Question

For each of the following production functions, solve for the marginal products of each input and marginal rate of substitution. Then answer the following for each: does this production function exhibit diminishing marginal product of labour? Does this production function exhibit diminishing marginal product of capital? Does this production function exhibit constant, decreasing, or increasing returns to scale? Show all your work.

(a) \(Q=L+K\)

(b) \(Q=2 L^{2}+K^{2}\)

(c) \(Q=L^{1 / 2} K^{1 / 2}\)

Answer 1

(a) \(Q=L+K\)

Marginal product of labor \(L\) is given by \(\frac{\partial Q}{\partial L}=\frac{\partial(L+K)}{\partial L}=1\)

Marginal product of labor \(\mathrm{K}\) is given by \(\frac{\partial Q}{\partial K}=\frac{\partial(L+K)}{\partial K}=1\)

Marginal rate of substitution is (Marginal product of labor L)/(Marginal product of capital \(\mathrm{K}\) ) \(=\frac{1}{1}=1\) So,marginal rate of substitution=1.

No this function does not exhibit diminishing marginal product of labor because marginal product of labor

is constant \(\frac{\partial^{2} Q}{\partial L^{2}}=\frac{\partial^{2}(L+K)}{\partial L^{2}}=0\)

No this function does not exhibit diminishing marginal product of capital because marginal product of capital is constant and \(\frac{\partial^{2} Q}{\partial K^{2}}=\frac{\partial^{2}(L+K)}{\partial K^{2}}=0\)

\(Q=L+K\)

When both L and \(K\) input are increased by same \(\alpha\) where \(\alpha>0\) then new \(Q\) will be such that:

\(Q=\alpha L+\alpha K\)

\(Q=\alpha(L+K)\)

Since the power of \(\alpha\) is 1 this imply \(Q=L+K\) exhibit constant returns to scale because output increases in same proportion as increase in inputs.

(b) \(Q=2 L^{2}+K^{2}\)

Marginal product of labor \(L\) is given by \(\frac{\partial Q}{\partial L}=\frac{\partial\left(2 L^{2}+K^{2}\right)}{\partial L}=2 * 2 L=4 L\)

Marginal product of labor \(\mathrm{K}\) is given by \(\frac{\partial Q}{\partial K}=\frac{\partial\left(2 L^{2}+K^{2}\right)}{\partial K}=2 K\)

Marginal rate of substitution is (Marginal product of labor L)/(Marginal product of capital \(\mathrm{K}=\frac{4 L}{2 K}=\frac{2 L}{K}\)

No this function does not exhibit diminishing marginal product of labor because when L is 1 marginal product of labor is \(4\left(4^{*} 1\right),\) L increase to 2 units then marginal product of labor is \(8\left(4^{*} 2\right),\) L increase to 3 units then marginal product of labor is \(12\left(4^{*} 3\right)\) which imply this function does not exhibit diminishing marginal product of labor as labor units increase marginal product of labor also increase.

No this function does not exhibit diminishing marginal product of capital because when \(\mathrm{K}\) is 1 marginal product of capital is \(2\left(2^{*} 1\right), \mathrm{K}\) increase to 2 units then marginal product of capital is \(4\left(2^{*} 2\right), \mathrm{K}\) increase to 3 units then marginal product of capital is \(6\left(2^{*} 3\right)\) which imply this function does not exhibit diminishing marginal product of capital as capital unit increase marginal product of capital also increase.

\(Q=2 L^{2}+K^{2}\)

When both \(L\) and \(K\) input are increased by same \(\alpha\) where \(\alpha>0\) then new \(Q\) will be such that:

\(Q=2(\alpha L)^{2}+(\alpha K)^{2}\)

\(Q=2 \alpha^{2} L^{2}+\alpha^{2} K^{2}\)

\(Q=\alpha^{2}\left(2 L^{2}+K^{2}\right)\)

Since the power of \(\alpha\) is greater than 1 this imply \(Q=2 L^{2}+K^{2}\) exhibit increasing returns to scale because output increases in more proportion than there is increase in inputs.

(c) \(Q=L^{\frac{1}{2}} K^{\frac{1}{2}}\)

Marginal product of labor L is given by \(\frac{\partial Q}{\partial L}=\frac{\partial\left(L^{\frac{1}{2}} K^{\frac{1}{2}}\right)}{\partial L}=\frac{1}{2} * \frac{K^{\frac{1}{2}}}{L^{\frac{1}{2}}}=\frac{K^{\frac{1}{2}}}{2 L^{\frac{1}{2}}}\)

Marginal product of labor \(\mathrm{K}\) is given by \(\frac{\partial Q}{\partial K}=\frac{\partial\left(L^{\frac{1}{2}} K^{\frac{1}{2}}\right)}{\partial K}=\frac{1}{2} * \frac{L^{\frac{1}{2}}}{K^{\frac{1}{2}}}=\frac{L^{\frac{1}{2}}}{2 K^{\frac{1}{2}}}\)

Marginal rate of substitution is (Marginal product of labor L)/(Marginal product of capital K)= \(\frac{\frac{K}{2 L}}{\frac{L}{2 K}}=\frac{K * 2 K}{2 L * L}=\frac{K}{L}\). So,marginal rate of substitution \(=\frac{K}{L}\)

Yes this function does exhibit diminishing marginal product of labor because \(\frac{\partial^{2} Q}{\partial L^{2}}=\frac{\partial^{2}\left(L^{\frac{1}{2}} K^{\frac{1}{2}}\right)}{\partial L^{2}}=\frac{-1 * K^{\frac{1}{2}}}{4 L^{\frac{3}{2}}}=\frac{-K^{\frac{1}{2}}}{4 L^{\frac{3}{2}}}\) is negative which means marginal product of labor fall when

Lincreases by 1 unit.

Yes this function does exhibit diminishing marginal product of capital because \(\frac{\partial^{2} Q}{\partial L^{2}}=\frac{\partial^{2}\left(L^{\frac{1}{2}} K^{\frac{1}{2}}\right)}{\partial L^{2}}=\frac{-1 * L^{\frac{1}{2}}}{4 K^{\frac{3}{2}}}=\frac{-L^{\frac{1}{2}}}{4 K^{\frac{3}{2}}}\) is negative which means marginal product of capital fall

when \(\mathrm{K}\) increases by 1 unit.

\(Q=L^{\frac{1}{2}} K^{\frac{1}{2}}\)

When both \(L\) and \(K\) input are increased by same \(\alpha\) where \(\alpha>0\) then new \(Q\) will be such that:

\(Q=(\alpha L)^{\frac{1}{2}}(\alpha K)^{\frac{1}{2}}\)

\(Q=\alpha^{\frac{1}{2}} L^{\frac{1}{2}} \alpha^{\frac{1}{2}} K^{\frac{1}{2}}\)

\(Q=\alpha L^{\frac{1}{2}} K^{\frac{1}{2}}\)

Since the power of \(\alpha\) is 1 this imply \(Q=L^{\frac{1}{2}} K^{\frac{1}{2}}\) exhibit constant returns to scale because output increases in same proportion as increase in inputs.