Two Period Agents with Production
A firm which makes investment decisions, and a consumer with both savings and leisure.
Consumers in a Two Period Economy with Production
\[\max_{c,c',l,l'}u(c,l)+\beta u(c',l')\] \[\begin{aligned} \text{s.t. }\ \ \ \ &c\geq0,\ \ c'\geq0,\ \ \ \ h\geq l \geq0, \ \ h\geq l'\geq0 \\ & c+\frac{c'}{1+r}\leq w(hl)  T +\frac{w^{\prime}(h^{\prime}l^{\prime})  T'}{1+r} \end{aligned}\]Characterizing equations
 Intertemporal Euler condition \(MRS_{cc'}=(1+r)\)
 Intratemporal Euler conditions \(MRS_{lc}=w \\ MRS_{l'c'}=w\)
 Budget \(c+\frac{c'}{1+r}=w(hl)T+\frac{w^{\prime}(h^{\prime}l^{\prime})  T'}{1+r}\)
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:
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 presentvalue 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 firstorderconditions 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 $K’$.