Question

Suppose John's utility function is 4XY, where X is consumption of beer and Y is consumption of pizza. For this utility function, the marginal utility of X is given by MUx= 4Y; the marginal utility of Y is given by MUY=4x. a. Suppose Y=3. Calculate John's utility for X=2, 3, 10, and 11. For a given level of Y, does good X display diminishing marginal utility? b. Suppose X = 3. Calculate John's utility for Y=2, 3, 10, and 11. For a given level of X, does good Y display diminishing marginal utility? c. Find three different bundles containing X and Y that give John 48 utils of satisfaction. Plot the three bundles and connect them with an indiffer- ence curve. What happens to the marginal rate of substitution between X and Y as consumption of X increases? d. Does the principle of diminishing MRS depend on the diminishing marginal utility of X and Y?

Answer 1

a. The utility for different values of X when Y=3 can be calculated by substituting the values into the utility function 4XY. b. The utility for different values of Y when X=3 can be calculated by substituting the values into the utility function 4XY. c. Three different bundles containing X and Y that give John 48 utils of satisfaction can be found by setting the utility function 4XY equal to 48 and solving for X in terms of Y.

Step-by-step explanation:

a. To calculate John's utility for different values of X when Y=3, we can substitute the values into the utility function 4XY. For X=2, we have 4(2)(3) = 24 utils. For X=3, we have 4(3)(3) = 36 utils. For X=10, we have 4(10)(3) = 120 utils. For X=11, we have 4(11)(3) = 132 utils. The marginal utility of X is 4Y, and since Y=3, the marginal utility of X is constant at 12.

b. To calculate John's utility for different values of Y when X=3, we can substitute the values into the utility function 4XY. For Y=2, we have 4(3)(2) = 24 utils. For Y=3, we have 4(3)(3) = 36 utils. For Y=10, we have 4(3)(10) = 120 utils. For Y=11, we have 4(3)(11) = 132 utils. The marginal utility of Y is 4X, and since X=3, the marginal utility of Y is constant at 12.

c. To find three different bundles that give John 48 utils of satisfaction, we can set the utility function 4XY equal to 48. Solving for X in terms of Y, we get X = 12/Y. Three possible bundles that satisfy the equation are (X,Y) = (12/4, 4), (12/6, 6), and (12/8, 8). Plotting these bundles on an indifference curve, we can see that as consumption of X increases, the marginal rate of substitution between X and Y gets smaller, indicating diminishing marginal rate of substitution.

d. The principle of diminishing marginal rate of substitution is not dependent on the diminishing marginal utility of X and Y. It is a separate concept that relates to the trade-off between two goods and the willingness of a consumer to substitute one good for another while maintaining the same level of satisfaction.