Question

Let the production function of a firm is given as

𝑞=(𝑥0.5 +𝑦0.5)2

Where 𝑥 and 𝑦 are inputs and 𝑤𝑥 is the price of input 𝑥 and 𝑤𝑦 is the price

of input 𝑦.

a) Assume the firm has a limited budget to spend on buying input. Find

the cost-conditional input demand function for each input.

b) Find the cost function of the firm

Answer 1

Solution:

a.). Derive MRTS = $\frac{MP_{X} }{MP_{Y}}$

MP_X = $\frac{\partial Q} {\partial X} = \frac{1} {x^{0.5} }$

MP_Y = $\frac{\partial Q} {\partial Y} = \frac{1} {y^{0.5} }$

$\frac{MP_{X} }{MP_{Y}} = \frac{w }{r}$

$\frac{\frac{1} {x^{0.5} } }{\frac{1} {y^{0.5} } } = \frac{w} {r }$

$\frac{y^{0.5}} {x^{0.5} } = \frac{w} {r }$

x = $\frac{r^{2}y} {w^{2} }$

y = $\frac{w^{2}x} {r^{2} }$

 

b.). The cost function of the firm:

TC = rY + wX

TC = r($\frac{w^{2}x} {r^{2} }$) + w($\frac{r^{2}y} {w^{2} }$)

TC = $\frac{w^{2}x} {r} +$ $\frac{r^{2}y} {w}$