# Does global maximum of CRS Cobb-Douglas profit exist

In most macroeconomic papers it is taken as given that the aggregate prodution function is $$Y=AK^{\alpha}L^{1-\alpha}$$, and that the optimality conditions for inputs determine input demands: $$\max_{K,L} AK^{\alpha}L^{1-\alpha}-WL-RK \\ \alpha A K^{\alpha-1}L^{1-\alpha}=R \\ (1-\alpha) A K^{\alpha}L^{-\alpha}=W$$ However, it looks like the profit function does not have a global maximum given input prices, hence it is impossible to determine the optimal choice of inputs.

In fact, even trying to reduce the dimensionality of the problem by using the tangency condition one gets $$K=\frac{\alpha}{1-\alpha}\frac{W}{R}L \\ \implies Y=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)^{\alpha}L\\ \max_{L}\left[A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)^{\alpha}-\frac{W}{1-\alpha}\right]L$$ which does not have a finite solution if the term in square brackets is positive.

What am I getting wrong? Why these conditions are widely used in macro even though they don't look correct from a mathematical point of view?

• Where are the power $\alpha$ and $1-\alpha$ in the expression of $Y=...$? Aug 30 at 19:20
• Sorry for the typo, I put them.
– Jsck
Aug 30 at 19:25
• The term $L^{1-\alpha}$ is missing in the last eq. Aug 30 at 19:29
• No it's not: $AK^{\alpha}L^{1-\alpha}=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}L\right)^{\alpha}L^{1-\alpha}=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)^{\alpha}L^{\alpha+1-\alpha}=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)^{\alpha}L$
– Jsck
Aug 30 at 19:34

## 1 Answer

It is generally true that a profit-maximizing firm with a constant-returns to scale technology can produce a positive output only if the profit is zero. Output prices are pinned down by the zero-profit condition.

It follows that the profit-maximizing output level is completely indeterminate at equilibrium prices. However, equilibrium output is not indeterminate. Supply has to equal demand; such a firm will just produce enough to satisfy the market demand.

• Do you not also need that the firm is a taker of output prices?
– BrsG
Aug 31 at 10:31
• My question was related more to the input choice. What is the point of taking FOC wrt inputs if there is not solution to that problem? I would understand this if the firm takes $Y$ as given at the beginning and optimizes constrained on the production function. Then we would have profits as function of $Y$, and then one should pin down $Y$ and input prices by equating their supplies and demands. Is this correct? I would still not get the point of taking FOC anyway, and it happens everywhere in macro.
– Jsck
Aug 31 at 14:37
• @Jsck But then you would solve the usual problem of minimizing $WL+RK$ subject to $AK^{\alpha}L^{1-\alpha}=y$, which is well defined. Aug 31 at 22:26
• @BrsG That is part of the definition of a competitive equilibrium. If the firm is not a price taker things are very different. Aug 31 at 22:27
• @MichaelGreinecker I know, the thing that bugs me about the uncostrained problem is that it does not have a solution, even keeping $Y$ indeterminate. What is the point of taking FOC if there is no solution even taking prices as given?
– Jsck
Sep 1 at 15:50