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I want to find the total demand for good r and the optimal price p for a monopolist using the following information:

Marginal cost per good = $c$ (constant)

All consumers have the following utility function: $u(r,g)=r^{0.1}g^{0.9}$, where $r$ is the good of interest and g represents all other goods.

The average income of a consumer is $y=20,000$, and there are $100,000$ possible consumers who would buy good $r$.

I need to compute the demand for $r$, the profits the monopolist would earn, and the optimal price for $r$ when $c=1$.

This is what I have done so far:

I used this Lagrange Multiplier: $L(r,g,\lambda)=r^{0.1}g^{0.9}+\lambda(20000-rp-gp_g)$ to find the demand curve for one consumer, which I found to be $r=\frac{2000}{p}$.

To find the demand for $100,000$ consumers, I multiplied the demand function above by $100,000$ and obtained $r=\frac{200,000,000}{p}$.

Then, I determined that the profit function for the monopolist would be $\Pi(p,r)=(p-c)\cdot r(p)=(p-1)\cdot\frac{200,000,000}{p}$.

To find the optimal price, I tried to find the critical point of the profit function above by differentiating it with respect to p, but this is where I ran into trouble. I believe that a critical point does not exist for the given profit function.

$\frac{\partial \Pi}{\partial p}=\frac{200,000,000}{p}+(p-1)(-200,000,000)\cdot p^{-2}=0$

$\frac{200,000,000p-200,000,000p+200,000,000}{p^2}=0$, which just simplifies to $200,000,000=0$. Hence, no critical point.

Where did I go wrong? I'd appreciate any guidance. I've looked over my work over and over again, so I'm quite sure it's not an arithmetic error, but I can't figure out how else to model this particular situation. Thank you in advance!

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    $\begingroup$ scanning your question very quickly : did you check for corner solutions? If there are no mistake in you algebra and there are no critical point, it is usually that the solution is a corner one... $\endgroup$ – Martin Van der Linden Dec 18 '13 at 13:28
  • $\begingroup$ Thank you for the suggestion! I just graphed the profit function, and I don't think that the solution is a corner solution as the profit function extends to negative infinity as p approaches 0 and as p goes to infinity, profit just approaches 200,000,000. $\endgroup$ – Jess Dec 19 '13 at 6:00
  • $\begingroup$ That was a bad guess then... I'll try to have a closer look at you problem later today :) $\endgroup$ – Martin Van der Linden Dec 19 '13 at 11:24
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Your work is fine. You got the right demand function (next time, just make it clear that you normalize $p_g = 1$), you expressed the profit function in the right way and you took your derivative right too.

You are also right to conclude that there is no critical point, as is confirmed by your graphical analysis.

Now the only way to conclude is to say that the maximization problem of the monopolist does not have a solution. This is again what comes out from your graphical analysis. There cannot be a price $p^*$ large enough to maximize the profit of the monopolist because for any price $p' > p^*$ you have that $\Pi(p')$ is closer from 200,000,000 than $\Pi(p^*)$. Although demand goes to zero as $p$ increases, it does not do so fast enough to prevent profit to always increase.

This is in fact a well know problem in the litterature, see for instance the very first paragraph of Yin, Xiangkang. "A Tractable Alternative to Cobb‐Douglas Utility for Imperfect Competition." Australian Economic Papers 40.1 (2001): 14-21.

If you really want to get a solution, you need to change your problem. You could for instance assume a maximal price $\bar{p}$. Then the solution would be the corner solution $p^* = \bar{p}$. Or you could choose another utility function (see the aforementioned paper).

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