Question

Suppose the representative consumer’s preferences are given by the utility function,

U(C, l) = aln C + (1- a) ln l

Where C is consumption and l is leisure, with a utility function that is increasing both the arguments and strictly quiescence, and twice differentiable.

Question:

The total quantity of time available to the consumer is h. The consumer earns w real wage from working in the market, receives endowment π from his/her parents, and pays the T lump-sum tax on the endowment. Setup the consumer’s optimization problem where he/she chooses C and l to maximize U(C, l). Obtain and interpret the optimization conditions and optimum consumption bundle for the consumer.

PS:

This is a constrained optimization problem.

There are two possible ways (that I am aware of); lagrangian and substitution to solve this.

Setup lagrangian, find first-order conditions and then solve for c and l.  

Answer 1