Cobb-Douglas1 utility function that measures utility for a bundle of goods x: ... 1The Cobb-Douglas function is invented by Paul Howard Douglas, the later U.S. ...
UTILITY MAXIMIZATION WITH NONLINEAR BUDGET CONSTRAINTS UNDER GAMS ERWIN KALVELAGEN
Abstract. This document describes a simple utility maximization problem with non-convex budget constraints resulting in an MINLP model.
1. Consumer demand and utility maximization Consumer choice theory is an important part of micro economics. In this section we will discuss the well known optimization problem of utility maximization under a budget constraint. Interesting, non-trivial variants can be constructed by assuming some non-linear pricing structures actually observed in daily life. 2. Linear budgets A simple model for optimal consumer behavior is as follows [7]. Consider a Cobb-Douglas1 utility function that measures utility for a bundle of goods x: (1)
u(x) = Kxa1 1 xa2 2 · · · xann = K
n Y
xai i
i=1
Consumers want to maximize utility subject to their budget function: n X (2) pi xi = m i=1
where pi is the price of good i and m is the available budget. I.e. the optimization model that follows is: n Y max u = K xai i i=1
(3)
subject to
n X
pi xi = m
i=1
xi ≥ 0 Such a model can be easily coded and solved in GAMS. The only thing one may want to watch out for is choosing a starting point different than zero. These models will typically fail to find the optimal solution if a default starting point (all zero) is used. Date: March 12, 2003. 1 The Cobb-Douglas function is invented by Paul Howard Douglas, the later U.S. Senator for the state of Illinois, and the mathematician Charles W. Cobb. The Cobb-Douglas function is also used as a production function for which purpose is was originally developed [1]. For an interesting account of other functional forms from before Cobb-Douglas see [5]. 1
2
ERWIN KALVELAGEN
Figure 1. Utility maximization with linear budget
As we can apply a monotone transformation on the objective function we can also write this model as:
max (4)
subject to
n X i=1 n X
ai ln(xi ) pi xi = m
i=1
xi ≥ 0 If we drop the requirement xi ≥ 0 (it is observed automatically in the optimal solution when pi , ai , m > 0), then a closed form solution is available: ai m xi = P j aj pi
(5)
This equation is called the demand function. The demand function can be derived using the lagrangean: (6)
L(x, λ) =
n X i=1
ai ln(xi ) − λ
n X i=1
! pi xi − m
UTILITY MAXIMIZATION WITH NONLINEAR BUDGET CONSTRAINTS UNDER GAMS 3
which yields: ∂L = 0 ⇒ ai = λpi xi ∂xi n (7) X ∂L =0⇒ pi xi = m ∂λ i=1 P P Adding up ai = λpi xi leads to ai = λ pi xi or n 1 X (8) λ= ai m i=1 which can be plugged back in ai = λpi xi to get our result. 2.1. Example: two goods. A two good example is considered. A graphical solution of this model is depicted in figure 1. The contours of the utility function are also known as indifference curves. 2.2. Model consumer1.gms. 2 In this model we choose as starting point xi = 1 and add lowerbounds xi ≥ 0.001. The default starting point x = 0 is quite problematic. First, the derivative of xai i is not defined for xi = 0 and 0 < ai < 1. GAMS will issue a message during generation time: **** Exec Warning 41 at line 27 .. Derivative for zero**b for 0 < b < 1 is undefined - set to 1e10
and **** ERRORS(S) IN EQUATION utility 3 instance(s) of DERIVATIVE OF X**Y NOT DEFINED FOR X=0, 0
