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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 $$K=\frac{\alpha}{1-\alpha}\frac{W}{R}L \\ \implies Y=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)L\\ \max_{L}\left[A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)-\frac{W}{1-\alpha}\right]L$$$$K=\frac{\alpha}{1-\alpha}\frac{W}{R}L \\ \implies Y=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)^{\alpha}L\\ \max_{L}\left[A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)^{\alpha}-\frac{W}{1-\alpha}\right]L$$ 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?

Does global maximum of CRS Cobb-Douglas 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 $$K=\frac{\alpha}{1-\alpha}\frac{W}{R}L \\ \implies Y=A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)L\\ \max_{L}\left[A\left(\frac{\alpha}{1-\alpha}\frac{W}{R}\right)-\frac{W}{1-\alpha}\right]L$$ 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?