Jacob: Sometimes the demand curve is written as Q in terms of P and sometimes it is written as P in terms of Q. Which way should we use?

Rachel: We can write supply and demand curves as P in terms of Q or Q in terms of P.

Jacob: Can you show how we compare elasticities at a given price?

Rachel: We write the demand curve as Q in terms of P:

At P = 10, the elasticities are

We chose P = 10 arbitrarily. For the homework assignment, you observe that this is true at all prices, or you observe that this is true at the equilibrium price and quantity.

Jacob: Is this what we expect in real life?

Rachel: That depends on the scenario. Some seniors are not working, so they have less current income and more free time. They are more likely to choose a lower cost product. Their higher price elasticity of demand leads to senior discounts for many products.

For some products, seniors have a lower price elasticity of demand. An older person may have less information about competing prices or may be disabled and is not able to price shop. The senior will remain with a given supplier even if a less expensive product is offered by another supplier.

Jacob: One candidate writes on the discussion forum that for P=21, the elasticity for adults is -5.25 and 3.5 for seniors, and for P = 19, the elasticity for adults is -3.167 and 4.75 for seniors. Is this correct?

Rachel: The price elasticity of demand is never positive. At P = 19 or P = 21, seniors don’t buy any of the good. At P = 15, the quantity demanded is zero. At P > 15, the quantity demanded remains zero.

Jacob: Is the price elasticity of demand zero for P > 15?

Rachel: The price elasticity of demand is not defined at P > 15.

Jacob: If the demand curve is linear, how we easily see which curve is more elastic?

Jacob: What is the total market demand curve?

Rachel: We add the two demand curves:

Q_total = Q_T = Q_A + Q_S = 150 – 6P + 120 – 8P = 270 – 14P

To add two demand curves, we must have Q in terms of P, not P in terms of Q.

Jacob: Do we write Q_T in terms of P_A + P_S or in terms of a single P?

Rachel: We write Q_T in terms of a single P. We ask: "What is the total quantity demanded in the market if we do not discriminate by group?" We are not asking: "What is the total quantity demanded by both groups if they each have their own price."

Jacob: How do we solve for the equilibrium price and quantity?

Rachel: We set marginal cost = marginal revenue. We write the demand curve as P in terms of Q. Both marginal cost and marginal revenue are in terms of Q, not in terms of P. They are the additional cost or revenue for one more unit, not for one more dollar of price.

Jacob: Is there a way to write marginal cost and marginal revenue in terms of P?

Rachel: Marginal cost has nothing to do with price, so we can’t write it in terms of P. For the demand curve, Q is a function of P, so we could write marginal revenue in terms of P. But marginal cost is function of Q, so we must have marginal revenue as a function of Q.

Jacob: How do we get the marginal cost curve for adults vs seniors?

Rachel: The marginal cost curve depends on the total quantity produced. The cost of a plane ticket, a train ticket, a theater seat, or a magazine is the same regardless who sits in the seat or reads the magazine.

Jacob: How do we get the marginal revenue curve?

Rachel: We use the demand curve for that market, which differs for seniors, adults, or the total. We write P in terms of Q. We then multiply P × Q, to get their product in terms of Q (not in terms of P).

The product of total revenue. The partial derivative of total revenue with respect to quantity is marginal revenue.

Jacob: Is there a simple formula for a linear demand curve?

Jacob: How do we solve the homework assignment for two separate groups?

Rachel: We write three equations in three unknowns. The equations are simple linear equations, so the arithmetic is easy.

The three unknowns are Q_A, Q_S, and Q_total. One linear equation is Q_A + Q_S = Q_total.

Jacob: For insurance we also set different prices by group, such as men vs women or old vs young. Do the same principles apply?

Rachel: Insurance is not price discrimination. Men and women have different costs, and we price each group separately.

Jacob: We have many pricing actuaries at my company, and no one estimates marginal revenue or marginal cost. Is Landsburg’s text realistic?

Rachel: Actuaries set rates for the next policy year in highly competitive markets. This is long-term competitive pricing. We set premium rates at the minimum total cost, including both variable and fixed costs.

Almost all insurance expenses are variables costs. They may vary with policy counts or exposures, not necessarily with premium, but they are still variable costs. This makes long-term pricing easy.

Short term pricing is done by underwriters, not by actuaries. (For reinsurance pricing, actuaries often work with the underwriters.) The underwriter must consider elasticities, sunk costs, and the other items that Landsburg discusses.

Good underwriters do this, but without the equations. Landsburg does not say that firms use these equations to price their products. Firms price this way by common sense, trial and error, and experience. If they price too high, they don’t sell the product. If they price too how, they lose money.