<p>A monopolist maximizes profit. For me, it is usually easier to do this in the quantity space. So you rearange the demand and maximize
<span class="math-container">$$\max_{Q_p,Q_r} \quad P_p(Q_p)Q_p + P_r(Q_r)Q_r - TC(Q_p+Q_r)$$</span>
<span class="math-container">$$\max_{Q_p,Q_r} \quad (Q_p-10)Q_p + (28-2Q_r)Q_r - 5-2(Q_r+Q_p) -\frac{(Q_r+Q_p)^2}{8}$$</span>
The FOC gives you two equations with two unknowns, <span class="math-container">$Q_r$</span> and <span class="math-container">$Q_p$</span>, which you solve for. Both FOC equations can be written in the form <span class="math-container">$MR=MC$</span>. You then plug these <span class="math-container">$Q$</span>s into the demand equations to get the prices. Next, you calculate consumer surplus and producer surplus and add them to get social welfare.</p>

CC BY-SA 4.0
2020-12-14T09:54:19.473Z
2020-12-15T07:48:48.710Z
2020-12-15T07:48:48.710Z
Bayesian
<p>Between two countries, Richland and Poorland, with a strict ban on cross-border sales agreed with the Poorlandian government. The respective demand functions for both countries are:</p>
<p><span class="math-container">$Q_{poor} = 10 - P_{poor}$</span> and <span class="math-container">$Q_{rich} = 14 - 0.5P_{rich}.$</span></p>
<p>The total cost, TC, in millions, depends on the total amount of units produced <span class="math-container">$Q = Q_{rich} + Q_{poor}$</span> and also includes 5 million of fixed costs.</p>
<p>TC <span class="math-container">$= 5+2Q + \frac{Q^2}{8}$</span></p>
<p>I. What is the profit-maximization strategy? Please indicate the prices and quantities, as well as the amount of social welfare generated.</p>
<p>II. Also are the prices above consistent with the rule of elasticities?</p>
<p>Can someone please help me with this?</p>
<p><span class="math-container">$\textbf{My Idea:}$</span> I took the derivatives of the TC to find MC. I isolated <span class="math-container">$P$</span> for poor and rich and multiplied it by <span class="math-container">$Q$</span> to find each revenue function, then subtracted it from TC to get the Total revenue, then I took the derivative to find MR = MC; which is this the maximization for a monopoly. Is this path the optimal one? Can someone please help me solve these? Thank you for your time and help.</p>

<microeconomics><elasticity><industrial-organisation><monopoly><profit-maximization>
Profit-maximization for a monopoly
rudinsimons12
# Profit-maximization for a monopoly

(Asked by: username_0 on 2020-12-13)

Between two countries, Richland and Poorland, with a strict ban on cross-border sales agreed with the Poorlandian government. The respective demand functions for both countries are:

$Q\_{poor} = 10 - P\_{poor}$ and $Q\_{rich} = 14 - 0.5P\_{rich}.$

The total cost, TC, in millions, depends on the total amount of units produced $Q = Q\_{rich} + Q\_{poor}$ and also includes 5 million of fixed costs.

TC $= 5+2Q + \frac{Q^2}{8}$

I. What is the profit-maximization strategy? Please indicate the prices and quantities, as well as the amount of social welfare generated.

II. Also are the prices above consistent with the rule of elasticities?

Can someone please help me with this?

$\textbf{My Idea:}$ I took the derivatives of the TC to find MC. I isolated $P$ for poor and rich and multiplied it by $Q$ to find each revenue function, then subtracted it from TC to get the Total revenue, then I took the derivative to find MR = MC; which is this the maximization for a monopoly. Is this path the optimal one? Can someone please help me solve these? Thank you for your time and help.

**Tags:** microeconomics, elasticity, industrial-organisation, monopoly, profit-maximization

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## 1 answers

### Answer by username_1 on 2020-12-14. Score: 3 | Accepted answer

A monopolist maximizes profit. For me, it is usually easier to do this in the quantity space. So you rearange the demand and maximize
$$\max\_{Q\_p,Q\_r} \quad P\_p(Q\_p)Q\_p + P\_r(Q\_r)Q\_r - TC(Q\_p+Q\_r)$$
$$\max\_{Q\_p,Q\_r} \quad (Q\_p-10)Q\_p + (28-2Q\_r)Q\_r - 5-2(Q\_r+Q\_p) -\frac{(Q\_r+Q\_p)^2}{8}$$
The FOC gives you two equations with two unknowns, $Q\_r$ and $Q\_p$, which you solve for. Both FOC equations can be written in the form $MR=MC$. You then plug these $Q$s into the demand equations to get the prices. Next, you calculate consumer surplus and producer surplus and add them to get social welfare.


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