# Marshallian demand function

A firm has a cost function given by the following:

c(w1, w2, y)= w1w2y^2/(w1+w2)

where the wi's are the prices of the factors (inputs) x1 and x2 respectively, and y is output.

a) Is this a legitimate cost function?

b) Find the firm's production function, y= f(x1, x2).

c) From the cost function derive the firm's conditional (output constant) demand function x1(w,y).

d) From the cost function derive the firm's profit function for output price p, pi(w,p). What is the firm's supply function? Use the firm's supply function results and the results from part (c) to derive the Marshallian demand function x1(w,p)

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#### Solution Preview

a) Is this a legitimate cost function?

This is a legitimate cost function, because MC = dc/dy = 2y w1w2/(w1+w2)

Thus, the supply curve is an upward sloping curve.

b) Find the firm's production function, y= f(x1, x2).

since the cost can be also written into:

c = w1 x1 + w2 x2

we can write

w1w2y^2/(w1+w2) = w1 x1 + w2 x2

y^2 = (w1 x1 + w2 x2)* (w1+w2)/ w1w2

y^2 = w1 x1 + w2 x2)* (w1+w2)/ w1w2

y^2 = (w1^2 x1 + w1w2 x1+ w1w2 x2 + w2^2 x2) / ...

#### Solution Summary

Marshallian demand function is found.