Managerial economics guide 4

Optimal pricing and mark-up pricing

Optimal pricing is normally called mark-up pricing in which, from differentiating one condition, we get to the rule of price over mark-up and marginal cost. This comes directly from the first condition. The optimal price will depend on the elasticity of demand, as you can see, we are considering the elasticity to price as a relative value that includes its sign, so if you’re writing this, it would normally be negative.

Elasticity and demand

Own-price elasticity is normally a negative value because the demand function is normally downward sloping, and in absolute value, the elasticity should be greater than 1. A monopoly has to face a demand which is elastic, so it shouldn’t be less than 1, which will give us a negative price greater than 1. Negative mark-up doesn’t make sense.

If you have an elastic demand, you will have a negative marginal revenue, which means to increase the revenue you have to increase the quantity offered to the market. This goes up to 0, but at 0, you don’t have anything, so there is a discontinuity. The optimum is 0, but 0 is senseless because you don’t get any revenue at all. So a monopolistic demand makes sense only if demand is elastic.

Second condition and elasticity

We should always check for the validity of the second condition which we didn’t talk about but it’s there. The first derivative should be equal to 0 to maximize profit, and the second derivative should be smaller than or equal to 0. Let’s suppose that the second condition is always fulfilled. What can be seen here is the higher the elasticity demand to price in absolute value, the smaller the mark-up.

The Cobb-Douglas utility function uses income and elasticity to own price and income equal to 1. If you have higher elasticity, this ratio goes down and the mark-up is small and almost equal to 1, which means you will have a low mark-up. On the other hand, if the elasticity is small (-2), then you can double your marginal cost. If it was 1.5, it would be even bigger. So, the closer the absolute value of elasticity to 1, the higher your mark-up (cannot be 1), and the higher the coefficient which multiplies the marginal cost to get the optimal price.

Rule of thumb and market discrimination

Rule of thumb: The lower the elasticity, the higher the mark-up that a monopoly will charge. If you have two markets and there’s no arbitrage, so people cannot buy the goods on a market where it’s cheaper then sell it on another market where it’s more expensive. Suppose there is no link between the two markets' prices. What happens is the market with lower elasticity always above 1 will have the higher price for the same good. So, a monopolist who can discriminate perfectly between two markets with two different demand functions will give a higher price in the market where elasticity is lower to maximize total profit.

Price discrimination

Price discrimination works by differentiating with respect to different demand elasticities. The price will be higher when the demand elasticity is lower, which is how price discrimination functions.

Problems and examples

In order to use excess end-of-the-year inventory, Henry Ford offered a 1% discount on the average price of the car.