(SHOW ALL YOUR WORK; GRAPHS ON GRAPH PAPER OR PRINTED FROM EXCEL PLEASE!)

1. PERCENTS and INDICES

To calculate a percent change, divide the change amount by the starting quantity.
If the retail price of 87-octane gasoline rises from $1.49/gallon in July
to $1.79/gallon in August, what PERCENT INCREASE is that?

If the price of gas falls from $1.79/gallon to $1.49/gallon again, what PERCENT
DECREASE is that?  (Percent changes depend on direction of change.)

An INDEX shows trends in time-series data relative to a base (reference) time
period.  All the data are re-scaled so that the index value for the base period
equals 100.  Using July as the base period so that I(Jul)=100, the price index
for August is calculated--

 I(Aug)/I(Jul) = I(Aug)/100 = $1.79/$1.49
 I(Aug) = 100 * $1.79/$1.49 = 120.1

So relative to the base value of 100 in July, how does the August index match the
percent change you calculated?

Using first-quarter 1998 as the base year, index the following quarterly US real
GDP data (in $billions, from Economic Report of the President 2003; try using a
spreadsheet program like MS-Excel if you want--it's quicker!)

 QUARTER  REAL US GDP
 1998: I  8,396.3
      II  8,442.9
     III  8,528.5
      IV  8,667.9
 1999: I  8,733.2
      II  8,775.5
     III  8,886.9
      IV  9,040.1
 2000: I  9,097.4
      II  9,205.7
     III  9,218.7
      IV  9,243.8
 2001: I  9,229.9
      II  9,193.1
     III  9,186.4
      IV  9,248.8
 2002: I  9,363.2
      II  9,392.4
     III  9,485.6
 
 

2. CPI and WELFARE MEASURES

The best-known economic index is the consumer price index (CPI), which is
calculated from the overall cost of a fixed bundle of consumer goods and services
as the prices of these vary over time.  Here's a simplified example showing how
the CPI is calculated.

GOOD      2001 QUANTITY  2001 PRICE   2002 PRICE
eggs      3 doz.         $1.25/doz.   $1.45/doz.
flour     4 lbs          $0.49/lb.    $0.55/lb.
milk      4 qts.         $0.64/qt.    $0.46/qt.
apples    4 lbs.         $0.39/lb.    $0.32/lb.
potatoes  8 lbs.         $0.19/lb.    $0.24/lb.
beer      3 6packs       $2.10/6pack  $2.45/6pack

Using 2001 quantities, calculate the total cost of this bundle in 2001 and 2002.
Using 2001 as the base period, calculate the overall cost index for the entire
bundle in 2002.  This is known as a LESPEYRES INDEX, because it compares new versus
old prices based on the OLD consumption bundle.

Suppose the overall CPI increased by the same percentage between 2001 and 2002,
while your salary increased from $25,000 to $26,000.  Were you better off with
2001's prices and a salary of $25,000, or with 2002's prices and a salary of
$26,000?  Explain.

In reality, price changes cause consumers to change the quantities they consume:
they substitute away from costlier goods, and toward relatively cheaper goods.
So we would expect the consumption bundle in 2002 to be different:

GOOD      2002 QUANTITY  2001 PRICE   002 PRICE
eggs      2 doz.         $1.25/doz.   $1.45/doz.
flour     3 lbs          $0.49/lb.    $0.55/lb.
milk      6 qts.         $0.64/qt.    $0.46/qt.
apples    6 lbs.         $0.39/lb.    $0.32/lb.
potatoes  5 lbs.         $0.19/lb.    $0.24/lb.
beer      2 6packs       $2.10/6pack  $2.45/6pack

Using 2002 quantities, calculate the total cost of the 2002 bundle in both 2001
and 2002.  Using 2001 as the base period, calculate the overall cost index for the
2002 bundle in 2002.  This is known as a PAASCHE INDEX, because it compares new
versus old prices based on the NEW consumption bundle.

Using this Paasche index, were you better off with 2001's prices and a salary of
$25,000, or with 2002's prices and a salary of $26,000?  Explain.

If you calculate the average of the Lespeyres and Paasche indices, does this
average indicate were you better off with 2001 prices and a salary of $25,000,
or with 2002 prices and a salary of $26,000?
 
 

3. LINEAR DEMAND EQUATIONS

The manager of your local supermarket has collected the following data, showing
monthly sales of Golden Delicious apples (quantity in cartons) at various prices
per pound through the year:

 MONTH  PRICE/LB QUANTITY
  1     1.49     240
  2     1.69     210
  3     1.99     180
  4     1.89     175
  5     1.79     210
  6     1.69     225
  7     1.79     200
  8     1.49     225
  9     1.29     245
 10     0.99     305
 11     1.09     275
 12     1.29     270

Construct a graph of these data points (on graph paper please!), with quantity on
the horizontal axis and price on the vertical axis.  (Your graph should show a
clear downward-sloping trend; if it jumps around, you probably put the quantity
values on the horizontal axis in monthly order rather than in correct numerical
order.)  These data include some “noise” and don’t trace out an exact straight
line. Use a straightedge to draw a straight trend line through the graphed data
points.

Calculate the slope and intercept of this trend line (demand schedule) as
accurately as you can.

Fill in the intercept (a) and slope parameter (b) for the demand equation:

 PRICE = ______ + ______ x QUANTITY
                 (a)      (b)

Now calculate the inverse of this equation (solve for Q in terms of P to get a*
and b*.)

 QUANTITY = ______ + ______ x PRICE
             (a*)     (b*)

(Note: Economists normally express demand for Q in terms of P, so Q = f(P).
Demand equations specified as P = g(Q) are called "inverse" demand equations.)
 
 
 

4. ARC and POINT ELASTICITIES

Elasticity is a unit-free index of the percent change in one variable
associated with a one percent change in another variable.  For example, the
elasticity of demand for apples with respect to the price of apples is calculated
as the percent change in quantity demanced divided by a specified percent change
in price.

If the price of apples is $1.50/lb., what demand quantity does your apple demand
equation from Question 3 predict?
If the price of apples is $2.00/lb., what demand quantity does your equation
predict?

If the price of apples rises from $1.50 to $2.00/lb., what is the price elasticity
of demand for apples?
If the price of apples falls from $2.00 to $1.50/lb., what is the price elasticity
of demand for apples?

(Like the percent change calculations in problem #1, "arc" elasticities calculated
for discrete price changes depend on the direction of the price change.)

But if we know the demand equation, we can calculate POINT elasticities that don't
depend on the direction of change:

 Elasticity = (% change in Q) / (% change in P)
     = [(change in Q)/Q] / [(change in P)/P]
     = [(change in Q)/(change in P)] * [P/Q]  (just rearranging terms)

Notice that the numerator of this last formula, (change in Q)/(change in P), is the
the slope of the demand equation, like the b* term you calculated in Question 3.

If the price of apples is $1.50/lb., calculate the predicted quantity demanded and
the point elasticity of demand.

If the price of apples is $2.00/lb., calculate the predicted quantity demanded and
the point elasticity of demand.

The slope of a linear demand schedule is constant, but the elasticity of demand
with respect to price ranges from zero to infinity as you move from the quantity
axis intercept to the price axis intercept.

At what point (Q,P) on the predicted demand schedule is the price elasticity equal
to one?
 

5. MARKET EQUILIBRIUM

Graph the following supply and demand functions with Q (quantity) on the
horizontal axis and P (price) on the vertical axis:

 Q(demand) = 24 - 2P  Q(supply) = 6 + P

Determine the market equilibrium point where Q(demand) = Q(supply) from the graph.
Calculate the equibrium price and quantity mathematically.

Suppose demand increases, so that Q(demand) = 36 - 2P.
What is the new equilibrium price and quantity?

Suppose consumers get angry about this price increase, and get the government to
pass a law making it illegal for producers to charge any more than $8 per unit of
Q. How much will producers be willing to supply at a price of $8?
How much will consumers want to buy at a price of $8?

The $8 price ceiling causes a market shortage.  Suppose the government tries to
control demand for Q by giving out (via lottery) a rationing coupon for each unit
of Q produced, so buyers pay sellers $8 plus one coupon for each unit of Q.  Do
the coupons have any value?  If so, how much would the holder of a coupon be able
to sell it for?  (If the law forbids coupon dealing, there will simply be a black
market in coupons!)
 

6. TOTAL and MARGINAL UTILITY

Suppose a consumer's total utility for beer is a quadratic function:

 U = 12B - B^2

where U is utility and B is daily beer consumption.  Graph this utility function
for integer values of B between 0 and 12.

Marginal utility is the increase in total utility from each additional beer.
Graph the marginal utility function for the same values of B on the same graph.

What level of daily beer consumption maximizes this consumer's total utility?
(Beyond this point, beer stops being a "good" and becomes a "bad.")

Solve for the roots of this quadratic equation.  At what non-zero level of beer
consumption does this consumer's total utility return to zero?