Representative Producer

The Producer in a one-period competitive equilibrium.

- Modified: Sep 27th, 2022

¶ The Firm’s Problem

Taking prices and capital $\lbrace w,K,z\rbrace$ as given, the firm chooses $\lbrace Y,N_{d}\rbrace$ to solve:

\begin{aligned} \max_{Y,N_{d}} & \left[\pi=Y-wN_{d}\right] \\ \text{s.t. } & N_{d}\geq0,\ Y\geq0 \\ & Y=zF(K,N_{d}) \end{aligned}

Note that we can simplify this problem by substituting away $Y$ to get

\begin{aligned} \max_{N_{d}} & \left[\pi=zF(K,N_{d})-wN_{d}\right] \\ \text{s.t. } & N_{d}\geq0,\ \ \ \ zF(K,N_{d})\geq0 \end{aligned}

¶ What decisions does the firm make?

The firms chooses:

• how much output to produce $Y$ and
• how much labor to hire $N_{d}$.

In general, we might also have the firm make investment decisions related to capital. But in this model (ch 4, 5), capital is exogenously given.

¶ What constraints do they operate under?

First of all, output and labor demand are non-negative:

$Y\geq0 \\ N_{d}\geq0$

Also, output is limited by the inputs. Inputs (labor,capital) are transformed into outputs using some production technology.

$\textcolor{#6c71c4}{Y} =\textcolor{#268bd2}{z} \textcolor{#dc322f}{F}(\textcolor{#859900}{K}, \textcolor{#b58900}{N_d})$

Output is equal to some total factor productivity multiplier times some function of the capital stock and the amount of labor hired.

Useful properties for this function to have:

• Constant Returns to Scale: Scaling up all inputs scales up output by the same amount. For any constant $x$, $zF(xK,xN_{d})=xzF(K,N_{d})$
• Strictly increasing in $N_{d}$ and in $K$.
$MP_{N}=zF_{N}^{\prime}(K,N)>0\\ MP_{K}=zF_{K}^{\prime}(K,N)>0$
• The marginal product of one input increases when we increase the quantity of the other input.
$\frac{\partial}{\partial N}\frac{\partial}{\partial K}zF(K,N)=\frac{\partial}{\partial N} MP_{K} > 0 \\ \frac{\partial}{\partial K}\frac{\partial}{\partial N}zF(K,N)=\frac{\partial}{\partial K}MP_{N} > 0$
• The marginal product of an input decreases as we increase the quantity of that input.
$\frac{\partial}{\partial K}\frac{\partial}{\partial K}zF(K,N)=\frac{\partial}{\partial K}MP_{K} < 0 \\ \frac{\partial}{\partial N}\frac{\partial}{\partial N}zF(K,N)=\frac{\partial}{\partial N}MP_{N} < 0$

¶ What is the firm’s goal?

Maximize Profit.

• Profit = Revenue - Costs
• Revenue in this model is simply output $Y$. (Because this is a “real” model, where the price of a unit of aggregate goods is 1).
• Costs = The money spent to hire labor. $= wN_{d}$

$\underbrace{w}_{realWage}\cdot\underbrace{N_{d}}_{laborDemand}$
• Capital is an input but not included in the cost, because with only time period, there is no investment, so capital is just exogenous.
• For simplicity, we’ll just assume that the firm owns the initial exogenous capital.
• We could also have consumers own it, and sell to the firm, but that wouldn’t change equilibrium allocations.
• Therefore profit is $\pi=Y-wN_{d}=zF(K,N_{d})-wN_{d}$

Putting it together, we get the rep firm’s constrained optimization problem.

¶ First Order Conditions

Assuming an interior solution, the firm’s profit maximization problem is characterized by

$\text{Marginal Cost of Labor}=\text{Marginal Benefit of Labor}$ $w=MP_{N}$

For Cobb Douglas ($zK^{\alpha}N^{1-\alpha}$), this is

$w=(1-\alpha)z\left(\frac{K}{N^{*}}\right)^{\alpha}$

which can be solved for $N^{*}$ to get

$N^{*}=\left(\frac{(1-\alpha)zK^{\alpha}}{w}\right)^{\frac{1}{\alpha}}$

¶ Shocks to our exogenous parameters:

How does the Firm’s decisions change in response to changes in exogenous parameters?

Exogenous parameters: $w,K,z,\alpha$

¶ Increase in real wage $w$

$N_{d}^{\ast}$ decreases and so $Y^{\ast}$ decreases. For example, with Cobb-douglass, this can be shown by:

$\frac{\partial}{\partial w}N_{d}^{\ast}=\frac{\partial}{\partial w}\left(\frac{(1-\alpha)zK^{\alpha}}{w}\right)^{\frac{1}{\alpha}} < 0$

¶ Increase in the exogenous capital stock $K$

$N_{d}^{\ast}$ increases and so $Y^{\ast}$ increases.

In Cobb-Douglas:

$\frac{\partial}{\partial K}N_{d}^{\ast}=\frac{\partial}{\partial K}\left(\frac{(1-\alpha)z}{w}\right)^{\frac{1}{\alpha}}K=\left(\frac{(1-\alpha)z}{w}\right)^{\frac{1}{\alpha}} > 0$

¶ Increase in total factor productivity $z$

In this model, without investment, this has the same effect as increase in K. $N_{d}^{\ast}$ increases and so $Y^{\ast}$ increases.

In Cobb-Douglas:

$\frac{\partial}{\partial z}N_{d}^{\ast}=\frac{\partial}{\partial z}\left(\frac{(1-\alpha)K^{\alpha}}{w}\right)^{\frac{1}{\alpha}}z^{\frac{1}{\alpha}}=\left(\frac{(1-\alpha)K^{\alpha}}{w}\right)^{\frac{1}{\alpha}}\frac{1}{\alpha}z^{\frac{1}{\alpha}-1} > 0$

¶ Increase in $\alpha$:

(This one only applies for Cobb-Douglas)

It becomes optimal to hire less labor because labor has less impact on output. $N_{d}^{\ast}$ decreases.