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Robert Winslow

Two Period Equilibrium

- Modified: Dec 30th, 2022

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:
\[\max_{c,c',l,'l} \left[u(c,l)+\beta u(c',l')\right]\] \[\begin{aligned} \text{s.t. }\ \ \ \ & c\geq0, \ \ \ \ c'\geq0, \ \ \ \ 0\leq l \leq h, \ \ \ \ 0\leq l' \leq h' \\ & c+\frac{c'}{1+r}\leq\left[w(h-l)+\pi-T\right]+\frac{w'(h'-l')+\pi'-T'}{1+r}\\ \end{aligned}\]
  • Representative Firm, taking prices as given, solves:
\[\max_{N_s, N_s', I, K'} \left[\pi + \frac{\pi'}{1+r}\right]\] \[\begin{aligned} \text{s.t. }\ \ \ \ & N_d\geq0, \ \ \ \ N_d'\geq0, \ \ \ \ K'\geq 0\\ & K' = (1-\delta)K + I\\ & \pi = zF(K,N_d) - wN_d - I\\ & \pi' = z'F(K',N_d') - w'N_d' + (1-\delta)K' \end{aligned}\]
  • Markets Clear:
\[\begin{aligned} N_d &= N_s = h-l\\ N_d' &= N_s' = h'-l'\\ zF(K,N_d) &= c + G + I\\ z'F(K',N_d') &= c'+ G' - (1-\delta) K'\\ \end{aligned}\]
  • Government Budget is balanced:
\[G+\frac{G^{\prime}}{1+r}=T+\frac{T^{\prime}}{1+r}\]

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: $w\downarrow$, $r\uparrow?$, $Y\uparrow$, $N\uparrow$,