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I have been stuck on this for a few hours (actually a few days). I can't find much useful information on the web that has made it any clearer.
Suppose that a firm’s production function of output Q is a function of two inputs, labour (L) and capital (K) and can be written Q = 25LK.
Letting the wage rate for labour be w and the price of capital be r, what is the equation for the firm’s demand for labour?
I understand that the answer is L=√(rQ/25w) but I don't know how to get there.
You could set up the profit function as suggested by @Kitsune , but here I take the expenditure minimization approach (which is mathematically equivalent).
Set up the minimization: $$\min_{L,K} wL + rK \\ s.t. \quad 25LK=Q$$
Using the Lagrange method: $$\mathcal{L}=(-wL - rK) + \lambda (25LK-Q)$$ Giving us the First-Order Conditions: $$(1)\quad \frac{\partial\mathcal{L}}{\partial L}=-w + \lambda 25K=0$$ $$(2) \quad \frac{\partial\mathcal{L}}{\partial Q}=-r + \lambda 25L=0$$ $$(3) \quad \frac{\partial\mathcal{L}}{\partial \lambda}=25LK-Q=0$$
Taking (1) and (2): $$(4) \quad K=\frac{w}{r}L$$ From (3): $$(5) \quad K=\frac{Q}{25L}$$
From (4) and (5): $$\frac{w}{r}L=\frac{Q}{25L}$$
Solving for L: $$L=\sqrt{\frac{rQ}{25w}}$$
