# From Cobb-Douglas Production Function to Profit Function

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.

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 $$\gamma=\frac{\alpha_K}{1-\alpha_L}$$.

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.

• 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? Jan 1 at 12:46
• 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
Jan 1 at 13:37
• What is $Z_t$? and why do you not include cost of capital in the profit function? Jan 1 at 23:07