Question

Use your knowledge about price-searching firms and two-part pricing to advise the company below.

The company has a bar and is trying to decide on the cover charge (if any) and price for each drink. It has done a modest survey to ask customers to classify themselves as light drinkers or heavy drinkers and to indicate the number of drinks they would typically consume during the evening at various possible prices.

The estimate from the study is that a change in the price equal to $1 per drink causes light drinkers to change their consumption on average by 0.5 drinks per night. However, a change in price of $1 causes heavy drinkers to change their consumption on average by 1.0 drink per night. For both groups a typical consumer will not consume anything once the price reaches $9 per drink. (Customers might instead go to another bar or not go to a bar at all.)

(Note the distinction between dQ/dP and dP/dQ, which is its inverse.) Draw an inverse demand curve for a typical light drinker and for a typical heavy drinker on the same diagram. Explain your diagram. Write equations for the curves in slope-intercept form.

If 300 people visit the bar on a typical evening, with 200 people being light drinkers and 100 people being heavy drinkers, draw an overall (inverse) demand curve for all of the consumers combined. (A good way to start is with a price of $9. Then determine what would happen if the price were reduced all the way to zero. You would then be able to plot on a diagram the total quantity demanded at $9 and the total quantity demanded at $0. Connect the two points involved with a straight line and determine its slope.)

What is the slope and what is the intercept for this (total) demand curve? Write an equation in slope-intercept form.

Recall that, in the case of a straight-line demand curve, the slope of the marginal revenue line for a company that does not practice price discrimination is double the slope of the (total) market demand curve.        

If the marginal cost of making drinks (the alcohol, the bartender’s labor, and the amortized cost of purchasing glasses and cleaning them repeatedly) is constant at $5 per drink, and if no cover charge is assessed, what is the best price to charge for drinks? How many drinks would be sold on a typical evening? What would your profits be? Show your work. What would be the point price elasticity of demand at the profit-maximizing price? (Find the quantity where marginal revenue equals marginal cost, and then use your equation for the total demand curve to determine the price to charge.)

However, our last two-part pricing slide tells us that a monopoly user charge is too high from the standpoint of two-part pricing. If you cut your price by $1 per drink AND assess the maximum possible cover charge without causing a typical light drinker to refuse to enter the bar, would your profits improve? How high would the cover charge be? Calculate both the cover charge and your total profits. Would the new pricing increase profits? Show your work.

(Using calculus). Maybe the best price cut is not exactly $1. Write a profit equation. Profits equal total revenue minus total cost. Total cost equals $5 times the number of drinks sold. Total revenue equals the price for drinks times the number of drinks sold, PLUS 300 people times the cover charge. The cover charge equals, for a light drinker, the triangle of consumer surplus above the price but below the demand curve for a light drinker. (The area of a triangle equals one half the base times the height.)

You will take the derivative of the profit equation with respect to P or Q and set it equal to zero. For example, use the equation for the total demand curve and solve for Q in terms of P. Then in the profit equation substitute in an expression involving P in place of every ‘Q’ that was in the original profit equation. Now you can take a derivative of profits with respect to P and set the derivative equal to zero. Eventually you can solve for the exact best P, Q, and cover charge.  

Answer 1