Why does $\frac{MP_L}{MP_K} = \frac{w}{r}$?

Question

I did a practice problem set. I was given a production function $F(K,L)$ and had to derive the firm's cost function. To do this, I used the following relationship:

\begin{equation} \frac{MP_L}{MP_K}=\frac{w}{r} \end{equation}

I found this formula in a set of lecture notes online. I have no idea where it comes from. I understand what $MP_K$, $MP_L$, $w$, and $r$ are but I don't know why these ratios are equal. I got the problem correct, but I want to understand this point better.

My Question:

Can someone explain why this ratio is true?

Answer 1

This is a result of firm maximization. Consider a problem that a profit maximizing firm faces --it must rent capital, $k$, at rate $r$ and hires labor, $l$, at wage $w$ to produce goods ($F(k, l)$) for which we can sell at a normalized price of 1.

$$ \max_{k, l} F(k, l) - w l - r k $$

Then if we just think about the firm's maximization, we can take derivatives w.r.t $k$ and $l$ to get:

$$ F_k(k, l) - r = 0 $$ $$ F_k(k, l) = r $$

and

$$ F_l(k, l) - w = 0 $$ $$ F_l(k, l) = w $$

Notice that by definition $MPL_K = F_k(k, l)$ and $MP_L = F_l(k, l)$. If we take the ratio of the two equations then we get what you initially introduced --that is:

$$ \frac{F_l(k, l)}{F_k(k, l)} = \frac{MP_L}{MP_K} = \frac{w}{r} $$

Edit: As mentioned in another answer, this is imposing some assumptions on the behavior of $F(k, l)$. These assumptions are $\frac{\partial F}{\partial k} > 0$, $\frac{\partial F}{\partial l} > 0$ and $\frac{\partial^2 F}{\partial k^2} < 0$ $\frac{\partial^2 F}{\partial l^2} < 0$

Answer 2

Mathematically it is a necessary condition in problem of profit maximisation and it can be easily derived under usual assumptions. However there is an intuitive explanation.

We make a usual assumption of decreasing productivity of capital and labour (i.e. each additional unit of the input produce some product, but less then previous one, $\frac{\partial^2F}{\partial K^2}<0, \frac{\partial^2F}{\partial KL^2}<0, \frac{\partial F}{\partial K}>0, \frac{\partial F}{\partial K}>0$).

Now, we believe that the firm is maximizing profit (another assumption, better get used to this sort of simplifications). Then the ratio on the right represents how much one additional unit of labor will produce in terms of production of additional unit of capital, while the ratio on the left shows how much more will additional unit of labour cost you in terms of cost of one unit of capital. Also, I will set price of the good the firm produces to be 1 without loss of generality.

Now, if the right side is greater, you may decrease you labour by $\frac{\Delta}{MP_L}$ and increase capital by$\frac{\Delta}{MP_K}$, and you profit will increase by $\frac{w\Delta}{MP_L}-\frac{r\Delta}{MP_K}>0$, so it was not maximised. If the left side was greater you decrese capital, increase labour and increase your profit by$\frac{w\Delta}{MP_K}-\frac{r\Delta}{MP_L}>0$, and so profit, again, was not maximised. It means, that profit can only be maximised at points that satisfy the equation.

Answer 3

Others have provided intuitive explanations; I thought I’d provide a short mathematical one.

Suppose a firm faces a production function $q = f(K,L)$. Assume $f_K, f_L > 0$ and $f_{KK}, f_{LL} < 0$. (The former is entailed by the assumption that firms wouldn't invest in capital or labor if it wasn't increasing their gains, while the latter is the law of diminishing marginal product.)

Let the cost of capital (per unit) be $r$ and the cost of labor (per unit) be $w$.

The firm’s problem is:

$$\max_{K,L} π = pf(K,L) - rK - wL$$

Now, a profit-maximizing firm’s first order conditions are:

$$π_K = pf_K - r = 0 \\ π_L = pf_L - w = 0$$

We can rearrange terms to represent these conditions as:

$$p = \frac{r}{f_K} \\ p = \frac{w}{f_L}$$

Equating the two:

$$\frac{r}{f_K} = \frac{w}{f_L} \implies \frac{f_L}{f_K} = \frac{w}{r} $$

Since $MP_L \equiv f_L$ and $MP_K \equiv f_K$, we get:

$$\frac{MP_L}{MP_K} = \frac{w}{r}$$

Hope that helps.