Question on monopolist (involves simple calculus)

Question

QUESTION: Consider the problem of a monopolist that sells its product in two different markets $m$, with $m=1,2$. Each market has an aggregate demand function given by $1200−α_m\cdot p_m$, where $p_m$ denotes the price in market $m$, and $α_m=m$ measures the responsivity of demand to prices in market $m$. The monopolist's cost function is given by $c(q)=12q^2$, where $q$ denotes the total amount produced for all markets.

The monopolist is owned by a foreign company, so none of the monopolist's profits are received by the consumers in these markets. The law allows the monopolist to charge different pricees in different markets, but does not allow any other forms of price discrimination.

1. What is the equilibrium level of production in market $2$?
2. What is total consumer surplus in the economy (i.e., taking both markets into account)?
3. Suppose that the government behind market $1$ introduces a tax of $\$100$ per unit on the monopolist's sales in its market (paid by the firm), and that the tax revenue is given back to consumers in market $1$ using lump-sum transfers. Suppose also that no such tax is introduced by the government behind market $2$. Then what is the new equilibrium level of production in market $2$?
4. What is the the new total level of consumer surplus in the economy (including the tax revenues)?

WHAT I TRIED:

For each market, (dropping $\_m$): $p^{*} = \text{argmax}_{p}\ (1200-ap)p - 12(1200-ap)^2$ or $q^{*} = \text{argmax}_{p}\ q(1200-q)/a - 12q^2.$ Took the FOC and tried to solve it. But could not get anywhere (did not get the result - my answer came out wrong)

If you don't understand anything or want me to add anything else, please add it as a comment. (There can be a problem with how I entered the symbol or my solution). Also, I am a little new to Stackexchange so any kind of input would help me. Thanks in advance.

EDIT:My setup was wrong. This is the right one: $$\max_{q_1,q_2}\ {p_1q_1 + p_2q_2 - 12(q_1+q_2)^2}$$ where $$p_1 = (1200-q_1)/a_1$$ $$p_2 = (1200-q_2)/a_2$$ Then take FCOs w.r.t $q_1$ and $q_2$ and solve.

But it's actually a pretty nasty expression since it expands into a polynomial with I think 7 terms.

I figured out the first two, but I need help with the last 2 now.

EDIT: The first answer came out to be 120. The second answer 68400. Please give a similar hint on the 3rd and 4th. I can't comment.

Answer 1

You really need to format the question above. I'll give a little sympathy to your cause and explain step by step what is happening. We have:

$q=1200−\alpha_mp_m$ for markets $m=1,2$

which can get us $p_m = \frac{1200 - q}{\alpha_m} = \frac{1200 - q_1 - q_2}{\alpha_m}$

Already there is something very fishy about this setup. As we go on, see if you can figure out what's happening.

$c(q)=12q^2$

$(q = q_1 + q_2)$

As per your question, the monopolist can charge different prices in different markets. Since the only thing possibly different about the two markets is the sensitivity index $\alpha$, keep an eye out on those terms.

How do we find the equilibrium production in market 2? Let's set up the profit equation, revenue minus costs.

\begin{align} \max_{q_1, q_2} \ \Pi & = \vec{p}\vec{q} - 12q^2 \\ & = p_1q_1 + p_2q_2 - 12(q_1 + q_2)^2 \\ & =p_1q_1 + p_2q_2 - 12(q_1^2 + 2q_1q_2 + q_2^2) \\ \text{We get first order conditions.}\\ \text{Just take the derivative with }\\ \text{respect to $q_1$ and $q_2$ and set to zero.}\\ \\ \frac{\partial{\Pi}}{\partial{q_1}} & = p_1 - 24q_1 - 24q_2 = 0 \\ \frac{\partial{\Pi}}{\partial{q_2}} & = p_2 - 24q_1 - 24q_2 = 0 \\ \text{Substitute each respective $p_m$ above:} \\ 0 & = \frac{1200 - q_1 - q_2}{\alpha_1} - 24q_1 - 24q_2 \\ 0 & = \frac{1200 - q_1 - q_2}{\alpha_2} - 24q_1 - 24q_2 \\ \end{align}

This is just a little system of equations. Two equations, two unknowns. Treat the $\alpha$'s as constants.

To make it a little more palatable to deal with, let's multiply both sides by their $\alpha$'s.

$$0 = 1200 - q_1 - q_2 - (24\alpha_1)q_1 - (24\alpha_1)q_2$$ $$0 = 1200 - q_1 - q_2 - (24\alpha_2)q_1 - (24\alpha_2)q_2$$

Combine terms and put the constants on one side.

$$-1200 = -(1 + 24\alpha_1)q_1 - (1 + 24\alpha_1)q_2$$ $$-1200 = -(1 + 24\alpha_2)q_1 - (1 + 24\alpha_2)q_2$$ $$\implies$$ $$\frac{1200}{1 + 24\alpha_1} = q_1 + q_2$$ $$\frac{1200}{1 + 24\alpha_2} = q_1 + q_2$$

Now wait a minute! There's no solution to this unless $\alpha_1 = \alpha_2$! And if that's true, there are infinite solutions. What gives?

Well, think about the original setup. The cost function doesn't penalize you for producing more in one market than another, and the demand for both markets are perfectly interdependent. So what's to stop a monopolist from just selling in the market with the higher price, that is, the market where $\alpha$ is the smallest? The producer should just produce all in one market unless $\alpha_1 = \alpha_2$

So we can redo the maximization problem, with only one market in consideration, whichever has the smallest $\alpha$. (You might be able to get away with using the above intuition, and not having to show the above result.)

$$\max_{q} \ \Pi = pq - 12q^2$$ $$\Pi = \frac{1200 - q}{\alpha}q - 12q^2$$ $$\frac{\partial{\Pi}}{\partial{q}} = \frac{1200 - 2q}{\alpha} - 24q = 0$$ $$(2 + 24\alpha)q = 1200$$ $$\boxed{q^* = \frac{1200}{2 + 24\alpha}}$$

That's how to get the first part. If you know about consumer theory and all that like you say in the comments, the next parts should not be too bad. Try those for yourself. I think this is enough help.