1. (18 points) In a two-commodity world, suppose that Winnie’s preference can be characterized by the utility function, UW(X, Y) = ln X + lnY + 2018, and Tigger’s utility is given by UT(X, Y) = X^(1/3)Y^(1/3). Good X costs $2 per unit, and Y is a composite good. Both Winnie and Tigger have an income of $24.
a. (12 pts) Solve step-by-step for the optimal consumption bundles of Winnie and Tigger.
b. (6 pts) What is the exact relationship between Winnie’s and Tigger’s utility functions? How can this relationship explain what you find in part (a)?
2. (16 points) Consider the following production function: Q = F(L, K) = L^4 * K^7 .
a. (8 pts) Does this production function exhibit diminishing or increasing marginal rate of technical substitution of labor for capital? Show your work.
b. (8 pts) Find the elasticity of substitution for this production function.
3. (16 points) Below are questions about elasticities that students see in Basic Microeconomics. With deeper knowledge in microeconomics, you are now confident to show the results step-by-step. Show your work to support your answers.
a. (4 pts) Consider the linear supply curve, Q = – a + bP, where a and b are positive real numbers. What is price elasticity of supply at the point intersecting with the P-axis?
b. (4 pts) Consider the linear supply curve, Q = – a + bP, where a and b are positive real numbers. What is (the limit of) price elasticity of supply as P approaches infinity?
c. (4 pts) Consider the linear demand curve, Q = a – bP, where a and b are positive real numbers. What is price elasticity of demand at the midpoint of the curve?
d. (4 pts) Consider the log-linear demand curve, lnQ = a – b lnP, where a and b are positive real numbers. What is price elasticity of demand?