Bob’s Bats produces baseball bats, and has the following costs: C(q)=5q2+720MC(q)=10q

and faces a market demand for bats: q=120−0.4p

where quantity is measured in thousands of bats

1. Write Bob’s Marginal Revenue function.

If we can find the inverse demand (of the form p=a+bx, we can simply double the slope (b) to get marginal revenue. We have the demand function, so solve it for p to get the inverse:

q=120−0.4pq+0.4p=1200.4p=120−qp=300−2.5q

This is the inverse demand function, so marginal revenue is

MR(q)=300−5q

2. Find the profit-maximizing quantity and price.

First, the quantity, we follow Rule #1 as always: the profit maximizing q∗ is where MR(q)=MC(q)

MR(q)=MC(q)300−5q=10q300=15q20=q⋆

Now that we know the profit-maximizing quantity, we need to find the maximum price consumers are willing to pay for 20 units. Plug this into the inverse demand function:

p=300−2.5qp=300−2.5(20)p=300−50p⋆=250

3. How much total profit does Bob’s Bats earn? Should Bob stay or exit this industry in the long run?

Total profit again can be found with Rule #2: π=[p−AC(q)]q

We first need to find the Average Cost function from total cost, by dividing it by q:

AC(q)=C(q)q=5q2+720q=5q+720q

Now we specifically need to find the average cost at 20 units:

AC(q)=5q+720qAC(20)=5(20)+720(20)AC(20)=100+36AC(20)=136

Now just plug in the price, average cost, and quantity:

π=[p−AC(q)]qπ=[250−136]20π=[114]20π=2,280

4. At what price would Bob’s Bats break even?

From before, we know that a firm’s break even price is at the minimum of its Average Cost curve, where Average Cost is equal to Marginal Cost. First let’s find the quantity where that happens:

AC(q)=MC(q)5q+720q=10q720q=5q720=5q2144=q212=q

This the quantity where AC is minimized and equal to MC. We need to find the price, so plug this quantity into either AC or MC. MC is easier here:

MC(q)=10qMC(12)=10(12)MC(12)=120

The firm breaks even at a price of $120.

5. How much of Bob’s price is markup (over marginal cost)?

Use the Lerner Index: L=p−MC(q)p. This will tell us what proportion of the price is markup above marginal cost.

First, we do need to find the marginal cost at q∗=20:

MC(q)=10qMC(20)=10(20)MC(20)=200

Now plug this and p∗ into the Lerner index:

L=p−MC(q)pL=250−200250L=50250L=0.20

The Lerner index says that 20% of the firm’s price ($250) is markup above marginal cost ($200).

6. Calculate the price elasticity of demand at Bob’s profit-maximizing price.

While you could calculate this manually, it’s a lot faster to use the full Lerner Index equation: L=p−MC(q)p=−1ϵ. Since we know L, we can set it equal to −1ϵ and solve for ϵ:

L=−1ϵ0.20=−1ϵ0.20ϵ=−1ϵ=−10.20ϵ=−5

Demand is elastic. For every 1% the price increases (decreases), consumers will buy 5% less (more).