1. To find the market demand curve for this monopolist we will need to sum the individual group demand curves together horizontally. For prices that are greater than or equal to $5 per unit, class two does not have any demand for the product since the product is too expensive for them. So, for prices greater than or equal to $5 per unit, the market demand curve is just class one’s demand curve or P = 10 – Q. For prices less than or equal to $5 we need to do a bit more work. I sketched the two groups demand curves on two separate graphs arranged horizontally next to each other and then found at $5 per unit, only 5 units of the good were demanded by class one and zero units by class two: so, one point on the market demand curve would therefore be (Q, P) = (5 units, $5 per unit). Then I thought about the two x-intercepts for these demand curves and added the two x-intercepts together: the x-intercept for class one is 10 units and the x-intercept for class two is 10 units as well. That gives me another point on the market demand curve: (Q, P ) = (20 units, $0 per unit). Use these two points to calculate the slope: m = rise/run = -5/15 = -1/3. Then use the general form of the y-intercept form of the equation and one of the given points to find the value of the y-intercept, b: Y = mX + b or P = mQ + b and then using (Q, P) = (20, 0) you have 0 = (-1/3)(20) + b or b = 20/3. Thus, the market demand curve for prices less than or equal to $5 per unit is therefore P = 20/3 – (1/3)Q.

1. From (a), we have two demand curves: P = 10 – Q for P greater than or equal to $5 and P = 20/3 – (1/3)Q for prices less than or equal to $5. That implies that there are two MR curves that we need to consider: MR = 10 – 2Q and MR = 20/3 – (2/3)Q. We know MC = 2.

So, let’s start by equation MC = MR and use the first MR equation: 2 = 10 – 2Q or Q = 4. Then, plug this quantity back into the relevant demand equation to find the single price: P = 10 – Q = 10 – 4 = $6 per unit. Let’s pause for a minute and think about this quantity and price combination: the quantity is less than 5 units and the price is greater than $5 per unit. This quantity and price combination is one that fits on the first segment of the market demand curve: the market demand curve P = 10 – Q where P is greater than or equal to $5 per unit.

Let’s look at the second MR equation and use it in a similar way: 2 = 20/3 – (2/3)Q or Q = 7. If we plug this quantity back into the demand equation to find the single price: P = 20/3 – (1/3)Q = 20/3 – (1/3)(7) = $4.33. Let’s pause for a minute and think about this quantity and price combination: the quantity is greater than 5 units and the price less than $5 per unit. This quantity and price combination is one that fits the lower segment of the market demand curve: the market demand curve P = 20/3 – (1/3)Q is for where P is less than or equal to $5 per unit. So, we can use this combination to figure out the firm’s profits and see if this results in a better outcome than the price of $6 and the quantity of 4 units. TR = (13/3)(7) = 91/3 (you will see in a minute why I am keeping this as a fraction). TC = 2(7) = 14. Profit with this price and quantity is equal to 91/3 – 42/3 = 49/3 = $16.33. This is a higher level of profits than the firm gets when it sells only to the first segment of the market demand curve.

So, the firm charges $4.33 per unit and produces 7 units.

1. If this firm practices third degree price discrimination we will need to figure out what total quantity of the good the firm will produce and then how the firm will allocate this production between the two classes of buyers. From (b) we know that the lower segment of the market demand curve intersected the MC curve at 7 units. Let’s call this the total output that the firm will produce. Then, let’s consider how to allocate these 7 units of output.

In class one, the MC of producing the good is $2 per unit. The MR curve for class one buyers is MR = 10 – 2Q. Setting MR = MC for this class of buyer we get 10 – 2Q = 2 or Q = 4 units. Class one should get 4 units of the total output of 7 units allocated to the buyers in this class: that implies that class two will get the remaining three units (we will check that in just a bit). What price will class one pay? Substitute Q = 4 into the demand curve for this class: P = 10 – Q = 10 – 4 = $6 per unit. That implies that TR from class one will be $24. TC for this level of production for class one will be ($2 per unit)(4 units) = $8. Profit from class one will be $16.

In class two, the MC of producing the good is $2 per unit. The MR curve for class two buyers is MR = 5 – Q. Setting MR = MC for this class of buyer we get 5 – Q = 2 or Q = 3 units. Class two should get 3 units of the total output of 7 units allocated to the buyers in this class: note that Qtotal = Q1 + Q2 where Q1 is the amount of the good provided to class one and Q2 is the amount of the good provided to class two. What price will class two pay? Substitute Q = 3 into the demand curve for this class: P = 5 – (1/2)Q = 5 – (1/2)(3) = $3.50 per unit. That implies that TR from class two will be $10.50. TC for this level of production for class two will be ($2 per unit)(3 units) = $6. Profit from class two will be $4.50.

Total profit from the two classes of buyers will equal $16 + $4.50 or $20.50.

Since total profit from third degree price discrimination is greater than total profit from selling the good at a single price we can conclude that third degree price discrimination is worthwhile to this monopolist.