Monopoly and monopolistic revenues, equations, elasticities, and price discrimination

This post is going over a question recently received:

You have been assigned the task of helping the Midland Milk Marketing (MMM),
the sole marketing agency for milk produced in the Island State of Midland. In the past the producers have been marketing their milk as a homogeneous commodity to all processors of milk. You have studied the market extensively and realize that the market can be segmented into two separate units: 

(1) the market for fluid milk (milk for drinking) and (2) the market for processing milk (for manufacture of cheese, etc.). Your preliminary analysis has generated the following demand curves for the two separate markets.

Fluid milk market:

Inverse demand curve: Pfluid= 22 – 2.5*Qfluid
Marginal revenue curve: MRfluid = 22 – 5*Qfluid

Processing milk market:
Inverse demand curve: Pprocessing = 20 – 3*Qprocessing
Marginal revenue curve: MRprocessing = 20 – 6*Qprocessing

Assume that individual firms have the same cost function which is as follows: TC = 10 + 2*Q

1. What is the profit-maximizing allocation of milk production in each of the markets? Assuming that MMM can practice price-discrimination in the market, what is the profit-maximizing price and quantity in each market?

2. What is the total revenue for the MMM using price discrimination?

3. Calculate the own price elasticities of demand for fluid and processing milk at the equilibrium values for P and Q.

4. Illustrate the effects of price discrimination with a graph of both the fluid milk market segment and the processed milk market segment. Your graph should include a demand curve, a marginal revenue curve, and the profit-maximizing price and quantity under price discrimination. If the price without market discrimination is $11.50, which segment of the market benefits from market discrimination?

The first thing we need to do is find the profit maximizing quantity.  To do this, we have to remember that for a any business (including monopolies) the profit maximizing point occurs when marginal revenue is equal to marginal cost (MR=MC).  So first we have to identify what marginal revenue is, and luckily that is given above.  In order to calculate marginal cost, you can either take the derivative of total cost (TC) or find the slope of the TC curve.  Either way you will end up with MC=2.  

The next step is to set MR=MC, so for the fluid milk market, we set:

MC=MR or
2 = 22-5*Q

Now solve for Q by subtracting 2, and adding 5Q to both sides.  Then divide by 5 and you will get:

So the equilibrium quantity for the fluid milk market is 4.  To get equilibrium price, plug in equilibrium quantity into the price equation:
P=22-2.5*4 =22-10=12
So equilibrium price will be 12.

Let’s do the same process for the next market:
MC=MR or
2 = 20-6Q

Now solve for Q by subtracting 2, and adding 6Q to both sides.  Then divide by 6 to get:

Plug our equilibrium quantity into the price function for the processing market to get equilibrium price:
So equilibrium price in the processing market will be 11.

If our producer is allowed to price discriminate, then the equilibrium price will be used in either market, so revenue will be price multiplied by quantity.  This results in revenue of 12*4=48 for the fluid market and revenue of 11*3=33 for the processing market.

To calculate the own price elasticities of demand for a static point, we can use a trick using the slope of the demand function.  In this case, the price elasticity of demand equation will be:

Elasticity = (P/Q)*(dQ/dP) or E=(P/Q)*(change in Q/ Change in P)
This can also be written as E = -bP/Q = -bP/(a-bP)
Where our linear demand curve is q=a-bp.

So first we need to make our inverse demand function into a normal demand function by solving for Q.  We get this by adding 2.5Q to both sides (of our fluid market) and subtracting P from both sides, then dividing by 2.5.  This gives us:

Q = 8.8 - 0.4P

So in the above equation, a = 8.8, and b = 0.4.  P and Q are still 12 and 4 respectively.
Plug these values into our price elasticity of demand measure to get:

E = -0.4*12/(8.8-0.4*12) = -4.8/4 = -1.2

So our point price elasticity of demand measure at equilibrium price and quantity for the fluid market is -1.2.

Using the same method for the processing market we get:
Q = 6.67 - .33 P

After plugged into our elasticity function gives us:
E = -.33P/(6.67-.33P) = -3.67/3 = -1.22

So the price elasticity of demand for both markets is elastic, this means that if revenue for the firms could be increased if prices went down (but just because revenue would rise, doesn’t mean profit would, because we have positive costs).

Finally, the graphs for the two markets would look like the graph to the right.  Here we can see the MR=MC at a point where P (in dollars) is $2, and quantity is 4.  However, when we figure out what the price paid by the consumer will be, by drawing the line up to intersect with the demand curve, we get a price paid of $12.  

 The next graph to the left shows the monopoly market for the processed milk.  Here, MC=MR again at a price of $2, but since the MR curve is steeper, the quantity is only at 3.  When we draw the line up to intersect the demand curve, we see that this intersection occurs at a price of $11.

And if we put the graphs together, we get a very complicated graph, be we can see how the firm is price discriminating in the two different markets.

And if the price of milk was $11.50 with the opportunity for price discrimination than the fluid market would lose revenue because they are currently charging $12 for their milk, while the processing market would receive $0.50 more because their current price is only $11.

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