# Two Period Equilibrium

Now that we’ve discussed the agents in this two-period economy (the competitive part of our competitive equilibrium), there are only a few missing pieces to our definition.

Those are the government’s budget constraint,

\[G + \frac{G'}{1+r} = T + \frac{T'}{1+r}\]and the market clearing conditions:

\[\begin{aligned} N_d &= N_s = h-l\\ N_d' &= N_s' = h'-l'\\ zF(K,N_d) &= c + G + I\\ z'F(K',N_d') &= c'+ G' - (1-\delta) K'\\ \end{aligned}\]Once we have all these pieces, we can put them together into a competitive equilibrium model.

## A Summary of the model from Chapter 11

## Definition of the Closed Economy 2-Period Competitive Equilibrium

Given the exogenous parameters $\lbrace K,h,h’,z,z’,G,G’\rbrace$, a competitive equilibrium is any set of endogenous prices $\lbrace w,w’,r\rbrace$ and allocations $\lbrace c,c’,l,l’,N_{s}=h-l,N_{s}^{‘}=h’-l’,N_{d},N_{d}^{\prime},I,K^{\prime},T,T^{\prime}\rbrace$ that satisfy the following conditions:

- Representative Consumer, taking prices as given, solves:

- Representative Firm, taking prices as given, solves:

- Markets Clear:

- Government Budget is balanced:

### Solving the Consumer’s Problem

Characterizing Equations are:

\[MRS_{lc} = w\] \[MRS_{l'c'} = w'\] \[MRS_{cc'} = (1+r)\] \[c+\frac{c^{\prime}}{(1+r)}=\left[w(h-l)+\pi-T\right]+\frac{\left[w^{\prime}(h^{\prime}-l^{\prime})+\pi^{\prime}-T^{\prime}\right]}{1+r}\]### Solving the Firms’s Problem

Characterizing Equations are:

\[MP_{N}=\frac{\partial}{\partial N_{d}}zF(K,N_{d})=w\] \[MP_{N^{\prime}}=\frac{\partial}{\partial N_{d}^{\prime}}z'F(K',N_{d}^{\prime})=w'\] \[r+\delta=\frac{\partial}{\partial K'}z'F(K',N_{d}^{\prime})=MP_{K^{\prime}}\]## Determining Equilibrium

For the Labor market (N,w):

- The labor demand curve is determined by $MP_N = w$
- The labor supply curve is a bit more complicated.
- For simplicity, we assume leisure goes down when w goes up (substitution effect is stronger), which makes the labor demand curve slope upward.
- Note that the position of the labor supply curve is determined by the interest rate $r$. An increase in $r$ means saving is more attractive, borrowing less attractive, so the labor supply curve shifts right.

- Given any particular $r$, we get a particular labor supply curve, which gives us a particular equilibrium $w$ such that $N_d = N_s$.

For the output market (Y,r):

- The output supply curve comes from the labor market outcome.
- Given $r$, some equilibrium is determined, which can be plugged into the production function to get the quantity of output supplied.

- The output demand curve comes from $Y_d(r) = c(r) + I(r) + G$. The book goes into more detail about this, but what I want students to remember is just:
- We know from the firm’s optimal investment rule that $I$ is a decreasing function of $r$.
- A higher interest rate means the consumer will tend to save more and consume less today. So $c$ is also a decreasing function of $r$.
- Thus the quantity of output demanded decreases when the interest rate goes up. And the output demand curve is downwards sloping.

Then given exogenous parameters, there is one specific equilibrium $r$ such that the output from the labor market at this interest rate is equal to $c+I+G$

## Shocks to the 2 Period Model

- Increase in $z$:
- $N_d$ curve shifts right.
- At any $r$, we have higher $Y$, so $Y_s$ curve shifts right.
- Equilibrium $r$ increases causing the $N_s$ curve to shift to the left.
- Results: $w\uparrow$, $r\downarrow$, $Y\uparrow$, $N\uparrow?$,

- Increase in $z’$:
- Firms want to invest more.
- $Y_d$ curve shifts right because of the increase in investment demand.
- Equilibrium $r$ and equilibrium $Y$ rise.
- As $r$ goes up, $N_s$ curve shifts right, increasing output and moving along the $Y_s$ curve until things come back into equilibrium.
- Results: $w\downarrow$, $r\uparrow$, $Y\uparrow$, $N\uparrow$,

- Increase in $K$:
- Like the shock to $z$, shifts $N_d$ and $Y_s$ curves shift to the right.
- $Y_d$ curve shifts left because the firm needs to invest less to meet their goals for $K’$ because they already start with lots of capital.
- Equilibrium $r$ decreases, but the change in $Y$ is technically ambiguous.
- The $N_s$ curve shifts left because of the change in $r$.
- Results: $w\uparrow$, $r\downarrow$, $Y\uparrow?$, $N\uparrow?$,

- Increase in $G$:
- Two direct effects:
- Increase in $G$ shifts $Y_d$ curve to the right.
- Increase in $G$ requires higher taxes, which makes the consumer work more, shifting $N_s$ curve to the right.

- This (non-interest-rate related) shift in the $N_s$ curve shifts the $Y_s$ curve to shift right as well.
- The change in $r$ causes a second shift in the $N_s$ curve.
- Like the shock to $z$, shifts $N_d$ and $Y_s$ curves shift to the right.
- $Y_d$ curve shifts left because the firm needs to invest less to meet their goals for $K’$ because they already start with lots of capital.
- Equilibrium $r$ decreases, but the change in $Y$ is technically ambiguous.
- Results: $w\downarrow$, $r\uparrow?$, $Y\uparrow$, $N\uparrow$,

- Two direct effects: