Question

For each of the following production functions, determine whether returns scales are decreasing, constant, or increasing
a. Q= 2K+3L+KL
b. Q=20K^0.6L^0.5
c. Q=100+3K+2L
d. Q=5K^a L^b, where a+b=1
e. Q= 10K^a L^b where a+b=1.2
f. Q= K/L

Answer 1

Answer on Question 50825, Economics, Microeconomics

Question:

For each of the following production functions, determine whether returns scales are decreasing, constant or increasing:

a. $Q = 2K + 3L + KL$

b. $Q = 20K^{0.6}L^{0.5}$

c. $Q = 100 + 3K + 2L$

d. $Q = 5K^{a}L^{b}$, where $a + b = 1$

e. $Q = 10K^{a}L^{b}$, where $a + b = 1.2$

f. $Q = \frac{K}{L}$

Answer:

Basically, the returns to scale refers to how much output changes given a proportional change in all inputs, where all the inputs change by the same factor. For example:

$$F(zK, zL) = AK^{a}L^{b} \left\{ \begin{array}{l} < zF(K, L) \rightarrow \text{Decreasing Returns to Scale} \\ = zF(K, L) \rightarrow \text{Constant Returns to Scale} \\ > zF(K, L) \rightarrow \text{Increasing Returns to Scale} \end{array} \right.$$

a. Let $Q_0 = F(K, L) = 2K + 3L + KL$ be the initial production function. Let us multiply it by factor $z$ and call it $Q_1$:

$$Q_1 = F(zK, zL) = 2(zK) + 3(zL) + (zK)(zL) = z(2K + 3L + zKL)$$

It is obvious that $F(zK, zL) > zF(K, L)$:

$$z(2K + 3L + zKL) > z(2K + 3L + KL)$$ $$2K + 3L + zKL > 2K + 3L + KL$$ $$zKL > KL$$ $$z > 1.$$

This production function represents increasing returns to scale.

. Let $Q_{0}=F(K,L)=20K^{0.6}L^{0.5}$ be the initial production function, then after multiplying it by factor $z$ we obtain:

$Q_{1}=F(zK,zL)=20(zK)^{0.6}(zL)^{0.5}=z^{0.6}z^{0.5}20K^{0.6}L^{0.5}=z^{1.1}Q_{0}$

In this case $F(zK,zL)>zF(K,L)$.

This production function represents increasing returns to scale.

. Let $Q_{0}=F(K,L)=100+3K+2L$ be the initial production function, then after multiplying it by factor $z$ we obtain:

$Q_{1}=F(zK,zL)=100+3(zK)+2(zL)$

In this case $F(zK,zL)<zF(K,L)$:

$100+3(zK)+2(zL)<100z+3(zK)+2(zL)$

This production function represents decreasing returns to scale.

d. Let $Q_{0}=F(K,L)=5K^{a}L^{b}$, where $a+b=1$, be the initial production function, then after multiplying it by factor $z$ we obtain:

$Q_{1}=F(zK,zL)=5(zK)^{a}(zL)^{b}=z^{a}z^{b}5K^{a}L^{b}=z^{a+b}Q_{0}=zQ_{0}$

In this case $F(zK,zL)=zF(K,L)$.

This production function represents constant returns to scale.

e. Let $Q_{0}=F(K,L)=10K^{a}L^{b}$, where $a+b=1.2$, be the initial production function, then after multiplying it by factor $z$ we obtain:

$Q_{1}=F(zK,zL)=10(zK)^{a}(zL)^{b}=z^{a}z^{b}10K^{a}L^{b}=z^{a+b}Q_{0}=z^{1.2}Q_{0}$

In this case $F(zK,zL)>zF(K,L)$.

This production function represents increasing returns to scale.

f. Let $Q_{0}=F(K,L)=\frac{K}{L}=KL^{-1}$ be the initial production function, then after multiplying it by factor $z=2$ we obtain:

$Q_{1}=F(zK,zL)=(2K)(2L)^{-1}=2\cdot 2^{-1}KL^{-1}=KL^{-1}$

In this case $F(2K, 2L) = 2F(K, L)$.

This production function represents constant returns to scale.

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