Question: Suppose Michael's preferences for goods 1 and 2 are given by: \(U=\frac{1}{2}\ln {{x}_{1}}+\frac{1}{2}\ln {{x}_{2}}\). Michael works 40 hours a week, earns a wage of $10 per hour and knows that the prices of goods 1 and 2 are, respectively, 10 and 5.

a. Obtain the marginal utilities for goods 1 and 2. Does the marginal utility of good 1 conform to the law of diminishing marginal utility? Explain. Obtain the MRS of x₁ for x₂

b Write Michael's budget constraint as a function of xi and represent it in a diagram What is the maximum quantity of good 2 Michael could purchase if he spent his complete income on good 2. What is the slope of Michael's budget line?

c. Write the Lagrangean function. Obtain the first order conditions, and derive the Marshallian demand functions.

d. What is the most preferred market basket? What is the highest level of utility Michael can reach given his incoMeand the prices of both goods? Represent Michael's indifference curves and the equilibrium point on your diagram in part b. (hint.: indifference curves have the regular shape) (3 points)

e. Obtain the cross-price elasticity for good 1. Based on this result, what. type of goods are 1 and 2?

f. Obtain the income elasticity for good 2. Based on this result, what type of good is 2?