San Jose State University Basic Principles and Extensions Paper

1.Tom (T) and Jerry (J) have identical incomes I. Their preferences can be described by the utility functions U_T = min{x₁, x₂}, and U_J = x₁x₂, respectively.

(i)Do Tom and Jerry consume an identical amount of good 1 (x₁), or different amounts? Explain!

(ii)Suppose both of their incomes double. By how much does consumption of good 1 (x₁) change for Tom, and by how much for

Jerry?

(iii)Suppose good 1 (x₁) becomes more expensive. In which direction does the consumption of good 2 (x₂) change for Tom, and in which direction for Jerry?

2.Mr. Odde Ball enjoys commodities x and y according to the utility function U(x, y) = (x²+ y²)2¹

(i)Maximize Mr. Ball’s utility if p_x = $3, p_y = $4, and he has $50 to spend.

(ii)Graph Mr. Ball’s indifference curve and its point of tangency with his budget constraint. What does the graph say about Mr. Ball’s behavior? What is the optimal bundle that maximizes Mr.

Ball’s utility?

3. Suppose that the only items you consume are bread and wine. If the price of bread were to increase tomorrow, and if simultaneously your income were to increase by just enough so that you were equally as happy tomorrow as today, what would happen to the level of your consumption of bread? Illustrate your answer with indifference curves.

4.The CES utility function is given by

x? + y? U(x, y) = ?

(i)Show that the first-order conditions for a constrained utility maximum with this function require individuals to choose goods in the proportion

(ii)Show that the result in part (i) implies that individuals will allocate their funds equally between x and y for the case ? = 0.

(iii)How does the ratio ^p_p^x_yy^x depend on the value of ?? Explain your results intuitively.

(iv)Derive the indirect utility and expenditure functions for this case and check your results by describing the homogeneity properties of the functions you calculated.

5.Suppose the utility function for goods x and y is given by

U(x, y) = xy + y

(i)Calculate the uncompensated (Marshallian) demand functions for x and y, and describe how the demand curves for x and y are shifted by changes in I or the price of the other good.

(ii)Calculate the expenditure function for x and y.

(iii)Use the expenditure function calculated in part (b) to compute the compensated demand functions for goods x and y. Describe how the compensated demand curves for x and y are shifted by changes in income or by changes in the price of the other good.