# Maximization Problem

1166 Words Nov 24th, 2011 5 Pages
CHAPTER 4 APPENDIX
DEMAND THEORY – A MATHEMATICAL TREATMENT
EXERCISES
1. Which of the following utility functions are consistent with convex indifference curves, and which are not?
a. U(X, Y) = 2X + 5Y
b. U(X, Y) = (XY)0.5
c. U(X, Y) = Min(X, Y), where Min is the minimum of the two values of X and Y
The three utility functions are presented in Figures 4A.1.a, 4A.1.b, and 4A.1.c. The first may be represented as a series of straight lines; the second as a series of hyperbolas; and the third as a series of “L’s.” Only the second utility function meets the definition of a strictly convex shape.
To graph the indifference curves which represent the preferences given by U(X,Y)=2X+5Y, set utility to some given level U0 and solve for Y to
Because there is no substitution as prices change with this type of utility function, the substitution effect is zero. The income effect is the shift from U1 to U2.

Figure 4A.3

4. Sharon has the following utility function: where X is her consumption of candy bars, with price PX=\$1, and Y is her consumption of espressos, with PY=\$3.
a. Derive Sharon’s demand for candy bars and espressos.
Using the Lagrangian method, the Lagrangian equation is To find the demand functions, we need to maximize the Lagrangian equation with respect to X, Y, and , which is the same as maximizing utility subject to the budget constraint. The necessary conditions for a maximum are Combining necessary conditions (1) and (2) results in You can now substitute (4) into (3) and solve for Y. Once you have solved for Y, you can substitute Y back into (4) and solve for X. Note that algebraically there are several ways to solve this type of problem, and that it does not have to be done exactly as we have done here. The demand functions are b. Assume that her income I=\$100. How many candy bars and espressos will Sharon consume?
Substitute the values for the two prices and income into the demand functions to find that she consumes X=75 candy bars and Y=8.3 espressos.
c. What is the marginal utility of income?
From part a . Substitute into either part of the equation to find that