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Robert Winslow

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

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

A Graphical Example

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.