Utility Maximization with Endowments

Budget constraint with endowment

$$p_1{x}_1+p_2{x}_2=\underset{\text{Endowments}}{\underbrace{p_1{e}_1+p_2{e}_2}}+I$$

⇒ Whether the change in price leads to a utility increase or decrease depends on where your original optimum is; i.e.

• Whether you are a net seller or net buyer of the good (optimum is left/right of endowment)
• Which way the price ratio changes

Leisure—Consumption Utility §

Same equation as above, but with Consumption-Positive Leisure tradeoff

$$\begin{array}{rl}\text{Consumption}& =\text{Exogenous Income}+\text{Endowment}\\ wL+pc& =I+wT+pe\end{array}$$

…where…

• Labor quantities:
  • $w$: wage (=price of leisure)
  • $L$: quantity of leisure, in unit hours
  • $T$: total time endowment, in unit hours
• Consumption quantities:
  • $p$: price of composite consumption good
  • $c$: quantity of composite consumption good
  • $e$: consumption endowment

Cobb-Douglas:

$${\text{max}}_{L,c}u=L^{\alpha }{c}^{(1-\alpha )}\text{such that}wL+pc=I+wT+pe$$

therefore we can plot the Ordinary Demand of Leisure (=Labor Supply)

• $c=\frac{\alpha (I+wT+pe)}{p}$ Ordinary Demand of Consumption
• $L=\frac{\alpha (I+wT+pe)}{w}$ Ordinary Demand of Leisure
  • $N:=T-L=T-\frac{\alpha (I+wT+pe)}{w}$ ==Ordinary Supply of Labor==

Taxation §

Percentage tax $t$ on wage (=price of leisure) ⇒ removing $t$ from the wage

$$w(1-t)L+pc=I+w(1-t)+pe$$

• Tax Revenue $\stackrel{\text{Time worked}}{\overbrace{(T-L)}}\times \stackrel{\text{Hourly tax}}{\overbrace{wt}}$
• Laffer Curve