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A monopoly can produce any output level at a constant marginal (and average) cost of θ per unit. Assume the monopoly sells its goods in two markets separated by some distance. The demand curve in the first market and the demand curve in the second market are given. Assume that monopoly price in these two markets is not same. So, initially monopolist choose to segment the market and do price discrimination(P1*>P2*)

Now if we are given that the demander has the option to incur a cost of $c to transport goods between these two markets. In the solution given they take the new constraint to be P1-P2=c and then solve for P1 and P2 with the aim to maximize profit. But I don't understand why are we doing this.

What I did was I said all the people in market 1(higher monopoly price) with valuation(P1) of (P2+c)<P1<(P1*) category can travel to market 2 and buy from there for this I created a new demand curve and people who value the good more than monopoly price can stay here in market 1. I know what I'm doing is wrong because I'm not telling what people valuation P1<P2+c are going to do.

For market 2 I said that they are not going to go to market 1 as there the monopoly price is higher.

The solution takes P1=P2+c they basically are forcing the monopoly price of market 1 to equal to P1=P2+c. How is it possible. They say "Producer wants to maximize price differential in order to maximize profits but maximum price differential is $c. So P1=P2+c."

I don't understand how is profit maximized by maximizing price differential. And how can we generalize the prices to be in such a way that P1=P2+c

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The monopolist has information on transportation costs. Therefore they try to maximise profit such that the price difference between $P1$ and $P2$ is less than or equal to $c$.The monopolist sets a price to maximise the total profit rather than profit for individual markets. This results in constraint maximization,

$ maximize ~ \pi= TR(P1,P2) - TC(P1,P2) ;~where ~\theta~is ~a ~constant \\ subject: P1-P2<=c \\\\$

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