Equilibrium levels of output
A monopolist produces two different goods. The (inverse) demand for each good is given as:
p_1 = theta - (alpha)q_1 + (beta)q_2
p_2 = theta + (alpha)q_1 - (beta)q_2
The monopolist produces both of these goods at a constant unit cost (c > 0) and maximizes the profit function below by setting the output of each good.
(pi) = p_1q1 + p_2q_2 - cq_1 - cq_2
You can interpret the parameters as follows. The theta parameter is the intercept term for both demand functions. Note that it is necessary for theta > c. The alpha and beta parameters are measures of differentiation. As alpha gets smaller, or as beta gets larger, the products are increasingly heterogeneous.
a. Find the equilibrium levels of output for goods 1 and 2.
b. Show whether these levels of output maximize profits.
c. Use comparative statics to show whether this monopolist's equilibrium profits will rise or fall if the monopolist makes these goods more similar (i.e. more homogeneous). That is, as alpha and beta change.