Question

Given the production function Q = L0.75 K0.25
1. If the fixed quantity of capital in the short run is 10000units, estimate SR product function
2. Find the value of L that maximize SR product function in 1above

Good luck

Answer 1

1:

given production function $Q = L^{0.75}K^{0.25}$

Suppose we fix the capital at K = 10000. To find the short run production function, we substitute this value of K in the above production function:

$Q = L^{0.75}K^{0.25}= L^{0.75}(10000)^{0.25}$ now since $0.25=\frac{1}{4}$ the production function can be written as

$Q=L^{0.75}(10000)^{0.25} =L^{0.75}(10000)^{\frac{1}{4}}$

Now, note that $10000 = 10^{4}$

so,

$Q=L^{0.75}(10000)^{\frac{1}{4}}=L^{0.75}(10^4)^{\frac{1}{4}}=L^{0.75}(10)^{4×\frac{1}{4}}=L^{0.75}10^1=10L^{0.75}$

Hence the short run production function is: $Q=10L^{0.75}$

2:

Now, given the short run production function, Q = 10L^0.75, we have to find L which maximizes this function. But note that, as long as we increase L, Q always increases. In other words, the short run production function is always increasing in L.

$\frac{dQ}{dL} =\frac{ d(10L^{0.75})}{dL}=10×0.75L^{0.75−1}=7.5L^{−2.5}>0$

This can also be seen as follows

The above derivative is positive for all values of L. Hence, the value of L that maximizes the short run production function is L = ∞