<p>A firm's output is given by the Cobb-Douglas production function <span class="math-container">$$Y_t=X_tK_t^{\alpha_K} L_t^{\alpha_L}$$</span>
where <span class="math-container">$\alpha_K\approx\frac{1}{3}$</span> is the capital share and <span class="math-container">$\alpha_L$</span> the labor share.</p>
<hr />
<p><strong>Question:</strong> By optimizing labor, is it possible to arrive at a profit function like <span class="math-container">$\Pi_t=Z_tK_t^\gamma$</span> which only depends on capial? I think the answer involves <span class="math-container">$\gamma=\frac{\alpha_K}{1-\alpha_L}$</span>.</p>
<hr />
<p>The firm's profit is output minus wage costs,<span class="math-container">$$\Pi_t=Y_t-wL_t$$</span>
Assume labor is frictionless. Choosing <span class="math-container">$w$</span> to maximize profits and solving the FOC gives <span class="math-container">$w^*_t=\alpha_LX_tK_t^{\alpha_K} L_t^{\alpha_L-1}=\alpha_L\frac{Y_t}{L_t}$</span>, which is the marginal product of labor. Thus, <span class="math-container">$$\Pi_t=(1-\alpha_L) Y_t=(1-\alpha_L) X_tK_t^{\alpha_K} L_t^{1-\alpha_L}$$</span>But that is still not the answer to my question.</p>

CC BY-SA 4.0
2024-01-01T12:46:21.130Z
Hi! Can you please explain what you mean by "*homogeneous profit function*"? Also, what do you mean when you write that $\gamma$ is "*typically*" close to one? In your example it is one half? Finally, why is capital free, how come your company does not pay interest?
Giskard
CC BY-SA 4.0
2024-01-01T13:37:42.267Z
Hi @Giskard I am looking into investment theory. The key choice variable is (physical) investment while labor doesn't feature in the models. The models often start with $\Pi=ZK^\gamma$ (where $\gamma$ is calibrated to values around 0.9) and go on to impose capital investment frictions and study firm dynamics. Some papers state that $\Pi=ZK^\gamma$ can actually be derived from a Cobb Douglas production function by optimizing over (frictionless) labor but I have never seen that written out.
Alex
CC BY-SA 4.0
2024-01-01T23:07:21.640Z
What is $Z_t$? and why do you not include cost of capital in the profit function?
Pekisch
<macroeconomics><production-function><profit-maximization><cobb-douglas><capital>
From Cobb-Douglas Production Function to Profit Function
Alex
# From Cobb-Douglas Production Function to Profit Function

(Asked by: username_0 on 2024-01-01)

A firm's output is given by the Cobb-Douglas production function $$Y\_t=X\_tK\_t^{\alpha\_K} L\_t^{\alpha\_L}$$
where $\alpha\_K\approx\frac{1}{3}$ is the capital share and $\alpha\_L$ the labor share.

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**Question:** By optimizing labor, is it possible to arrive at a profit function like $\Pi\_t=Z\_tK\_t^\gamma$ which only depends on capial? I think the answer involves $\gamma=\frac{\alpha\_K}{1-\alpha\_L}$.

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The firm's profit is output minus wage costs,$$\Pi\_t=Y\_t-wL\_t$$
Assume labor is frictionless. Choosing $w$ to maximize profits and solving the FOC gives $w^\*\_t=\alpha\_LX\_tK\_t^{\alpha\_K} L\_t^{\alpha\_L-1}=\alpha\_L\frac{Y\_t}{L\_t}$, which is the marginal product of labor. Thus, $$\Pi\_t=(1-\alpha\_L) Y\_t=(1-\alpha\_L) X\_tK\_t^{\alpha\_K} L\_t^{1-\alpha\_L}$$But that is still not the answer to my question.

**Tags:** macroeconomics, production-function, profit-maximization, cobb-douglas, capital
> **Comments:**
> **username_1**: Hi! Can you please explain what you mean by "*homogeneous profit function*"? Also, what do you mean when you write that $\gamma$ is "*typically*" close to one? In your example it is one half? Finally, why is capital free, how come your company does not pay interest?  
> **username_0**: Hi @Giskard I am looking into investment theory. The key choice variable is (physical) investment while labor doesn't feature in the models. The models often start with $\Pi=ZK^\gamma$ (where $\gamma$ is calibrated to values around 0.9) and go on to impose capital investment frictions and study firm dynamics. Some papers state that $\Pi=ZK^\gamma$ can actually be derived from a Cobb Douglas production function by optimizing over (frictionless) labor but I have never seen that written out.  
> **username_2**: What is $Z_t$? and why do you not include cost of capital in the profit function?  

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## 0 answers

