Two-Period Firm's Problem

Setup: Inputs

To make the firm's problem intertemporal, we let capital persist across time. The firm starts period one with an endowment of \(K\) units of capital. This is determined exogenously.

The firm can choose how much capital they want to have in the second period, but they have to plan ahead and buy it in the first period. In addition, some portion of their capital today can be held onto and reused tomorrow. This is represented by the following constraint: \[K'\leq (1-\delta)K + I\] where I is the amount of investment, and \(\delta\) is the portion of capital which depreciates. If I is negative, then they sell off some of their capital.

At the end of the second period, the firm will sell off all their leftover capital, \((1-\delta)K\). The firm only exists for two periods, so they are going to liquidate their assets at the end of that time.

Labor hiring decisions are the same as in the one period setup. The firm hires \(N\) units of labor at wage \(w\) in the first period, and (\N'\) units at wage \(w'\) in the second period. The firm can freely choose \(N\) and \(N'\).

Setup: Outputs and Profit

The firms output for period one is \(Y=zF(K,N)\), and for period two is \(Y'=z'F(K',N')\). Notice that F is the same, but the technology multiplier might change over time. For the rest of this example, we will have \(F\) be a "Cobb-Douglass Production Function", which is a special case of CES production: \[F(K,L)=K^\alpha N^{1-\alpha}\] This production function has several convinient properties. Namely, it is strictly increasing, concave, and the partial derivatives are infinite at zero.

It also exhibits constant returns to scale. This is not necessarily realistic, but it makes the math easier to set up in the Competitive Equilibirum later on. Otherwise, this kind of production function is a decent approximation of real-world industry.

For simplicity, all prices are expressed in terms of units of consumption. In the first period, the firm earns \(Y\) units of revenue, and pays costs \(wN+I\). In the second, they earn \(Y'+(1-\delta)K'\) and pay cost \(w'N'\). So \[\pi=Y-wN-I\] \[\pi'=Y-w'N'+(1-\delta)K'\] The firm wants to maximize the present value of their profits, expressed as \(V\equiv\pi+{\pi'\over 1+r}\)

The optimization problem

Putting this all together, the firm takes exogenous parameters \((K,z,z',\alpha)\) for granted. The prices \((w,w',r)\) will be endogenous when we set up the competitive equilibrium, but the firm will treat them as given. The firm is a price taker, in other words.

Given the above, the firm chooses \((N,N',K',I)\) to solve \[\max_{N,N',K',I} zK^\alpha N^{1-\alpha} - wN - I + {1 \over 1+r} \left[ z'{K'}^{\alpha}{N'}^{1-\alpha} \right]\] \[s.t. K' \leq (1-\delta)K + I\] \[K',N,N'\geq 0\]

Solving

Because of the properties of our production function, we know that the cooefficients for the non-negativity constraints will be 0; the constraints will be non-binding. The lagrangian is then: \[ℒ = zK^\alpha N^{1-\alpha} - wN - I + {1 \over 1+r} \left[ z'{K'}^{\alpha}{N'}^{1-\alpha} \right] - \lambda(K'-(1-\delta)K-I)\]

To find the optimum, we set the partial derivatives equal to zero giving us the following first order conditions: \[\begin{split} 0 = {{\partial ℒ}\over{\partial N}} & = (1-\alpha)zK^\alpha N^{-\alpha} - w \\ 0 = {{\partial ℒ}\over{\partial N'}} & = {1 \over 1+r} \left[ (1-\alpha)z'{K'}^\alpha {N'}^{-\alpha} - w' \right] \\ 0 = {{\partial ℒ}\over{\partial I}} & = -1 + \lambda \\ 0 = {{\partial ℒ}\over{\partial K'}} & = {1 \over 1+r} \left[ \alpha z'{K'}^{\alpha-1} {N'}^{1-\alpha} +(1-\delta) \right] - \lambda \\ 0 = {{\partial ℒ}\over{\partial \lambda}} & = K'-(1-\delta)K-I \\ \end{split}\]

Now we want to use the equations above to find expressions which allow use to calculate the choice variables based on all of the variables which the firm takes for granted.

First note that the condition for \(I\) says that \(\lambda = 1\). Plugging this into the condition for \(K'\) yields: \[1 = {1 \over 1+r} \left[ \alpha z'{K'}^{\alpha-1} {N'}^{1-\alpha} +(1-\delta) \right] \] \[1+r = \alpha z'{K'}^{\alpha-1} {N'}^{1-\alpha} +(1-\delta) \] \[r = \alpha z'{K'}^{\alpha-1} {N'}^{1-\alpha} -\delta \] So at the optimum, \(r\) will be equal to the marginal product of period-two capital, accounting for depreciation.

Likewise, the conditions for labor: \[w = (1-\alpha)zK^\alpha N^{-\alpha}\] \[w' = (1-\alpha)z'{K'}^\alpha {N'}^{-\alpha}\] say that wages will be equal to the marginal product of labor in each period.

Rearranging the above to solve for our choice variables yields \[N = \left({(1-\alpha)zK^\alpha \over w}\right)^{1 \over \alpha} = \left({(1-\alpha)z \over w}\right)^{1 \over \alpha} K \] \[N' = \left({(1-\alpha) z' \over w'}\right)^{1\over\alpha} K' \] \[ K' = \left({ z' \alpha \over r + \delta}\right)^{1 \over 1-\alpha} N' \]

The first of these equations is simply the formula for N in terms of variables which the firm takes for granted, and so uniquely determines period-one labor demand. In particular, note that \(N\) decreases as \(w\) increases, as expected of a demand curve.

Alas, the latter two equations are linearly dependent, and so cannot be used to solve for a unique value of period-two inputs. However, we can note that as the price of a factor input increases, if we hold the quantity of the other input constant, the demand for that factor input will decrease. Ie, holding \(N'\) constant, as \(r\) increases, optimal \(K'\) will decrease.

And because \(I=K'-(1-\delta)K\), an increase in \(r\) will be associated with a decrease in investment.

Finally, although we cannot explicitly solve for period-two inputs, we can find their ratio by dividing their first-order conditions: \[ {r+\delta \over w'} = {\alpha z'{K'}^{\alpha-1} {N'}^{1-\alpha} \over (1-\alpha)z'{K'}^\alpha {N'}^{-\alpha}} = {\alpha \over (1-\alpha)}{N' \over K'} \] \[{K' \over N'} = {\alpha w' \over (r+\delta)(1-\alpha)}\]

An equivalent setup

Instead of solving the constrained optimization problem above, we could also use our intuition to note that there is no reason for the firm to invest if they are not going to use that additional period-two capital, and so the investment constraint will bind with equality. If we solve for \(I\) to get \(I=K'-(1-\delta)K\), we can simplify the problem to the following unconstrained optimization: \[\max_{N,N',K'} zK^\alpha N^{1-\alpha} - wN - (K'-(1-\delta)K) + {1 \over 1+r} \left[ z'{K'}^{\alpha}{N'}^{1-\alpha} -w'N' + (1-\delta)K' \right]\] Verify for yourself that the first order conditions are equivalent to those above.