Find the optimal demand functions for capital and labour for this firm

Question

I'm trying to solve this question which states:

Suppose that a profit maximizing producer has a production function described by Q = K^3/4 L^1/4 and faces the general isocost line (TC = rK + wL).

Find the optimal demand functions for capital and labor for this firm.

Here's what I've done and know:

I've solved for the MP1(labor) and MP2(capital): MP1 = 1/4 (K^3/4 * L^-3/4 ) MP2 = 3/4 (K^-1/4 * L^1/4 )

I also know that MP1 / MP2 = w / r.

Can someone point out a suggestion to solve for K and L, the optimal demand functions?

Thanks

Answer 1

I think you may need some more information to find an expression for L (or K). From what we have, we know MP1/MP2 = w/r, and that lets us say K = aL. Now we look at our profit, the thing we want to maximize. Substituting K = aL into Q - TC, we get L(pb - 4w), where $b = pa^{3/4}$, and p is the price. As long as p > 4r/3, we can increase profit forever by increasing L, as long as we increase K proportionally. This assumes p is fixed, and labor and capital are infinitely available at r and w. So that's where our constraint has to come from. (Unless we're simply given a fixed value for Q, in which case we solve from K = aL.) If we have a demand curve, and we're the only supplier, we increase L and K until Q becomes such that p = 4r/3, or we hire people until we can't get aL units of capital, or we increase K until we can't get K/a units of labor. If we're not the only supplier, we play a game with the other suppliers and fix Q ourselves. But I don't think we can maximize profit without that extra information, because the functional form of Q together with the fact that L and K are linearly related means Q is linearly related to L or K, and the cost function is as well.

Answer 2

A firm requieres capital $K\in\mathbb{R}_+$ and labor $L\in\mathbb{R}_+$ to produce the final good $Q$. The technology is of Cobb-Douglas type $Q:(K,L)\mapsto K^{\alpha}L^{1-\alpha}$, here $\alpha=0.75$. Denote the capital rent by $r$ and the worker's wage by $w$. Production costs are then given by $C:(K,L)\mapsto rK+wL$. Instead of maximizing profits, the firm wants to minimize costs to produce a given quantity $\bar{Q}>0$ (look up the duality of $\min-\max$ problems in producer theory). We end up with the following programm \begin{align} \min_{K,L}~C(K,L)\quad\text{s.t.}~~Q(K,L)\geq\bar{Q}. \end{align}
Which is \begin{align} \min_{K,L}~(rK+wL)\quad\text{s.t.}~~K^{.75}L^{.25}\geq\bar{Q}. \end{align} Set up the lagragian \begin{align} \mathcal{L}=rK+wL+\lambda(\bar{Q}-K^{.75}L^{.25}). \end{align} Note that the constraint is binding (Why? Look that up too!) The FOCs are given by \begin{align} \frac{\partial \mathcal{L}}{\partial K}=0\\ \frac{\partial \mathcal{L}}{\partial L}=0\\ \frac{\partial \mathcal{L}}{\partial\lambda}=0 \end{align} Solve for $K$ and $L$.