Question

Modified version of problem 7.12 from Nechyba text Assume Mr. T’s preferences are captured by the utility function U(x, y) = x1/2⋅y1/2, where x stands for miles driven and y stands for other consumption, with Py = 1. Suppose that Mr. T has a monthly income of $600 to spend on goods X and Y. a. Suppose that Mr. T (with his bank) owns a gas guzzler. His monthly payment to the bank on the gas guzzler is $200. In addition, for every mile driven, Mr. T pays $0.20 per mile for the gas guzzler. Find Mr. T’s utility-maximizing choice of x and y under these circumstances. Show your work and explain. b. Now suppose that Mr. T trades in his gas guzzler for a fuel-efficient car. After taking out a bigger car loan and paying off the old loan, his monthly payment to the bank increases to $F, but the price that he pays to drive per mile decreases to $0.10 per mile for the fuel-efficient car. Suppose that following these changes, Mr. T is just as happy as he was before he traded in the gas guzzler. Based on that information, how large is his new car payment, $F? c. Does he drive more than, less than or the same as before with the fuel-efficient car? Show your work and explain. Does he consume half as much fuel each month with the fuel-efficient car? d. Suppose that the price of gasoline doubles, so that the price per mile doubles for the gas guzzler to $0.40 per mile and for the fuel-efficient car to $0.20 per mile. Does Mr. T’s new utility-maximizing choice still only make him equally as happy as he would have been had he kept the gas guzzler and paid the higher price?

Answer 1

To find Mr. T's utility-maximizing choice, we set up budget constraints and equalize the marginal utility per dollar for miles driven and other consumption. When trading in for a fuel-efficient car, we solve for the new monthly payment $F by keeping the utility level constant. Subsequent parts involve comparing driving choices and fuel consumption after changing to a fuel-efficient car and considering changes in utility due to a potential increase in fuel prices.

Step-by-step explanation:

To determine Mr. T's utility-maximizing choice of miles driven (x) and other consumption (y), we must first define his budget constraint with the given information. Mr. T's income is $600, with a fixed payment of $200 towards the gas guzzler. The cost of driving per mile is $0.20. His budget constraint for miles driven and other consumption can be expressed as $600 - $200 - $0.20x = y, simplifying to $400 = y + $0.20x. To maximize utility, Mr. T will select an x and y such that the marginal utility per dollar spent on each is equal, according to the equation MUx/Px = MUy/Py, where Px and Py are the prices of x and y, respectively.

For part b, we know that Mr. T's utility level remains the same with the fuel-efficient car. The budget constraint changes to $600 - $F - $0.10x = y, where $F is the new monthly payment. Since Mr. T is as happy as before, we can say that the utility obtained from the new budget constraint with the fuel-efficient car is equal to the utility obtained from the initial budget constraint with the gas guzzler. Solving for $F requires equating the utility levels before and after the car trade-in.

For part c, to determine if Mr. T drives more or less, we need to examine how the change in car payment and the price per mile affects his choices. We also check if fuel consumption is halved by analyzing the change in the cost per mile from $0.20 to $0.10.

For part d, if the price of gasoline doubles the budget constraints would further change, affecting Mr. T's utility levels and decisions. We would again compare utility levels with the new prices to determine if Mr. T would have the same level of happiness or not.