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

a .)

Demand condition for the factors of production is achieved at the point where the production is at the optimal level for a given price ratio of the factors . 

Optimal Production condition for a firm is where the MRTS is equal to the factor price ratio .

& MRTS = Marginal Rate of technical Substitution $=\frac{MU(x) }{MU(y)}$  

In our example , two inputs : x & y 

Price of x = Wx 

Price of y = Wy 

Hence the Costs will be in the form :

$Wx\times x + Wy\times y = C$  

Factor price ratio $= \frac{Wx}{Wy}$

 

Production function : $q = 2 (x^{0.5} + y^{0.5} )$

$MP(x) = \frac{dq}{dx} = x\\ MP(y) =\frac{ dq}{dy} = y\\ MRTS = \frac{x}{y}$

Optimal Condition : MRTS = factor Price Ratio 

$\frac{x}{y} = \frac{Wx}{Wy }\\ x = y×\frac{Wx} {Wy} -----(i)$

Putting this value of (x) into the Cost condition faced by producer :

$Wx\times x + Wy\times y = C \\ Wx (y\times \frac{Wx}{ Wy} + Wy\times y = C \\ y* = \frac{C }{\frac{W^2}{Wy}+Wy}\\ y*=\frac{Wy×C}{Wx^2 + Wy^2}$    (Demand function for input y )

Putting this value of y* into the optimal equation (i) 

$x* = y\times ×\frac{Wx}{Wy}\\ x*=\frac{Wx×C}{Wx^2 + Wy^2}$ (Demand function for input y ) 

b .)

Cost function for a given technology refers to the minimum expenditure which would be made to achieve certain production level . 

Hence , cost function could be depicted as the optimal cost that would be incurred . So , we put demand functions into the cost condition for firm :

Cost Function $= Wx(x*) + Wy(y*)$

$= Wx (\frac{Wx×C}{Wx^2 + Wy^2}) + Wy(\frac{Wy×C}{Wx^2 + Wy^2}\\ = \frac{C}{Wx^2+Wy^2}(Wx^2 + Wy^2)$

Cost function = C = Wx(x) + Wy(y)