Question

Sally consumes two goods, X and Y. Her utility function is given by the expression = 3XY^2. The current market price for X is $10, while the market price for Y is $5. Sally's current income is $500. a. Sketch a set of two indifference curves for Sally in her consumption of X and Y. b. Write the expression for Sally's budget constraint. Graph the budget constraint and determine its slope. c. What is the Marginal Utility for goods X and Y? Determine the X, Y combination which maximizes Sally's utility, given her budget constraint. Show her optimum point on a graph. d. Calculate the impact on Sally's optimum market basket of an increase in the price of X to $15. What would happen to her utility as a result of the price increase?

Answer 1

Sally's utility function is given by 3XY^2. To sketch indifference curves, set utility at different levels and solve for Y in terms of X. The budget constraint is 10X + 5Y = 500. The slope is -2. The marginal utility for X is 3Y^2 and for Y is 6XY. The combination that maximizes utility is where the budget constraint is tangent to an indifference curve. Without more information, we cannot calculate the impact on utility from a price increase.

Step-by-step explanation:

Sally's utility function is given by 3XY^2. In order to sketch two indifference curves, we can fix the utility function at different levels and solve for Y in terms of X. For example, if we set utility at 1, we get X = (1/3)Y^-2. By varying the utility levels, we can plot different indifference curves on a graph where X is on the x-axis and Y is on the y-axis.

Sally's budget constraint can be written as: 10X + 5Y = 500. To graph the budget constraint, we can solve for Y in terms of X, which gives Y = 100 - 2X. We can then plot this line on a graph. The slope of the budget constraint is -2, which represents the opportunity cost of X in terms of Y.

The marginal utility for X is given by the derivative of the utility function with respect to X, which is equal to 3Y^2. Similarly, the marginal utility for Y is given by the derivative of the utility function with respect to Y, which is equal to 6XY. To find the combination of X and Y that maximizes Sally's utility, we need to find the point where the budget constraint is tangent to an indifference curve. This is the point where the ratio of marginal utilities is equal to the ratio of prices. However, without the specific prices and income level, we cannot determine the exact X, Y combination or draw the graph.

If the price of X increases to $15, Sally's budget constraint will change to: 15X + 5Y = 500. To find the new optimum market basket, we need to find the point where the new budget constraint is tangent to an indifference curve. This point may be different from the previous optimum, depending on the slope of the new budget constraint and the new prices. Without more information, we cannot determine the impact on Sally's utility as a result of the price increase.