Problem $1.$ Labor Supply $($optional$)$

This problem serves two key purposes. First, it famil

iarizes you with the general equilibrium framework, illustrating how households or families

function as both consumers of goods and providers of labor, which are essential inputs for pro

ducing those goods. Second, it demonstrates how consumer theory can be applied to analyze

the relationship between labor supply and wages. Empirical labor economists have found that,

based on real$-$world data on wage rates and hours worked, labor supply initially increases with

rising wages. However, beyond a certain wage threshold, labor supply tends to decrease as

wages continue to rise. This is called backward$-$bending labor supply curve. We will show

how simple consumer theory can help explain these observations.

$1.$ To build the simplest model to understand labor supply, consider two goods: leisure $($x$1,$

measured by the number of hours$)$ and a composite consumption goods $($x$2).1$ Denote

the utility function as u$($x$1,$ x$2).$ Please discuss why we focus on these two goods. Specifi$-$

cally, explain why we prefer to include leisure rather than labor supply itself in the utility

function, and why it is essential to consider an additional consumption good to explain

the backward$-$bending labor supply curve.

$2.$ To define the budget constraint, we need to consider the prices of two goods, denoted

as p$1$ and p$2.$ Explain what the price of leisure represents in reality. $($Hint: The price of

leisure should be interpreted as the opportunity cost of time spent not working.$)$ Given

that each person has $24$ hours available, the budget constraint can be written as

p$2$x$2=$ p$1(24$ x$1).$