Answer to Question #186673 in Microeconomics for Bornface Kandunda

Question

You are given the production function: 𝑄(𝐾, 𝐿) = 10𝐾 𝛼𝐿 𝛽 a) What is the average product of labour, holding capital fixed at K? Simplify fully [6 marks] b) What is the marginal rate of technical substitution (MRTS)? Simplify fully [13 marks] Does the above function exhibit increasing, decreasing or constant returns to scale? Illustrate why and explain what this means [6 marks

Answer 1

Average product of labor (APL), is total product (TP) divided by the total quantity of labor.

Given;

production function

$Q(K,L)=10K^{\alpha}L^{\beta}$

(a)

Average product of labor holding capital fixed at K.

$=\frac{Production(Q)}{Labor(L)}$

$=\frac{10K^{\alpha}L^{\beta}}{L}$

Average Product of Labor (APL)$=10K^{\alpha}L^{\beta -1}$

(b)

Marginal rate of technical substitution illustrates the rate at which one factor must decrease so that the same level of productivity can be maintained when another factor is increased.

$MRTS {K \atop L} =\frac{MP_L}{MP_k}=\frac{\delta Q \delta L}{\delta Q \delta K}=\frac{10\beta K^{\alpha}L^{\beta -1}}{10\alpha K^{\alpha -1}L^{L^{\beta}}}$

$MRTS=\frac{\beta}{\alpha}\frac{K}{L}$

As, production $f^n:Q=10K^{\alpha}L^{\beta}.$

${let \lambda >1}$

$Q(\lambda K, \lambda L)=10(\lambda K)^{\alpha}(\lambda L)^{\beta}$

$=10\lambda ^{\alpha +\beta}(K^{\alpha L^{\beta}})$

$[Q(\lambda K, \lambda L) =\lambda ^{\alpha +\beta}Q(K,L).]$

If $\alpha + \beta=1,$ the function exhibits constant returns to scale.

If $\alpha + \beta >1,$ the function exhibits increasing returns to scale.

if $\alpha + \beta<1,$ the function exhibits decreasing returns to scale.

It means:

1. If inputs are increased by $\lambda$ (some) amount, then the output increases $\lambda$ amount., then it is called constant returns to scale.

2. If inputs are increased by $\lambda$ (some) amount, then the output increases by more than $\lambda$ amount., then it is called increasing returns to scale.

3. If inputs are increased by $\lambda$ (some) amount, then the output increases by less than $\lambda$ amount., then it is called decreasing returns to scale.