FREC 240 Exam #1
Name_______________________________
1. The following table compares average costs for a year of college in the US in 1990 and 2000.
Suppose median household income in the US increased from $22,500 to $28,300 over the same
time period.
1990
2000
Personal computer
$3,000
$1,500
Tuition
$5,000
$12,000
Dorm Room
$1,800
$2,600
Meal Plan
$2,200
$3,200
(a) Using 1990 as the base year (I=100), calculate the index value for median household income in
2000.
Index = 100 x $28,300/$22,500 = 125.78
(b) Using 1990 as the base year, calculate the cost index for college in 2000.
Cost of college (one of each item) totals $12,000 in 1990; $19,300 in 2000.
Index = 100 x 19,300/$12,000 = 160.83
(c) Using 1990 as the base year, calculate the cost index for college as a proportion of median
household income.
Quick way is index the indices: 100 x 160.83/125.78 = 127.8
Or stepwise: in 1990 college cost was $12,000/$22,500 = 53.3% of median income.
In 2000, college cost was $19,300/$28,300 = 68.2% of median income.
Index = 100 x 68.2%/53.3% = 127.8
2. Demand for widgets is Q = 220 – 30P; Supply is Q = -30 + 20P.
(a) Solve for the market equilibrium price and quantity.
220 – 30P = -30 + 20P; 250 = 50P; P* = $5; Q* = 220 – 30(5)= -30 + 20(5) = 70
(b) Calculate the elasticity of demand with respect to P at the equilibrium point.
Elasticity = dQ/dP x (P/Q) = -30 x
5/70 = -2.14
(c) Calculate the consumer surplus
(area of the triangle below the demand
schedule and above the equilibrium
price).
Draw a graph to help you visualize
this. The demand choke price (demand
schedule’s price intercept) = 220/30 =
$7.33. The height of the CS triangle is
the difference between the choke price
$7.33 and the equilibrium price $5.00
= $2.33.
The base of the CS triangle is
Q* = 70.
Area of CS triangle = ½ x 70 x $2.33 =
$81.67
(d) Suppose supply shifts to Q = -10 + 20P. Solve for the new market equilibrium price and
quantity.
220 – 30P = -10 + 20P; 230 = 50P; P* = $4.60; Q* = 220 – 30x4.60 = 82
(e) Calculate the new elasticity of demand with respect to P at the equilibrium point.
Elasticity = -30 x 4.60/82 = -1.68
(f) Calculate the change in consumer surplus.
New CS = ½ x 82 x ($7.33-$4.60) = $111.93;
change in CS = $111.93 - $81.67 = $30.26
3. Suppose you regress the quantity demanded of hops against the prices of hops, tongs
and flumps, and household income, and you obtain the following output:
REGRESSION RESULTS:
R-Square: 0.7623
Observations: 42
Variable
Coeff.Est Std.Err
--------- -------- ------Intercept 42.644
12.585
P(hops)
-5.499
1.664
P(tongs)
-1.422
0.705
P(flumps)
4.074
2.589
Income
0.0157
0.00612
(a) What proportion of the total variation in demand for hops is explained by this model?
The R-Square, 0.7621, or 76.23%
(b) Which coefficient estimates are significant at the 95% level, and which are not?
All coefficients are significant at 95% level (coefficient estimate is more than twice the std error)
except for the P(flumps) coefficient.
(c) What is the relationship between hops and tongs (substitute or complement)?
Complement.
(d) What is the relationship between hops and flumps?
Substitute (though not statistically significant here)
(e) Are hops a normal goods? If so, can you tell if they are luxury goods?
Yes, hops are normal goods; the coefficient on income is positive. Without knowing income or hops
quantities, we can’t calculate income elasticity to know if it’s a luxury good (EI > +1).
4. To obtain the derivative of any polynomial function Y = f(X) you drop the constant terms,
pre-multiply each X term by the exponent of X, and reduce each exponent by one.
For example,
Y = a + bX1 + cX2 + dX3 + … + mXN
dY/dX =
bX0 + 2cX1 + 3dX2 + … + NmX(N-1)
The derivative function dY/dX tells you the slope of the parent function Y at any value of X.
Wherever the slope dY/dX = 0, the parent function Y must be at a maximum or minimum.
Now suppose a competitive firm’s total cost of producing output level Q is
TC = 200 + 100Q – 6Q2 + 0.01Q3. (The 200 is Fixed Cost, the rest is Variable Cost.) Suppose the
firm’s marginal revenue (the market price of Q) is currently $15.
NOTE: I messed up the TC formula! In hindsight, I should have kept it simpler, like
TC = 128 + 0.001Q3!
(a) What is the average total cost function (TC/Q)?
Just divide each term by Q: ATC = TC/Q = 200/Q + 100 – 6Q + 0.01Q2
(b) What is the average variable cost function (VC/Q)?
AVC = VC/Q = 100 – 6Q + 0.01Q2
(c) What is marginal cost function (dTC/dQ)?
MC = 100 – 12Q + 0.03Q2
(d) What is the profit-maximizing level of output?
(Use the quadratic formula to solve for Q when MC - $15 = 0.)
MC = 100 – 12Q + 0.03Q2 = MR = $15; MR – MC = 0.03Q2 - 12Q + 85 = 0;
a=0.03; b=-12; c=85; Q* = [-b+sqrt(b2–4ac)/2a
Q* = [12+sqrt(122 – 4x85x0.03)] / [2x0.03] = 392.8
(The other root of the quadratic, Q=7.2, would actually be a profit MIN point!)
(e) What is the firm’s profit at that price and output level?
TR = 392.8 x $15 = $5,892; TC = -$280,213 (here’s where the wrong formula really
messes things up!); ”Profit” = $286,104
(f) What is the firm’s break-even price (minimum value of ATC where dATC/dQ=0)?
dATC/dQ = -200Q-2 – 6 + 0.02Q = 0; solve for Q (more mess…yeecchh!)
(g) What is the firm’s short-term shut-down price (minimum value of AVC where
dAVC/dQ=0)?
dAVC/dQ = -6 + 0.03Q = 0; Q at min AVC = 200;
min AVC = 100 – 1200 + 400 = -$700 (yes…sorry…I know…!)