Microeconomics exercise on returns to scale

Question

I have an exercise in microeconomics that exhibits the following function Y = L^(1/2) * (W^(1/2) + K^(1/2))

With L = labor, K = capital, W = water

In a first question, we are asked to find what the returns to scale are for this function and I found that this function has constant returns to scale. In fact, multiplying all the inputs by t (with t>1), we get that it is equivalent to multiplying the whole production function by t.

However, in the second question, we assume that the level of water is fixed to the level Wf. I understand that now the function only depends on K and L so I tried to multiply K and L by t, but I can't simplify the function with respect to the factor t.

I am stuck at the following expression : t^(1/2)*L^(1/2)Wf^(1/2) + tL^(1/2)*K^(1/2).

I hope my question is not too confused and I thank anyone who tries to help me :)

Answer 1

If $W$ is fixed, the production function is no longer constant returns to scale. The only think you can do is to determine the returns to scale at a certain point for the inputs. If you have a production function $Y(K,L)$ then the returns to scale at a particular value $(K,L)$ is simply the sum of the input elasticities: $$ Y_L \frac{L}{Y} + Y_K \frac{K}{Y}. $$ In your case you get: $$ \frac{1}{2} + \frac{1}{2} \frac{L^{1/2} K^{1/2}}{L^{1/2}(K^{1/2} + W_f^{1/2})} $$ which will be strictly smaller than 1 if $W_f$ is larger than zero.

Answer 2

Definition 1. Production function $f(L, K, W)$ exhibits Constant Returns to Scale at $(L,K,W)$ if for all $t>1$, $f(tL,tK,tW) = tf(L,K,W)$.

Definition 2. Production function $f(L, K, W)$ exhibits Decreasing Returns to Scale at $(L,K,W)$ if for all $t>1$, $f(tL,tK,tW) < tf(L,K,W)$.

Clearly, Production function $f(L,K,W) = L^\frac{1}{2}(W^\frac{1}{2}+K^\frac{1}{2})$ exhibits Constant Returns to Scale at all $(L,K,W)$ because for any $t>1$, $f(tL,tK,tW) = (tL)^\frac{1}{2}((tW)^\frac{1}{2}+(tK)^\frac{1}{2})=t\left(L^\frac{1}{2}(W^\frac{1}{2}+K^\frac{1}{2})\right)=tf(L,K,W)$

Now, suppose the water is fixed at $W_f$, so that the production function is $g(L,K)=f(L,K,W_f)=L^\frac{1}{2}(W_f^\frac{1}{2}+K^\frac{1}{2})$. Now consider any $(L,K)$ where $L>0$, $W_f>0$, and $K\geq 0$, for $t>1$, we have $g(tL, tK)=f(tL,tK,W_f)=(tL)^\frac{1}{2}(W_f^\frac{1}{2}+(tK)^\frac{1}{2})<(tL)^\frac{1}{2}((tW_f)^\frac{1}{2}+(tK)^\frac{1}{2})=t\left(L^\frac{1}{2}(W_f^\frac{1}{2}+K^\frac{1}{2})\right)=tf(L,K,W_f)=tg(L,K)$.

Therefore, $f(L,K,W_f)$ exhibits decreasing returns to scale at $(L,K,W_F)$ where $L>0$, $W_f>0$, and $K\geq 0$. You can check that it exhibits constant returns to scale elsewhere i.e. if either $L=0$ or $W_f=0$.