#Q4. (36 points + 4 bonus) Consumer Choice for Cobb-Douglas Utility
Consider two goods x and y. Suppose the utility function is $u(x, y) = x^{0.2}y^{0.8}$ which gives
you the marginal rate of substitution formula in this specific case $MRS = \frac{y}{4x}$.
Suppose the price of both goods are $p_x = \$6$ and $p_y = \$12$ per unit. The budget is $I = \$90$.
Parts A and B: Solve the consumer choice problem by setting up two equations.
• A. (30 points) How many units of x and y will the consumer buy? i.e., compute the optimal $x^*$ and $y^*$.
• B. (6 points) Based on your answer in Part A, compute how much the consumer will spend on good x and good y, i.e., compute $p_x x^*$ and $p_y y^*$.
Parts C and D (Bonus): For the following bonus questions, you can use the short cut $x^* = 0.2I/p_x$ and $y^* = 0.8I/p_y$ to answer.
• C. (Optional, 2 bonus points) Re-do Part A if the budget rises from $90 to $I = $180$. Based on your answer, are goods x and y normal goods or inferior goods? Why?
• D. (Optional, 2 bonus points) Re-do Part A if the price of good x rises from $6 to $12. How many units of x will the consumer buy? Based on your answer, does good x follow the law of demand? Is this change in consumption a movement along the demand curve or a shift of demand curve?
