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QUESTION: Consider the problem of a monopolist that sells its product on two different markets $m$, with $m=1,2$. Each market has an aggregate demand function given by $1200−α_mp_m$, 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 $c(q)=12q2$, 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.

(i) What is the equilibrium level of production in market $2$?

(ii) What is total consumer surplus in the economy (i.e., taking both markets into account)?

(iii) 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 $2$.

What is the new equilibrium level of production in market $2$?

(iv) 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* = argmax_{p} (1200-ap)p - 12(1200-ap)2
          or
          q* = argmax_{q} q(1200-q)/a - 12q2.
          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_{q1,q2} {p1q1 + p2q2 - 12(q1+q2)^2} where p1 = (1200-q1)/a1 p2 = (1200-q2)/a2 Then take FCOs w.r.t q1 and q2 and solve. But it's actually a pretty nasty expression since it expands into a polynomial with I think 7 terms.

Any further solution would be helpful.

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  • $\begingroup$ I think this question is more appropriate and relevant for the Economics SE. economics.stackexchange.com $\endgroup$ – rocinante Mar 9 '16 at 13:40
  • $\begingroup$ I initially tried asking it there, but no one answered so I thought I should ask it here, because it seems that there are more users here. $\endgroup$ – John Millibar Mar 9 '16 at 14:01

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