MB $515$: Economic Theory and Application

Problem Set $6$

$6\mathrm{.}1$

At Silverado Golf Club, the demand for rounds of golf by each one of the $140$ identical senior golf members is given by $\backslash ($ D$\_\{\backslash$mathrm$\{$sp$\}\}\backslash )$ in the figure below. Silverado's annual fixed costs are $\backslash (\backslash$$ $1,250,000\backslash ),$ and variable costs are constant and equal to $\backslash (\backslash$$ $25\backslash )$ per round.

a$.$ If the manager of Silverado charges a uniform green fee to all senior golfers, the profitmaximizing green fee is $\backslash (\backslash$$ $\backslash )$ per round. Under this uniform pricing plan, Silverado's annual total revenue is $\backslash (\backslash$$ $\backslash )$ and total variable cost is $\backslash (\backslash$S $\backslash )$ Silverado's profit under uniform pricing is $\backslash (\backslash$$ $\backslash )$ per year.

b$.$ If the manager instead decides to employ a two$-$part pricing plan, the profit$-$maximizing green fee is $\backslash (\backslash$$ $\backslash )$ per round, and the annual membership charge is $\backslash (\backslash$$ $\backslash )$ The two$-$part pricing plan results in total annual profit of $\backslash$$

c$.$ Which pricing plan$$uniform pricing or two$-$part pricing$-$generates more profit for Silverado's owner? Is this the pricing plan you expected to be more profitable? Why or why not? $6\mathrm{.}2$

Suppose Silverado Golf Club in question $6\mathrm{.}1$ also has a second group of $140$ identical golfers, weekend players, who wish to play at the club. The demand for rounds of golf by each one of the $140$ identical weekend golfers is given by $\backslash ($ D$\_\{\backslash$text $\{$m $\}\}\backslash )$ in the figure below. Assume the same cost structure as given in question $6\mathrm{.}1.$ The manager designs an optimal two$-$part pricing plan for these two groups of golfers.

a$.$ The optimal green fee to set for each round of golf is $\backslash (\backslash$$ $\backslash )$

b$.$ The optimal annual membership charge is $\backslash (\backslash$$ $\backslash )$ for senior golfers and $\backslash (\backslash$$ $\backslash )$ for weekend golfers.

c$.$ Under this two$-$part pricing plan, Silverado's annual profit is $\backslash (\backslash$$ $\backslash )$

$6\mathrm{.}3$

A manager faces two separate markets and decides to price$-$discriminate.

The estimated demand functions for the two markets are

$\backslash [$

$\backslash$begin$\{$array$\}\{$l$\}$

Q$=2,250-40$ P $\backslash \backslash$

Q$=3,600-90$ P $.$

$\backslash$end$\{$array$\}$

$\backslash ]$

The long$-$run marginal cost is estimated to be

$\backslash [$

L M C$=2\mathrm{.}4+0\mathrm{.}001$ Q

$\backslash ]$

a$.$ How many units should the manager produce and sell?

b$.$ How should the manager allocate the profit$-$maximizing output between the two markets?

c$.$ What prices should the manager charge in the two markets?