 1 Suppose that when a firm produces and sellsx units of a commodity, it has revenue R(x)=rx, while the cost isC(x)=x2, whereris a positive parameter. The profit is then
π(x, r)=R(x)−C(x)=rx−x2 Find the optimal choicex∗ofx, and verify (1) in this case.
Solution: The quadratic profit function has a maximum whenπ1 =r−2x=0, that is for x∗ =r/2. So the maximum profit as a function ofris given byπ∗(r)=rx∗−(x∗)2 = r(r/2)−(r/2)2 =r2/4, and thendπ∗/dr =r/2. Using formula (1) is much more direct:
becauseπ2(x, r)=x, it implies thatdπ∗/dr =π2(x∗(r), r)=x∗(r)= 12r.
E X A M P L E 2 In Example 8.6.5 we studied a firm with the profit functionπ (Q, t )ˆ =R(Q)−C(Q)−t Q, wheretdenoted a tax per unit produced. LetQ∗=Q∗(t )denote the optimal choice ofQ as a function of the tax ratet, and letπ∗(t )be the corresponding value function. Because
ˆ
π2= −Q, formula (1) yields dπ∗(t )
dt = ˆπ2(Q∗(t ), t )= −Q∗(t ) which is the same result found earlier.
It is easy to generalize (1) to the case with many choice variables and many parameters. We letx=(x1, . . . , xn), andr=(r1, . . . , rk). Then we can formulate the following result:
E N V E L O P E T H E O R E M
Iff∗(r)=maxxf (x,r)and ifx∗(r)is the value ofx that maximizesf (x,r),
then ∂f∗(r)
∂rj =
∂f (x,r)
∂rj
x=x∗(r)
, j =1, . . . , k provided that the partial derivative exists.
(2)
Again,f∗(r)is thevalue function. It is easy to prove (2) by using the first-order conditions to eliminate other terms, as in the argument for (1). The same equality holds if we minimize f (x,r)w.r.t.xinstead of maximize (or even ifx∗(r)is any stationary point).
y ⫽ f*(r)
Kx⬘
Kx*
Kx⬙ y
r r
Figure 1 The curvey=f∗(r)is the envelope of all the curvesy=f (x, r)
Figure 1 illustrates (2) in the case where there is only one parameterr. For each fixed value ofxthere is a curveKxin thery-plane, given by the equationy =f (x, r). Figure 1 shows some of these curves together with the graph ofy = f∗(r). For allxand allr we have f (x, r)≤maxxf (x, r)=f∗(r). It follows that none of theKx-curves can ever lie above
S E C T I O N 1 3 . 7 / C O M P A R A T I V E S T A T I C S A N D T H E E N V E L O P E T H E O R E M 493 the curvey =f∗(r). On the other hand, for each value ofrthere is at least one valuex∗ ofxsuch thatf (x∗, r) =f∗(r), namely the choice ofx∗which solves the maximization problem for the given value of r. The curveKx∗will then just touch the curvey =f∗(r)at the point(r, f∗(r))=(r, f (x∗, r)), and so must have exactly the same tangent as the graph ofy =f∗(r)at this point. Moreover, becauseKx∗ can never go above this graph, it must have exactly the same tangent as the graph off∗at the point where the curves touch. The slope of this common tangent, therefore, must be not onlydf∗/dr, the slope of the tangent to the graph off∗at(r, f∗(r)), but also∂f (x∗, r)/∂r, the slope of the tangent to the curve Kx∗ at the point(r, f (x∗, r)). Equation (2) follows becauseKx∗ is the graph off (x∗, r) whenx∗is fixed.
As Fig. 1 suggests, the graph ofy =f∗(r)is the lowest curve with the property that it lies on or above all the curvesKx. So its graph is like an envelope or some “cling film” that is used to enclose or wrap up all these curves. Indeed, a point is on or below the graph if and only if it lies on or below one of the curvesKx. For this reason we call the graph off∗the envelopeof the family ofKx-curves. Also, result (2) is often called theenvelope theorem.
E X A M P L E 3 In Example 13.1.3,Q=F (K, L)denoted a production function withKas capital input andLas labour input. The price per unit of the product wasp, the price per unit of capital wasr, and the price per unit of labour wasw. The profit obtained by usingKandLunits of the inputs, then producing and sellingF (K, L)units of the product, is given by
ˆ
π (K, L, p, r, w)=pF (K, L)−rK−wL
Here profit has been expressed as a new function πˆ of the parametersp, r, andw, as well as of the choice variables K and L. We keep p, r, and w fixed and maximizeπˆ w.r.t.K andL. The optimal values ofK andLare functions of p,r, andw, which we denote byK∗=K∗(p, r, w)andL∗=L∗(p, r, w). The value function for the problem is
ˆ
π∗(p, r, w)= ˆπ (K∗, L∗, p, r, w). Usually,πˆ∗is called the firm’sprofit function, though it would be more accurately described as the “maximum profit function”. It is found by taking prices as given and choosing the optimal quantities of all inputs and outputs.
According to (2), one has
∂πˆ∗
∂p =F (K∗, L∗)=Q∗, ∂πˆ∗
∂r = −K∗, ∂πˆ∗
∂w = −L∗ (∗) These three equalities are instances of what is known in producer theory as Hotelling’s lemma. An economic interpretation of the middle equality is this: How much profit is lost if the price of capital increases by a small amount? At the optimum the firm usesK∗units of capital, so the answer isK∗per unit increase in the price. See Problem 4 for further interesting relationships.
P R O B L E M S F O R S E C T I O N 1 3 . 7
1. (a) A firm produces a single commodity and getspfor each unit sold. The cost of producing xunits isax+bx2and the tax per unit ist. Assume that the parameters are positive with p > a+t. The firm wants to maximize its profit. Find the optimal productionx∗and the optimal profitπ∗.
(b) Prove that∂π∗/∂p=x∗, and give an economic interpretation.
⊂SM⊃2. (a) A firm uses capitalK, labourL, and landT to produceQunits of a commodity, where Q=K2/3+L1/2+T1/3
Suppose that the firm is paid a positive pricepfor each unit it produces, and that the positive prices it pays per unit of capital, labour, and land arer,w, andq, respectively. Express the firm’s profits as a functionπof(K, L, T ), then find the values ofK,L, andT (as functions of the four prices) that maximize the firm’s profits (assuming a maximum exists).
(b) LetQ∗ denote the optimal number of units produced andK∗the optimal capital stock.
Show that∂Q∗/∂r= −∂K∗/∂p.
3. (a) A firm producesQ = aln(L+1)units of a commodity when labour input isLunits.
The price obtained per unit isPand price per unit of labour isw, both positive, and with w < aP. Write down the profit functionπ. What choice of labour inputL=L∗maximizes profits?
(b) ConsiderL∗as a function of all the three parameters,L∗(P , w, a), and defineπ∗(P , w, a)= π(L∗, P , w, a). Verify that∂π∗/∂P =πP(L∗, P , w, a),∂π∗/∂w=πw(L∗, P , w, a), and
∂π∗/∂a=πa(L∗, P , w, a), thus confirming the envelope theorem.
4. With reference to Example 3, assuming thatFis aC2function, prove the symmetry relations:
∂Q∗
∂r = −∂K∗
∂p , ∂Q∗
∂w = −∂L∗
∂p , ∂L∗
∂r =∂K∗
∂w
(Hint:Using the first result in Example 3 andYoung’s theorem, we have the equalities∂Q∗/∂r= (∂/∂r)(∂πˆ∗/∂p)=(∂/∂p)(∂πˆ∗/∂r). Now use the other results in Example 3.)
⊂SM⊃5. (a) With reference to Example 3 we want to study the factor demand functions—in particular how the optimal choices of capital and labour respond to price changes. By differentiating the first-order conditions(∗∗)in Example 13.1.3, verify that we get
FK(K∗, L∗) dp+pFKK (K∗, L∗) dK+pFKL (K∗, L∗) dL=dr FL(K∗, L∗) dp +pFLK (K∗, L∗) dK+pFLL (K∗, L∗) dL=dw
(b) Use this system to find the partials ofK∗andL∗w.r.t.p,r, andw. (You might find it easier first to find∂K∗/∂pand∂L∗/∂pby puttingdr=dw=0, etc.)
(c) Assume that the local second-order conditions(∗∗)in Example 13.3.3 are satisfied. What can you say about the signs of the partial derivatives? In particular, show that the factor demand curves are downward sloping as functions of their own factor prices. Verify that
∂K∗/∂w=∂L∗/∂r.
⊂SM⊃6. A profit-maximizing monopolist produces two commodities whose quantities are denoted by x1andx2. Good 1 is subsidized at the rate ofsper unit and good 2 is taxed attper unit. The monopolist’s profit function is therefore given by
π(x1, x2)=R(x1, x2)−C(x1, x2)+sx1−t x2
R E V I E W P R O B L E M S F O R C H A P T E R 1 3 495 whereRandCare the firm’s revenue and cost functions, respectively. Assume that the partial derivatives of these functions have the following signs:
R1 >0, R2>0, R11 <0, R12=R21<0, R22 <0 C1>0, C2>0, C11 >0, C12=C21 >0, C22>0 (a) Find the first-order conditions for maximum profits.
(b) Write down the local second-order conditions for maximum profits.
(c) Suppose thatx∗1 = x1∗(s, t ),x2∗ =x2∗(s, t )solve the problem. Find the signs of∂x1∗/∂s,
∂x∗1/∂t,∂x2∗/∂s, and∂x2∗/∂t, assuming that the local second-order conditions are satisfied.
(d) Show that∂x1∗/∂t = −∂x∗2/∂s.
R E V I E W P R O B L E M S F O R C H A P T E R 1 3
1. The functionf defined for all(x, y)byf (x, y)= −2x2+2xy−y2+18x−14y+4 has a maximum. Find the corresponding values ofxandy. Use Theorem 13.2.1 to prove that it is a maximum point.
⊂SM⊃2. (a) A firm produces two different kindsAandBof a commodity. The daily cost of producing Q1units ofAandQ2units ofBisC(Q1, Q2)=0.1(Q21+Q1Q2+Q22). Suppose that the firm sells all its output at a price per unit ofP1=120 forAandP2=90 forB. Find the daily production levels that maximize profits.
(b) IfP2remains unchanged at 90, what new price (P1) per unit ofAwould imply that the optimal daily production level forAis 400 units?
3. (a) The profit obtained by a firm from producing and sellingxandyunits of two brands of a commodity is given by
P (x, y)= −0.1x2−0.2xy−0.2y2+47x+48y−600 Find the production levels that maximize profits.
(b) A key raw material is rationed so that total production must be restricted to 200 units. Find the production levels that now maximize profits.
⊂SM⊃4. Find the stationary points of the following functions:
(a) f (x, y)=x3−x2y+y2 (b) g(x, y)=xye4x2−5xy+y2 (c) f (x, y)=4y3+12x2y−24x2−24y2
5. Definef (x, y, a)=ax2−2x+y2−4ay, whereais a parameter. For each fixeda=0, find the point(x∗(a), y∗(a))that makes the functionf stationary w.r.t.(x, y). Find also the value functionf∗(a)=f (x∗(a), y∗(a), a), and verify the envelope theorem in this case.
⊂SM⊃6. (a) Suppose the production function in Problem 13.7.2 is replaced byQ=Ka+Lb+Tc, for parametersa, b, c∈(0,1). Assuming that a maximum exists, find the values ofK,L, and T that maximize the firm’s profits.
(b) Letπ∗ denote the optimal profit as a function of the four prices. Compute the partial derivative∂π∗/∂r.
(c) Verify the envelope theorem in this case.
7. Definef (x, y)for all(x, y)by
f (x, y)=ex+y+ex−y−32x−21y
(a) Find the first- and second-order partial derivatives off, then show thatf (x, y)is convex.
(b) Find the minimum point off (x, y).
⊂SM⊃8. (a) Find and classify the stationary points of
f (x, y)=x2−y2−xy−x3
(b) Find the domainSwheref is concave, and find the largest valuef inS.
⊂SM⊃9. Consider the functionfdefined for all(x, y)byf (x, y)=12x2−x+ay(x−1)−13y3+a2y2, whereais a constant.
(a) Prove that(x∗, y∗)=(1−a3, a2)is a stationary point off. (b) Verify the envelope theorem in this case.
(c) Where in thexy-plane isf convex?
⊂SM⊃10. In this problem we will generalize several of the economic examples and problems considered so far. Consider a firm that produces two different goodsAandB. If the total cost function is C(x, y)and the prices obtained per unit ofAandBarepandqrespectively, then the profit is
π(x, y)=px+qy−C(x, y) (i)
(a) Suppose first that the firm has a small share in the markets for both these goods, and so takespandqas given. Write down and interpret the first-order conditions forx∗>0 and y∗>0 to maximize profits.
(b) Suppose next that the firm has a monopoly in the sale of both goods. The prices are no longer fixed, but chosen by the monopolist, bearing in mind the demand functions
x=f (p, q) and y=g(p, q) (ii)
Suppose we solve equations (ii) forpandqto obtain the inverse demand functions
p=F (x, y) and q=G(x, y) (iii)
Then profit as a function ofxandyis
π(x, y)=xF (x, y)+yG(x, y)−C(x, y) (iv) Write down and interpret the first-order conditions forx∗ >0 andy∗ >0 to maximize profits.
(c) Supposep=a−bx−cyandq=α−βx−γ y, wherebandγare positive. (An increase in the price of either good decreases the demand for that good, but may increase or decrease the demand for the other good.) If the cost function isC(x, y) =P x+Qy+R, write down the first-order conditions for maximum profit.
(d) Prove that the (global) second-order conditions are satisfied provided 4γ b≥(β+c)2.
14 C O N S T R A I N E D O P T I M I Z A T I O N
Mathematics is removed from this turmoil of human life, but its methods and the relations are a mirror, an incredibly pure mirror, of the relations that link facts of our existence.
—Konrad Knopp (1928)
The previous chapter 13 introduced unconstrained optimization problems with several vari- ables. In economics, however, the variables to be chosen must often satisfy one or more constraints. Accordingly, this chapter considers constrained optimization problems, and studies the method of Lagrange multipliers in some detail. Sections 14.1–14.7 treat equality constraints, with Section 14.7 presenting some comparative static results and the envelope theorem. More general constrained optimization problems allowing inequality constraints are introduced in Sec- tions 14.8–14.10. A much fuller treatment of constrained optimization can be found in FMEA.