1.           Partial Derivatives Practice:

Find  fx   , fy  ,  fxx  ,  fyy,   fxy    , and  fyx    for the following functions:

a.          f ( x , y ) =  xy

b.          f ( x , y ) =  exy

c.          f ( x , y ) =  5x2   -  4xy2   +  3x2y5   - 2x  +  y

2.          The market for tickets to this year’s “Live With Barry Manilow” concert is characterized by the following supply and demand schedules:

QD   =  20,000 – 100P                                          P:  $/ticket

QS   =   10,000  +  100P                                       Q:  tickets

Quantity supplied can respond to price because it’s an outdoor concert, and the organizers can add bleachers if necessary.

a.          Find the equilibrium price and quantity in the Manilow concert market in the absence of government intervention.

b.          The government, deluged by loyal Manilow fans (Fanilows) who were shut   out of last year’s extravaganza because of high ticket prices, decides to put a ceiling of $40 on the price of each ticket to this year’s show.  Show both the   ceiling situation and the original equilibrium in a graph.  Identify any              shortage or surplus.  Is the ceiling situation an “equilibrium” situation?  Why or why not?  What unintended consequences might cause us to question the government’s actions on efficiency grounds?  Explain.

c.          Bucky the Brilliant, having just finished studying economics at UW-Madison, knows that the price ceiling offers a unique opportunity for profit.  He buys  up ALL the available seats at the $40 ceiling price and scalps them for as        much as he can.  Assuming he charges only one price for his scalped tickets,  what will his profit be?  Show this situation in a supply/demand diagram.      Can you label areas representing his revenue, cost, and profit?

d.          With your understanding of my love of “synthesis,” could Bucky do better by flushing a few of his scalped tickets down the toilet?  Why or why not?  What other concept might allow you to figure an even greater gain for Bucky than  the one you calculated in (c)?  Explain.  [Hint:  When Bucky buys up all the     tickets available at the ceiling price, is he not now a "monopolist" in the          market for tickets?]

3.          The table below shows the retail price and sales for instant coffee and for roasted coffee for two years.

a.          Using these data alone, estimate the short-run own-price elasticity of demand for roasted coffee.   Derive a linear demand curve for roasted coffee.

b.          Now estimate the short-run own-price elasticity of demand for instant coffee.  Derive a linear demand curve for instant coffee.

c.          Compare your answers to (a) and (b).  Which type of coffee has the higher own-price elasticity (is more elastic)?  Explain your results  with reference to those factors that drive demand elasticity values.

Year

Retail Price

of Instant

( $ / lb. )

Sales of

Instant

(  lbs. )

Retail Price

of Roasted

( $ / lb. )

Sales of

Roasted

( lbs. )

1                         $10.35                         75 m.                       $4.11                            820 m.

2                         $10.48                         70 m.                       $3.76                            850 m.

Note:    Sales are in millions of pounds.  So, “75 m.” denotes 75 million pounds sold.

4.          Draw indifference curves that represent the following individuals’                  preferences for hamburgers and soft drinks.  Indicate the direction in which the individuals’ satisfaction/utility/happiness is increasing.

a.          Joe dislikes both hamburgers and soft drinks and has convex preferences for them.