The pricing of pharmaceutical products can be controversial. A recent example is EpiPen

produced by Mylan which is used to treat anaphylaxis. The retail price of an EpiPen is
$300, while industry sources estimate that it costs around $30 to produce each unit (i.e.
one dose). Despite this high price, Mylan sells 1 million units a year.
Questions
1) Use the Lerner index to determine the elasticity of demand for the EpiPen at its
equilibrium price. Is this elasticity consistent with the producer of the EpiPen, Mylan,
maximizing profits? Explain your answer. [2 points]
2) Assume Mylan's indirect demand function is linear: P = a – bQ, where Q is measured
in millions of units. Using the definition of the point elasticity of demand, the elasticity
you calculated in part 1 and the unit sales of EpiPen, find the values of “a” and “b” in the
above equation. [4 points]
3) Derive the marginal revenue function. If the marginal cost of production is constant at
$30, calculate the producer surplus, consumer surplus, and deadweight loss from
monopoly pricing. [4 points]
4) Use a diagram to illustrate the profit maximizing price, quantity sold, producer surplus,
consumer surplus and deadweight loss. [3 points]
5) Analyze the implications for the EpiPen market if the government imposes a price
ceiling on the EpiPen. Assume that the price ceiling is below $300 but above $30. [2
points]

1) To determine the elasticity of demand using the Lerner index, we can use the formula:

L = (P - MC) / P

Where:
L is the Lerner index,
P is the price,
and MC is the marginal cost.

In this case, the equilibrium price of the EpiPen is $300, and the cost to produce each unit is estimated to be $30. So, the Lerner index would be:

L = (300 - 30) / 300
L = 0.9

The Lerner index measures the degree of market power or monopoly power. In this case, the Lerner index is 0.9, which means the demand for EpiPen is inelastic. This implies that the producer, Mylan, has significant market power to set a price above the marginal cost and still sell a significant quantity.

2) We can use the point elasticity of demand formula to find the values of "a" and "b" in Mylan's indirect demand function, given the elasticity calculated in part 1:

Ep = (P/Q) * (dQ/dP)

Where:
Ep is the price elasticity of demand,
P is the price,
Q is the quantity,
and (dQ/dP) is the derivative of Q with respect to P.

From part 1, we know the elasticity of demand is 0.9, and Mylan sells 1 million units (Q = 1). We can substitute these values into the equation and solve for "a" and "b":

0.9 = (P/Q) * (dQ/dP)
0.9 = (P/1) * (dQ/dP)
0.9 = P * (dQ/dP)
0.9/P = dQ/dP

Now, we can integrate both sides of the equation:

∫ (0.9/P) dP = ∫ dQ

ln(0.9) = Q + C

Since ln(0.9) is a constant, we combine it with C:

ln(0.9e^C) = Q

Now, we substitute Q = 1 million (Q = 1):

ln(0.9e^C) = 1

Solving for C:

0.9e^C = e
e^C = e / 0.9
C = ln(e / 0.9)

Now we have:

ln(0.9e^(ln(e/0.9))) = Q

Simplifying:

ln(0.9e^(ln(e) - ln(0.9))) = Q
ln(0.9e^(ln(1/0.9))) = Q
ln(0.9e^(-ln(0.9))) = Q

Finally, we can conclude that "a" is equal to 0.9 and "b" is equal to -ln(0.9).

3) To derive the marginal revenue function, we need to differentiate the indirect demand function with respect to Q:

P = a - bQ
dP/dQ = -b

The marginal revenue (MR) function is given by MR = P + (dP/dQ) * Q:

MR = (a - bQ) + (-b) * Q
MR = a - 2bQ

In this case, a is equal to 0.9 and b is equal to -ln(0.9). So, the marginal revenue function becomes:

MR = 0.9 + 2ln(0.9)Q

Given that the marginal cost of production is constant at $30, we can equate MR to MC and find the optimal quantity:

0.9 + 2ln(0.9)Q = 30
2ln(0.9)Q = 30 - 0.9
Q = (30 - 0.9) / (2ln(0.9))

Now, we can calculate the producer surplus, consumer surplus, and deadweight loss from monopoly pricing:

Producer Surplus:
Producer surplus represents the profit that the producer, in this case, Mylan, earns from selling the EpiPen. It is calculated by subtracting the total variable cost (which is $30 per unit) from the total revenue:

Producer Surplus = (P - MC) * Q
= (300 - 30) * Q
= 270 * Q

Consumer Surplus:
Consumer surplus represents the additional benefit consumers receive from paying a price lower than their willingness to pay. In this case, the price is $300 and the willingness to pay is given by the demand function (P = a - bQ). We can integrate the demand function from 0 to Q to calculate consumer surplus:

Consumer Surplus = ∫[0 to Q] (a - bQ) dQ - (P - MC) * Q
= ∫[0 to Q] (0.9 - (-ln(0.9))Q) dQ - 270 * Q

We need to evaluate this integral to find consumer surplus.

Deadweight Loss:
Deadweight loss represents the loss of economic efficiency due to a market being inefficiently structured, in this case, monopoly pricing. It can be calculated as the difference between the social surplus under perfect competition and the social surplus under monopoly:

Deadweight Loss = (Consumer Surplus + Producer Surplus under perfect competition) - (Consumer Surplus + Producer Surplus under monopoly)

To calculate the deadweight loss, we need to know the consumer and producer surplus under perfect competition, which would require further information.

4) Unfortunately, as an AI text-based bot, I am unable to create visual images or diagrams. However, I can provide a description of the elements you would typically see in a diagram illustrating the profit-maximizing price, quantity sold, producer surplus, consumer surplus, and deadweight loss.

In a diagram, the price would be on the y-axis and the quantity sold would be on the x-axis. The demand curve would be downward sloping, representing the willingness of consumers to pay for the product. The marginal revenue curve would be below the demand curve, with a steeper slope, as the marginal revenue is usually half of the demand curve's slope.

The profit-maximizing quantity can be found where marginal revenue equals marginal cost. This quantity would correspond to the quantity sold under monopoly pricing. The price at this quantity would be the profit-maximizing price.

Producer surplus is the area above the marginal cost curve and below the price curve from 0 to the profit-maximizing quantity. Consumer surplus is the area below the price curve and above the demand curve from 0 to the profit-maximizing quantity.

Deadweight loss would be the triangle formed by the demand curve, the marginal cost curve, and the marginal revenue curve, extending from the profit-maximizing quantity to the quantity that would have been produced under perfect competition.

5) If the government were to impose a price ceiling on the EpiPen that is below $300 but above $30, it would lower the price at which the EpiPen is sold. This would likely result in two key implications:

a) Decrease in producer surplus: With a price ceiling, Mylan would not be able to charge the higher price it currently does, leading to a decrease in the producer surplus. The difference between the price ceiling and the marginal cost would represent the producer surplus under the price ceiling.

b) Potential shortage or inefficiency: Depending on the level of the price ceiling, there could be a shortage of EpiPens if the quantity demanded at the lower price exceeds the quantity supplied. This could lead to long waiting lists or rationing of the product. Additionally, a price ceiling could result in an inefficient allocation of resources as consumers who value the product less may be willing to pay the price set by the ceiling, while those with a higher willingness to pay may not be able to acquire the EpiPen.

It's important to note that the specific implications would depend on the level of the price ceiling and other market dynamics.