A note on Lagrange multipliers with several binding constraints

Economics Letters 59 (1998) 71–75

A note on Lagrange multipliers with several binding constraints Christian E. Weber* Dept. of Economics and Finance, Albers School of Business and Economics, Seattle University, Seattle, WA 98122, USA Received 23 April 1997; accepted 23 January 1998

Abstract When a function is optimized subject to one constraint, the Lagrange multipliers for the primal and dual problems are reciprocals. Here, I show how this result generalizes to the case where a function is optimized subject to several binding constraints.  1998 Elsevier Science S.A. Keywords: Lagrange multipliers; Multiple constraints; Dual problems JEL classification: C61; D11

1. Introduction Recently, analysis of household decision making under multiple constraints has been applied to a number of different questions. Cornes, 1992 (Chap. 7) provides a thorough survey of recent applications of optimization subject to several constraints to problems ranging from household allocation of time to choice under uncertainty to constraints on hours of labor supplied.1 He also notes that if an objective function is optimized subject to m binding constraints, then the maximization problem has associated with it m different dual problems, one for each of the m constraints in the primal. In the single constraint case, Silberberg, 1990 (Section 10.3) shows that the Lagrange multipliers for the primal and dual problems are reciprocals.2 In the multiple constraint case, however, the primal problem and its duals have m different Lagrange multipliers each. In this note, I show how the Lagrange multipliers in the primal problem and its duals are related to each other. That is, I show how the reciprocal result of the single constraint case generalizes to the multiple constraint case. The analysis is presented in Section 2. Section 3 presents a specific example, and Section 4 contains a brief summary. *Tel.: (1-206) 296-5725; fax: (1-206) 296-2486; e-mail: [email protected] 1 On the allocation of time, see Baumol (1973); Atkinson and Stern (1979). On decision making under uncertainty, see Diamond and Yaari (1972). On quantity constraints in the labor market, see Deaton and Muellbauer (1981); Blundell and Walker (1982). Historically, studies of household behavior subject to multiple constraints include the discussions of rationing of Samuelson (1947); Tobin and Houthakker (1951); Tobin (1952), and others. 2 Silberberg (1990) remarks that the reciprocal relationship between the multipliers ‘is a matter of some curiosity.’ 0165-1765 / 98 / $19.00  1998 Elsevier Science S.A. All rights reserved. PII S0165-1765( 98 )00021-4

C.E. Weber / Economics Letters 59 (1998) 71 – 75

72

2. Analysis To examine the relationships among the Lagrange multipliers, let x be a 1 3 n vector of goods consumed by the household. The household is assumed to maximize a quasi-concave utility function, U 5 U(x), subject to the m-vector of constraints: g(x) 5 0,

(1)

where g(x) is an m31 vector of functions of x. To ensure that Eq. (1) does not determine x alone, I assume m,n. The primal problem with all constraints binding is thus: maxp L 5 U(x) 2 L p g(x) ,

(2)

[x, L ] p

p

where L is a 13m vector of Lagrange multipliers, with typical element li , ie h1,..., mj. The superscript p indicates that L p and the l pi ’s apply to the primal problem. With all m constraints binding, the first-order conditions for this problem are the m constraints, Eq. (1), and the n vector of conditions: =L 5=U(x) 2 L p=G(x) 5 0 ,

(3)

where =L and =U(x) are the 13n gradients of the Lagrangean and utility functions, respectively, and =G(x) is an m3n matrix with typical element, ≠gi (x) / ≠x j . That is, the ith row of =G(x) is =gi (x), while the jth column is the vector of derivatives of each element of g(x) with respect to good j. To ensure that none of the constraints is a combination of the other constraints (i.e., to ensure that none of the constraints is redundant), I assume that =G(x) has full row rank: rank[=G(x)]5m. For the primal problem, Eq. (2), there are m dual problems, one for each constraint. The kth dual problem takes the level of utility and all m21 constraints in Eq. (1) except the kth as given and binding. If U 0 is the target level of utility, then the kth dual problem is: min

[x, L k , m k ]

L k 5 gk (x) 1 m k (U 0 2 U(x))) 1 L k g k (x) , i ± k,

k

(4) k

where m is the Lagrange multiplier on the utility constraint and L is a 13(m21) vector of Lagrange multipliers, with typical element l ki . The k superscripts indicate that the multipliers on the utility constraint and the other constraints are different for each of the m dual problems. Finally, g k (x), an (m21)31 vector of constraints, is g(x) with the kth constraint from the primal problem eliminated. The first-order conditions for this problem are the m constraints (including the utility constraint) and the n conditions: =L k 5=gk (x) 2 m k =U(x) 1 L k =G k (x) 5 0 ,

(5)

where =gk (x) is the kth row of =G(x), and =G k (x) is an (m21)3n matrix with typical element, ≠gi (x) / ≠x j , i ±k. Thus, the ith row of =G k (x) is =G i (x), for i ,k; for i .k, the ith row of =G k (x) is k =gi 11 (x). Defining L * as: k L k * ; [ l 1k l 2k ... l k21 1l kk11 ... l km ] ,

(6)

C.E. Weber / Economics Letters 59 (1998) 71 – 75

73

we can rewrite Eq. (5) as:

L k * =G(x) 2 m k =U(x) 5 0 .

(59)

I turn now to the relationships between the m(m11) multipliers, l pi , m k , and l ik for i,ke h1,...,mj, i ±k. These are summarized in: Theorem. For l pi , m i , and l ki as defined above, we have: l pi 51 /m i and l ip 5l ik /m k for i ± k. Proof. Write the first m of the f.o.c.’s for the primal problem (Eq. (3)) in vector matrix form as: =Um (x) 5 L p=Gm (x)

(7) p

where =G m (x) is an m3m matrix containing the first m columns of =G(x), L was defined above, and =Um (x) is a 13m vector containing the first m elements of =U(x). Assume that the elements of x are numbered such that =G m (x) has rank m.3 Next, divide each of the first m of the f.o.c.’s for the kth dual problem, equation Eq. (5)9, by m k and write the results as: =Um (x) 5 [L k * /m k ]=Gm (x)

(8) k

k

where =G m (x) and =Um (x) are as defined above, and L * /m is a 13m vector whose ith element for i ±k is l ki /m k ; its kth element is 1 /m k . Combining Eqs. (7) and (8) yields:

L p=Gm (x) 5 [L k * /m k ]=Gm (x) ,

(9)

or, postmultiplying both sides by [=G m (x)] 21 (which exists since =G m (x) has full rank):

L p 5 L k * /m k ,

(10)

which, for all possible combinations of i and k, is the result to be shown. In the single constraint case, the Lagrange multipliers for the primal and dual problems reciprocals. This result generalizes to the multiple constraint case, where the multiplier on the constraint in the primal problem is the reciprocal of the multiplier on the utility constraint in the dual problem. The theorem also shows how to calculate all of the m 2 Lagrange multipliers from dual problems once the m Lagrange multipliers from the primal problem are known.

are kth kth the

3. An example As an example, consider the problem of maximizing a quasi-concave utility function, U 5U(x), subject to both an income constraint, Si pi x i 5I, and a coupon rationing constraint, Si c i x i 5C. I is money income, pi is the money price of good i, C is the household’s coupon endowment, and c i is the coupon price of good i. The primal problem is: 3

Since =g(x) has rank m by assumption, this is nothing more than an issue of numbering the elements of x.

C.E. Weber / Economics Letters 59 (1998) 71 – 75

74

max

[x, l p1 , l p2 ]

L 5 U(x) 2 l p1 [Si pi x i 2 I] 2 l p2 [Si c i x i 2 C] .

(11)

The f.o.c.’s for this problem are the two constraints and the n conditions: ≠U(x) ]] 5 l p1 pi 1 l p2 c i , ieh1, ..., nj . ≠x i

(12)

The dual problem of minimizing money expenditure subject to the constant utility constraint and the coupon constraint 4 is: min

[x, m 1 , l 12 ]

L 1 5 Si pi x i 1 m 1 (U 0 2 U(x)) 1 l 12 [Si c i x i 2 C] .

(13)

Here, the f.o.c.’s are the two constraints and: ≠U(x) m 1 ]] 5 pi 1 l 12 c i , ieh1, ..., nj, or: ≠x i 1

l ≠U(x) 1 ]] 5 ]1 pi 1 ]21 c i , ≠x i m m

(14)

Denote the value functions for the problems in Eqs. (11) and (13), respectively, as V(p, I; c, C) and E(p; U 0 ; c, C). The Lagrange multipliers show the impact on the value functions of slightly relaxing the constraint associated with the multiplier: 5 ≠V( ) l p1 5 ]] , ≠I

≠V( ) l p2 5 ]] , ≠C

≠E( ) m 1 5 ]] 0 , ≠U

≠E( ) l 12 5 2 ]] . ≠C

(15)

The analysis of Section 2 implies l p1 51 /m 1 and l p2 5 l 12 /m 1 . Using Eq. (15), these results imply, among other things, the following ‘chain rule’ type envelope results:

l1 ≠V( ) ≠V( ) 2 ≠E( ) ]] 5 l 2p 5 ]21 5 l 1p l 21 5 ]] ]]] , ≠C ≠I ≠C m

D

(16a)

≠E( ) 2 ≠E( ) ≠V( ) ]] 5 2 l 12 5 2 m 1 l p2 5 ( ]]] )]] . 0 ≠C ≠C ≠U

(16b)

S

From Eq. (16a), the gain in utility from increasing the household’s coupon endowment by one unit is the gain in utility from a one unit increase in income times the money income equivalent of an increase in the coupon endowment, ≠E( ) / ≠C. From Eq. (16b), the decline in money expenditure made possible by an increase in the household’s coupon endowment is the decline in expenditure which 4

The second dual problem is to minimize coupon expenditure subject to the constant utility constraint and the money income constraint. 5 ≠E( ) / ≠C is negative. With the coupon constraint binding, an increase in the coupon endowment permits the household to rearrange its purchases to achieve the same level of utility while spending less money income.

C.E. Weber / Economics Letters 59 (1998) 71 – 75

75

occurs when the target utility level falls, 2≠E( ) / ≠U 0 , times the gain in utility which results ceteris paribus from increasing the coupon endowment.

4. Conclusion This paper has developed the relationships among the Lagrange multipliers for the primal problem and each of the dual problems in the case where a function is maximized subject to two or more binding constraints. These relationships generalize to the multiple constraint case the simple reciprocal relationship between the multipliers for the primal problem and its single dual in the single constraint case. Section 3 provided an example of the relationships among the multipliers for a specific two-constraint problem.

Acknowledgements I would like to thank an anonymous referee for helpful comments on an earlier draft of this paper. Any remaining errors are my own responsibility.

References Atkinson, A.B., Stern, N., 1979. A note on the allocation of time. Economics Letters 3, 119–123. Baumol, W.J., 1973. Income and substitution effects in the Linder theorem. Quarterly Journal of Economics 87, 629–633. Blundell, R., Walker, I., 1982. Modelling the joint distribution of household labour supplies and commodity demands. Economic Journal 92, 351–364. Cornes, R., 1992. Duality and Modern Economics, Cambridge University Press, New York. Deaton, A., Muellbauer, J., 1981. Functional forms for labor supply and commodity demands with and without quantity restrictions. Econometrica 49, 1521–1532. Diamond, P.A., Yaari, M.E., 1972. Implications of the theory of rationing for consumer choice under uncertainty. American Economic Review 62, 333–343. Samuelson, P.A., 1947. Foundations of Economic Analysis, Harvard University Press, Cambridge. Silberberg, E., 1990. The Structure of Economics: A Mathematical Analysis, 2nd ed., McGraw-Hill, New York. Tobin, J., 1952. A survey of the theory of rationing. Econometrica 20, 512–553. Tobin, J., Houthakker, H.S., 1951. The effects of rationing on demand elasticities. Review of Economic Studies 18, 140–153.