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Two Period Agents with Production

A firm which makes investment decisions, and a consumer with both savings and leisure.

- Modified: Jan 2nd, 2022

Consumers in a Two Period Economy with Production

\[\max_{c,c',l,l'}u(c,l)+\beta u(c',l')\] \[\begin{aligned} \text{s.t. }\ \ \ \ &c\geq0,\ \ c'\geq0,\ \ \ \ h\geq l \geq0, \ \ h\geq l'\geq0 \\ & c+\frac{c'}{1+r}\leq w(h-l) - T +\frac{w^{\prime}(h^{\prime}-l^{\prime}) - T'}{1+r} \end{aligned}\]

Characterizing equations

  • Intertemporal Euler condition \(MRS_{cc'}=(1+r)\)
  • Intratemporal Euler conditions \(MRS_{lc}=w \\ MRS_{l'c'}=w\)
  • Budget \(c+\frac{c'}{1+r}=w(h-l)-T+\frac{w^{\prime}(h^{\prime}-l^{\prime}) - T'}{1+r}\)

Quick Note about utility across time:

2 period version is

\[U(c,c',l,l')=u(c,l)+\beta u(c',l')\]

Infinite period version is typically written

\[U(c_{0},c_{1},...,l_{0},l_{1},...)=\sum_{t=0}^{\infty}\beta^{t}u(c_{t},l_{t})\]

Note that this is “exponential” time preferences, experimentally, it seems people have “hyperbolic” time preferences.

(You don’t need to worry about this for this class. We’re sticking to 2 time periods.)

The Two Period Firm

Refresher: Firms in the one period economy

  • Firms own exogenous capital $K$ at the start of the only period.
  • The firm’s profit maximization problem is:
\[\max_{N_{d}}\left[zF(K,N_{d})-wN_{d}\right]\]

subject to $N_{d}\geq0$

Firms in an intertemporal economy

  • Firms own exogenous capital $K$ at the start of the first period.
  • Second period capital is determined by $K^{\prime}=K\cdot(1-\delta)+I$, where $I$ is the firm’s investment in the first period.
  • The firm also chooses the amount of labor to hire in each period $N_d, N_d’$
  • The firm’s goal is to maximize present-value profits $\pi + \frac{\pi’}{1+r}$
  • In the first period, profits are output minus the cost of labor and investment.
  • In the second period, the firm must still hire workers, but is no need to invest because there is no third period.
  • Any capital left over after period two, $(1-\delta)K’$ will be sold as units of output.

The firm’s problem is thus:

\[\max_{N_{d},N_{d}^{\prime},I,K^{\prime}}\pi+\frac{\pi^{\prime}}{1+r}\] \[\begin{aligned} \text{s.t. }\ \ \ \ &N_{d}\geq0,\ \ N_{d}^{\prime}\geq0,\ \ K^{\prime}\geq0 \\ &\pi=zF(K,N_{d})-wN_{d}-I \\ &\pi^{\prime}=z^{\prime}F(K^{\prime},N_{d}^{\prime})-w^{\prime}N_{d}^{\prime}+K^{\prime}\cdot(1-\delta) \\ &K^{\prime}=(1-\delta)K+I \end{aligned}\]

Solve for $I$ and plug into profit equations:

\[I = K^{\prime}-(1-\delta)K\] \[\pi = zF(K,N_{d})-wN_{d}-K^{\prime}+(1-\delta)K\]

If we set up the firm’s problem with these substitutions:

\[\max_{N_{d},N_{d}^{\prime},I,K^{\prime}}\pi+\frac{\pi^{\prime}}{1+r}\] \[\begin{aligned} \text{s.t. }\ \ \ \ &N_{d}\geq0,\ \ N_{d}^{\prime}\geq0,\ \ K^{\prime}\geq0 \\ &\pi=zF(K,N_{d})-wN_{d}-K^{\prime}+(1-\delta)K \\ &\pi^{\prime}=z^{\prime}F(K^{\prime},N_{d}^{\prime})-w^{\prime}N_{d}^{\prime}+K^{\prime}\cdot(1-\delta) \\ \end{aligned}\]

On even more compact on one line:

\[\max_{N_{d},N_{d}^{\prime},K^{\prime}}zF(K,N_{d})-wN_{d}-K^{\prime}+(1-\delta)K+\frac{z^{\prime}F(K^{\prime},N_{d}^{\prime})-w^{\prime}N_{d}^{\prime}+K^{\prime}\cdot(1-\delta)}{1+r}\] \[\text{s.t. }\ \ \ \ N_{d}\geq0,\ \ N_{d}^{\prime}\geq0,\ \ K^{\prime}\geq0\]

Assuming an interior solution, then the first-order-conditions are:

\[0=\frac{\partial}{\partial N_{d}}\mathcal{L}=MP_{N}-w\\ 0=\frac{\partial}{\partial N_{d}^{\prime}}\mathcal{L}=\frac{MP_{N^{\prime}}-w^{\prime}}{1+r}\\ 0=\frac{\partial}{\partial K^{\prime}}\mathcal{L}=-1+\frac{MP_{K^{\prime}}+1-\delta}{1+r}\]

Simplify and rearrange to get the characterizing equations for this firm:

  • First period optimal hiring rule:

    \[MP_{N}=w\]
  • Second period optimal hiring rule:

    \[MP_{N^{\prime}}=w'\]
  • Optimal Investment rule:

    \[r+\delta=MP_{K^{\prime}}\]

How does the firm respond to changes in exogenous variables?

  • If $w$ increases, the firm hires a smaller amount of labor in the first period, and so output decreases as well.
  • If $z$ increases, then $MP_N$ increases for any given quantity of labor. And so for any given $w$, the firm will want to hire more labor.
  • If $K$ increases, then $MP_N$ increases for any given quantity of labor. And so for any given $w$, the firm will want to hire more labor. But also, the firm will want a lower amount of investment because they need less investment to reach any target amount of $K’$.