# Analyising Monopolist's demand and cost functions

Suppose the demand curve faced by a monopolist is:

q=p^-a where a>0

The total cost for the monopolist are C=sq

a) Find the profit maximizing level of output.

b) Specify the first and second order condition for profit maximization.

c) What is the price elasticity of demand faced by this monopolist

d) What happens to the profit maximizing level of output as s increases

e) What happens to the profit maximizing level of output as a increases.

f)Explain results

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#### Solution Preview

Please refer the attached file for complete description. Work done with the help of equation writer may not print here.

a) Find the profit maximizing level of output.

A monopolist will produce output to maximize profits in such a way that MC=MR

Total cost =C=sq

Marginal cost = MC=dC/dq=s

Total Revenue = price*output

Given q=p^-a

Total revenue = p*q=

For profit maximization MR=MC

b)Specify the first and second order condition for profit maximization.

Profit =Total revenue-Total cost

First order condition:

Put

d(Profit)/dq=0

d(Profit)/dq=d(TR)/dq-dC/dq=0

we know that d(TR)/dq=MR

and dC/dq=MC

d(Profit)/dq=d(TR)/dq-dC/dq=0

=MR-MC=0

i.e. MR=MC

We lead to same condition as we have used in part (a), so we can directly use the results here and can say that first order ...

#### Solution Summary

Solution describes the steps for determining profit maximizing output and elasticity of demand. It also discusses first and second order conditions for profit maximization.