Need help with pt 3 in particular - T is fixed and labor supply is dependent on

Question

Need help with pt 3 in particular - T is fixed and labor supply is dependent on T and leisure, with leisure dependent on wage. I'm unsure if we're supposed to use partials or the slutsky equation or... thank you :) Also need help on pt 5 especially!

Problem 1 - Labor supply Two brothers, Mario (M) and Luigi (G), have an endowment of T-12 hours per day, which they can devote to leisure I or to work L as plumbers at a wage of $10 per hour. They get utility from consumption and disutility from working. The cost of a unit of consumption good is normalized to $1. Below are the two brother's utility functions: Mario Luigi 1) Solve the two brothers' UMPs by calculating Mario's and Luigi's Marshallian demands for r"(L, c) = ln(12-L) + 3 . In(c) uG(L, c) = ln(12-L) + 5 . In(c) leisure and consumption and their labor supply, which we will call L for j-M,G 2) Comment on the economic intuition behind the two brothers' preferences and their resulting labor supply and consumption choices 3) How does the hourly wage affect Luigi's and Mario's labor supply? 4) Now assume that each sibling receives a Sy (undefined amount) bequest from an aunt What happens to their labor supply? Explain the economic intuition behind this fact. 5) How does the hourly wage affect Luigi's and Mario's labor supply now? Explain the economic intuition behind this fact.

Explanation / Answer

point 3 & 5:

An hourly wage makes leisure more expensive and Luigi and Mario reduce the consumption of leisure due to substitution effect. Similarly an increase in wage induces an income effect and as leisure is a normal good, consumers will demand more. In case of the two brothers, change in utility varies inversely with change in work, i.e. their utility level increases as work hours decrease. Also when aunt gives an extra amount to them, this makes them less relucant to work. Here income effect dominates the substitution effect.

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