The Mathematics of Utility Maximization

In the text a graphical approach was used to explore utility maximization and comparative statics. This appendix provides a brief introduction to a more rigorous analysis based on calculus. A knowledge of multivariate differentiation is needed to understand this section.

The problem facing a consumer is to maximize utility subject to a budget constraint. Utility is achieved through consuming goods, with the level of utility actually reached determined by the utility function U=f(X,Y), where for simplicity we restrict ourselves to two goods. The marginal utility for each good is the partial derivative of the function f with respect to each argument. That is, the marginal utility of X is given by the formula MUX=¶ f(X,Y)/¶ X. Similarly, for the marginal utility of Y, MUY=¶ f(X,Y)/¶ Y. We assume that the second partial derivatives, ¶ ²f(X,Y)/¶ X² and ¶ ²f(X,Y)/¶ Y² are negative to ensure diminishing marginal utility in each good. Also assume the f(X,Y) is a smooth, continuous function and that an interior solution to the problem exists.

The consumer maximizes utility subject to his budget constraint, M=P_xX+P_yY. To find the utility maximizing consumption we use a technique called Lagrangian multipliers which is a technique for finding constrained maximums. The Lagrange equation has the form L=f(X,Y)+l (M-P_xX-P_yY). L becomes the new function to be maximized, l is called the Lagrangian multiplier, and the budget constraint is written in implicit form.

The constrained maximum is found by treating l as a third variable in the problem, taking the partial derivatives of L with respect to each variable, and setting the results equal to zero. That is, derive each of the following equations:

¶ L/¶ X = ¶ f/¶ X - l P_x = 0 (1)

¶ L/¶ Y = ¶ f/¶ Y - l P_y = 0 (2)

¶ L/¶ l = M-P_xX-P_yY = 0 (3)

These are called the first-order necessary conditions. Equation (3) is just the budget constraint, indicating that to maximize the consumer should spend all his income. Equation (1) requires that the MUX=l P_x, and likewise equation (2) requires that MUY=l P_y. Rearranging gives that MUX/P_x=l and MUY/P_y=l . Thus, we find the second condition, that MUX/P_x=MUY/P_y. The conditions we put on f(X,Y), smoothness, continuity, and that the function exhibit diminishing marginal utility, ensures that this is a maximum, not a minimum.

¶ L/¶ l = 200 - 2X - Y = 0 (6).

Reducing (4) and (5) gives Y=2X which can be substituted into (6) to find 200-Y-Y=0, or Y=100, which means that X=50.

The easy way to do comparative statics is to just substitute alternative values for income or prices into the first-order necessary conditions. For example, if income were to fall to 100, but prices remained at 2 for X and 1 for Y, consumption would change to Y=50 and X=25. Alternatively, an increase in the price of Y to 2, holding income at 200 and the price of X at 2 means that X=Y, X and Y both equal 50.

Application Problems

1. Using the example utility function from the appendix, show that both goods exhibit diminishing marginal utility, and determine if the goods are normal or inferior.

2. Try various prices for X and Y in the example from the appendix, and determine a general rule for allocation of income across goods with this type of utility function.