Reference no: EM133202854

Each week, Mara works l hours per week to consume x1 and x2 quantities of two goods. She has L hours of time at her disposal per week. She choses how many hours she works and the quantities of these two goods she consumes to maximise her weekly utility function given by U(x1, x2, l) = αln(x1) + βln(x2) + (1 - α - β)ln(L - l), which is defined for 0 ≤ l < L and for x1, x2 > 0. Here α and β are positive parameters satisfying α + β < 1. She faces the budget constraint p1x1 + p2x2 = wl, where w is the wage per hour, p1 is the unit price of good 1, and p2 is the unit price of good 2. 

(b) Assume that α is equal to 0.25 and β is equal to 0.5 and that Mara has 16 hours at her disposal per day. Mara can buy either one unit of x1 or two units of x2 with the income she generates by working for 4 hours. Find Mara's demands of x ∗ 1 and x ∗ 2 , and her labor supply l ∗ that maximises her weekly utility.

(d) Imagine that the unit-price of good 1 increases by 25% due to a negative supply shock, all else equal. Discuss how and why this price increase affects the optimum demands of x ∗ 1 and x ∗ 2 and labor market supply l ∗.

(e) Assume now that β is equal to 0.25 instead of 0.5, all else equal. How does this change affect the optimum demands of x ∗ 1 and x ∗ 2 and labor market supply l ∗ . Please comment on what β parameter might signify and interpret how and why a shift in β leads to the changes you observe in x ∗ 1 , x ∗ 2 , and l *.