Question

Consider a monopolist which produces two different products, each having the following demand functions: q1 = 14-1/4 p1; q2 = 24-1/2 p2; where q1 and q2 represent the output levels of product 1 and product 2 and p1 and p2 represent their prices. The monopolistís joint cost function is given as C (q1; q2) = q1^2 + 5q1q2 + q2^2 :

(a) Write out the monopolist's profit function.

(b) Show the Hessian, H; for this problem. What does the second-order condition require for this problem? Show if it is satisfied

Answer 1

(a)

Monopolist's profit function is given as

Given:

Q1 = 14 - 1/4 P1 ------------------------ eq1

Converting this demand curve into inverse demand curve, we get

P1 = 56 - 4Q1

Similarily, Q2 = 24 - 1/2P2 will become P2 = 48 - 2Q2

Now, TR = P.Q

TR1 = P1.Q1

from eq1

TR1 = Q1( 56 - 4Q1) = 56Q1 - 4Q1 ^2

Similarily, TR2 = 48Q2 - 2 Q2^2

Now, Given that,

Profit Function =

==>  

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(b)

Hessian is a type of square matrix of second ordered partial derivatives of a scalar function.

From (a) we have the profit function as  

Taking first order partial derivatives of profit function, we get

Taking second order partial derivatives,

Now, hessian matrix is given as

From the obtained profit function, the hessian matrix can be given as :

det(H) = (-10*-6) - (-5*-5) = 35

Second order condition suggests that second order derivatives < 0.

And in case of hessian matrix det(H) > 0.

Here, det(H) = 35 > 0.

Hence, second order condition is not satisfied.

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