Intertemporal Consumption-Leisure Optimization (Full General Equilibrium)

Household Intertemporal §

$$\begin{array}{rlrl}\underset{C_t,{\ell }_t,C_{t+1},{\ell }_{t+1}}{\max }& u(\cdot )& & \\ \text{ s.t. }& C_t+S_t^P& & =w_t(h-{\ell }_t)+{\pi }_t-T_t\\ & C_{t+1}& & =w_{t+1}(h-{\ell }_{t+1})+{\pi }_{t+1}-T_{t+1}+S_t^P(1+r)\end{array}$$

Equating the savings $S_t^P$ for the budget constraint we have:

$$\underset{\text{ NPV consumption }}{\underbrace{C_t+\frac{C_{t+1}}{1+r}}}=\stackrel{\text{ Lifetime Wealth W }}{\overbrace{\underset{\text{ NPV labor income }}{\underbrace{w_t(h-{\ell }_t)+\frac{w_{t+1}(h-{\ell }_{t+1})}{1+r}}}+\underset{\text{ NPV capital income }}{\underbrace{{\pi }_t+\frac{{\pi }_{t+1}}{1+r}}}-\underset{\text{ NPV taxes }}{\underbrace{\left(T_t+\frac{T_{t+1}}{1+r}\right)}}}}$$

Using Lagrangian Method we have optimality conditions:

1. Intratemporal Constraint: $\frac{\partial u/\partial {\ell }_t}{\partial u/\partial C_t}={\text{MRS}}_{C_t,{\ell }_t}=w_t$
2. Intertemporal Leisure Constraint: $\frac{\partial u/\partial {\ell }_{t+1}}{\partial u/\partial {\ell }_t}\cdot ={\text{MRS}}_{{\ell }_t,{\ell }_{t+1}}=\frac{w_{t+1}}{w_t(1+r)}$
3. Intertemporal Consumption Constraint: $\frac{\partial u/\partial C_t}{\partial u/\partial C_{t+1}}=:{\text{MRS}}_{C_t,C_{t+1}}=1+r$ We observe that assuming a Cobb-Douglas Utility, labor supply and consumption depends on $r$ as well as $w$. Consider:

\tau=t

$N_t^s$	$w\uparrow$ ($C_t$ vs. ${\ell }_t$)	$r\uparrow$ (${\ell }_t$ vs ${\ell }_{t+1}$)
S.E.	$\oplus$ ∵leisure expensive	$\oplus$ ∵ work today
I.E.	$\ominus$ ∵lifetime wealth up	$\ominus$ ∵ easy money
Total	$\oplus$	$\oplus$

$C_t^d$	$w\uparrow$ ($C_t$ vs. ${\ell }_t$)	$r\uparrow$ (${\ell }_t$ vs ${\ell }_{t+1}$)
S.E.	$\oplus$, leisure expensive	$\ominus$ ∵ consume tomorrow
I.E.	$\oplus$, lifetime wealth up	$\oplus$ ∵ consume more total
Total	$\oplus$	$\ominus$

Thus we have (1) Labor supply and (2) Goods demand

Firm Intertemporal §

$$\begin{array}{rl}\underset{N_t^d,N_{t+1}^d,I}{\max }& V={\pi }_t+\frac{{\pi }_{t+1}}{1+r}\\ \text{s.t. }& K_{t+1}=(1-d)K_t+I\\ \text{where }& {\pi }_t:=zF(K_t,N_t)-w_t{N}_t-\underset{\text{ investment }}{\underbrace{I}}\\ & {\pi }_{t+1}:=zF(K_{t+1},N_{t+1})-w_{t+1}{N}_{t+1}+\underset{\text{ sell leftover capital }}{\underbrace{(1-d)K_{t+1}}}\end{array}$$

$V$ is the NPV of the firm. Optimal labor supply. $N^s$ vs Wage $w$. Recall from 2. Household Optimization that the optimal point is ${\text{MP}}_N=\frac{\partial Y}{\partial N}=w_t$. Assuming the Cobb-Douglas Utility, labor supply is proportional to wage. $\oplus$ Goods Supply $Y^s$vs Interest rate $r$.

1. labor demand $N^d$ being proportional to interest rate $r$ as seen above
2. Given the optimal labor supply, observe the equilibirum labor quantity $N^{\ast }$ is proportional to interest rate $r$ (See graph)
3. The goods supply $Y^s$ is proportional to quantity of labor supplied in Cobb-Douglas Utility.
4. Thus goods supply is proportaionl to interest rate. $\oplus$

Investment Decision §

We haven’t considered the firm decision to invest, $I$. First observe that the return from the firm considers the marginal cost of investment…

$$\text{MC}(I)=\frac{\partial I}{\partial I}=1$$

…against the marginal benefit (PV) of investment using the capital law of motion constraint…

$$\text{MB}(I):=\frac{\partial V}{\partial I}=\frac{{\text{MP}}_{K_{t+1}}+1-d}{\underset{\text{ PV }}{\underbrace{1+r}}}$$

The firm’s optimal investment decision is when $\text{MB}(I)=\text{MC}(I)$:

$$\begin{array}{rl}\frac{{\text{MP}}_{K_{t+1}}+1-d}{1+r}& =1\\ {\text{MP}}_{K_{t+1}}-d& =r\end{array}$$

Intuition. This has a nice interpretation. The firm will invest exactly as much capital such that the (marginal product of capital) less (the depreciation) will beat the risk-free rate $r$. We also see, more importantly, in the Cobb-Douglas production function that the optimal investment is inversely proportional to $r$. $\ominus$

General Intertemporal Equilibirum §

Goods demanded is a combination of consumption and investment. Thus we have: $Y^d=\stackrel{\ominus r}{C^d}+\stackrel{\ominus r}{I}$ where both $C^d$ and $I$ are inversely proportional to $r$ making $Y_d$ also so. For time $\tau =t$, we have the following general equilibrium in both the labor and goods markets. We list

• Labor market
  • $N^s(w^{\oplus },r^{\oplus },W^{\ominus })$
  • $N^d(w^{\ominus },z^{\oplus })$
• Goods market
  • $Y^s=z^{\oplus }F(K,\stackrel{\oplus }{N^{\star }}(w,r^{\oplus },W))$
  • $Y^d=C^d(r^{\ominus },\dots )+I(r^{\ominus },\dots )+G$

Exogenous Changes §