2. Consider the Cobb-Douglas utility function u(x, y) = x2y2. Let the budget constraint be p.,x + p,y = I, where p, p, are the prices and I denotes the income.

(a) Write the Lagrangian for this utility maximization problem. (b) Solve the first-order conditions to find the demand functions for both good x and good y. [Hint: Your results should only depend on the parameters pr, Py, 1.]

(c) In the optimal consumption bundle, how much money is spend on x? How much on y? Express your answer in proportion of the total income

I.

(d) OPTIONAL: Check the second order conditions.

Comparative statics:

(e) Assume the price p, increases. How does the optimal consumption bun-

dle change?

(f) Assume the price of good x doubles. How does the optimal consump-

tion bundle change?

(g) Are the goods complements/substitutes? (g) (h) Assume the income doubles. What is the new optimal consumption bundle? How does your answer in 2c change? i.e., How does the spending proportion change?

(i) Solve now this maximization problem using the method of substitution. Compare your result with those derived using the method of Lagrange.

(j) Now, consider the utility function

u(x, y) = min{ax, by},

where a, b > 0. We can only employ the method of substitution. Find the optimal solution.

2. Consider the Cobb-Douglas utility function u(s, y] = 321:2. Let the budget constraint be pg + pug = I. where papa, are the prices and I denotes the income. {a} Write the Lagrangian for this utility maximization problem. {in} Solve the rst-order conditions to nd the demand hmctions for both good a: and good y. [Hint Your results should only depend on the pa- rameters :3... pg. L] {c} In the optimal consumption bundle, how much money is spend on :c? How much on 3;? Express your answer in proportion of the total income I. {d} OPTIONAL: Check the second order conditions. Comparative statics: (e) Assume the price p1, increases. How does the optimal consumption bun- dle change? {f} Assume the price of good .1: doubles. How does the optimal consump- tion bundle change? {g} Are the goods complements! substitutes? (h) Assume the income doubles. What is the new optimal consumption bundle? How does your answer in 2c change? i.e., How does the spend- ing proportion change? {i} Solve now this maximization problem using the method of substitution. Compare your result with those derived using the method of Lagrange. (j) Now, consider the utility function \"(3314) = min {Mr by} . where o, b s- 0. Web can only employ the method of substitution. Find the optimal solution