Econ 301 – F07
Answers to PROBLEM SET 4 - due in class on Thursday Oct 25
Wissink
1. Derive/recall the demand functions for x & y for the utility function u(x, y) = xy. Calculate the ownprice elasticity of demand for x using the demand function for x you derived. Calculate the cross-price
elasticity of demand for x with respect to Py. Calculate the income elasticity of demand for x.
[ANSWER]
With the utility function, u(x,y)=xy, we get x*=I/2Px and y*=I/2Py.
The own price elasticity of demand for x, the cross-price elasticity of demand for x with respect to Py,
and the income elasticity of demand for x are in the followings:
x
Px
I
Px
 xD, Px 

 

1
2
Px
x
I / 2 Px
2 Px
x Py
Py
 xD, y 

 0
0
Py x
x
x I
1
I
 xD, I   

1
I x 2 Px I / 2 Px
2. Suppose there are two markets: the market for Butter, B, and market for Guns, G. Suppose the
demand curve in each market is linear where P=price and Q=quantity. That is: PB = K – sQB and
PG = F - tQG . Suppose these two demand curves intersect the price axis at the same value, i.e., K = F.
Using the exact point elasticity formula, show that if quantities are selected such that prices are equal in
these two markets, then so are their own-price elasticities of demand.
[ANSWER]
With K=F, we get sQB=tQG. from the following: PB=PG => K-sQB = F – tQG => sQB=tQG.
The own-price elasticity of demand for Butter is |-1/s|*PB/QB = PB/(s*QB) and the own-price elasticity of
demand for Guns is |-1/t|*PG/QG = PG/(t*QG) = PB/(s*QB) since sQB=tQG and PB=PG.
3. Suppose that on January 1st 2007, the Ithaca Transit Authority increased fares on all its bus lines
from 40 cents to 60 cents. Private bus companies in Ithaca did not change their fares. The Daily Sun of
January 8th reports that the total number of rides on city-owned buses decreased from 2200 to 1800 per
day; on those city lines competing with private bus lines, the number of rides decreased from 1150 to
850 per day.
a. Compute the own price elasticity of demand (over the one-week period) for all city-owned bus lines.
Compute the own price elasticity of demand for city-owned lines competing with private companies.
b. Is the demand for bus rides elastic or inelastic? Explain.
c. Why is the demand for rides on those city-owned lines competing with private lines more elastic
(than on other city lines)?
d. The Ithaca Transit Authority expects that, solely as a result of the fare increase on January 1st fewer
rides will be purchased in March than in January. Explain.
[ANSWER]
All city-owned bus
Old
New
Midpoint
Change
Percent change
Elasticity
Ticket sold
Price ($/ticket)
2200
0.40
1800
0.60
2000.00
0.50
-400.00
0.20
-20.00%
40.00%
-0.50
City-owned lines competing
with private companies
Price
Ticket sold ($/ticket)
1150
0.40
850
0.60
1000.00
0.50
-300.00
0.20
-30.00% 40.00%
-0.75
(a) The absolute value of the own price elasticity of demand (over the one-week period) for all cityowned bus lines is 0.5 and the own price elasticity of demand for city-owned lines competing with
private companies is 0.75. (b) In both cases the demand for bus rides is price inelastic. This is because
in both cases the own-price elasticity of demand is less than one. (c) It's more elastic because there are
more substitutes available, namely the privately owned bus lines. (d) Over time people can find
alternatives to Ithaca's city buses.
4. Suppose there are two states of the world: good and bad. In the good state of the world your income
is $250,000. In the bad state of the world your income is $40,000. The good state occurs with probably
.25 and the bad state occurs with probably .75. Suppose your utility function is: u = y1/2 where y =
income. What would an “income insurance” contract look like if it took away all your income risk and
left you indifferent between buying the insurance and taking the income gamble? How would your
answer change if the good state and bad state were equally likely?
[ANSWER] First find the $CE. Note that the Eu = .25*500 + .75*200 = 275 so that $CE=$75,625.
Now construct the contract. Consider the good state: 75,625 = 250,000 - Prem, so that Prem = 174,375.
Now consider the bad state: 75,625 = 40,000-174,375 + Ben, so that Ben = 210,000. Note that expected
profit = 174,375 - .75*210,000 = 16875.
When the good state and bad state were equally likely, Eu = .50*500 + .05*200 = 300 so that
$CE=$90,000. In the good state, you get 90,000=250,000 – NEWPrem, so that NEWPrem = 160,000.
In the bad state, you get 90,000=40,000-160,000+NEWBen, so that NEWBen = 210,000. The expected
profit = 160,000-0.5*210,000 = 55,000.
5. Contrast the concepts of diminishing returns to scale and diminishing returns to a factor of
production. Can a production function exhibit diminishing returns to scale but NOT have diminishing
returns to a factor?
[ANSWER] Diminishing returns to scale implies that if x=f(L,K) and y=f(sL, sK) where s>1, then y<sx.
All inputs must be scaled by the same factor, s, in the concept of returns to scale. With diminishing
returns to a factor of production, look at how the marginal product of a factor varies, holding the other
factors constant. The concept of diminishing returns to a factor of production implies that eventually the
marginal product of the factor declines. A production function CAN exhibit diminishing returns to scale
but NOT have diminishing returns to a factor. For example, consider the production function: x=L +
√K. Let L=1 and K=4x=3. Now let s=2, so that L=2 and K=8 and x=4.83 which is less than 2*3=6,
so you doubled inputs and less than doubled output, so we have decreasing returns to scale. But we DO
NOT have diminishing returns to the factor L. Its marginal product is constant and equal to 1, in this
case.
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6. Suppose the production function for widgets is given by:
Q=KL - .8K2 - .2L2, where Q represents the annual quantity of widgets produced, K represents annual
capital input, and L represents annual labor input. Using this production function, answer the following:
a. Suppose K=10. Graph the total and average product of labor curves. At what level of labor input
does this average product reach a maximum? How many widgets are produced at that point?
b. Again assuming that K=10 and using the information that the MPL curve is a straight line with an
intercept of 10 (for L=0), graph this curve. At what level of labor input does MP L=0?
c. Suppose capital inputs were increased to K=20, how would your answer to parts (a) and (b) change?
[ANSWER] With K=10. TPL = 10L – 80 -.2L2 and APL = Q/L = 10 – 80/L -.2L and also note that MPL
= dQ/dL = 10 - .4L. NOTE: that the economically relevant section of TPL is where it is both positive
and still increasing, so from L=10 to L=25. All other graphs would only be relevant from L=10 to L-25,
then, too. TPL reaches a max when MPL = 0. So setting MPL = 0 
10 -.4L = 0  L = 25. APL is at its max when APL=MPL, so set the two equal to each other and solve
for L  10 – 80/L - .2L = 10 - .4L  L=20. And note that when L=20, Qtotal=40. And APL = 0 at
L=10 and L=40. Also note that TPL=0 at L=10 and L=40.
From above, MPL=0 at L=25.If K=20, then: TP=20L-320-.2L2; AP=20 – 320/L -.2L and reaches a max
at L=40 and Qtotal=160; MP = 20 - .4L and MP=0 at L = 50. (You can draw the pictures for this, right?)
7. Suppose both Abe and Betty have very nicely behaved Cobb-Douglas production functions.
Suppose that Abe’s production function is: uAbe=f(X, Y). Suppose that Betty’s production function is a
monotonic transformation of Abe’s. That is uBetty = g(f(X, Y)) where (dg/df) is positive. Ima Smart
insists that Abe and Betty have the same underlying technology then. Evan Smarter insists that is not
the case. Who is correct and why?
[ANSWER]
Evan Smarter is correct! Production functions measure the relationship between inputs and outputs. If
we scale up a production function, we are generating more output for every input and we are altering the
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technology in a fundamental way. Note that with utility functions, the number itself doesn’t matter –
they are ordinal in content. However, the output of production functions is meaningfully measurable so
there is only one production function that will represent a given technology.
8. For each of the two production functions below discuss how the values for marginal product of labor
and average product of labor compare and contrast as the labor input expands:
a. Apples harvested (a) depend on hours of labor employed (L) as: a = 10 + 5L
b. Books dusted (b) depend on hours of labor employed (L) as: b = -10 + 5L
[ANSWER]
Apples: MPL = 5 and APL = 10/L + 5
Books: MPL = 5 and APL = -10/L + 5
MPL for apples doesn’t depend on the labor input. It is constant. Thus, the values for MPL don’t change
as the labor input expands. MPL for books is the same as MPL for apples. However, APL for apples
decreases and APL for books increases when the labor input expands. For example, the labor inputs
increase from 5 to 10. Then, APL for apples decreases from 7 to 6 and APL for books increases from 3
to 4.
9. The following chart shows some information about the demand function for oranges. Three distinct
sets of market data information were gathered on three separate occasions.
a. There are two elasticities that can be calculated from this table of information. Which two are they?
b. Set up the numbers and calculate these two elasticities.
c. What broad generalizations can you make about oranges from examining your answers?
Date 1
Date 2
Date 3
Qorange
10 lbs
15 lbs
8 lbs
Porange
$5/lb
$5/lb
$5/lb
Papple
$1/lb
$1/lb
$0.5/lb
Income
$100
$200
$100
[ANSWER] (a) You would be able to calculate two elasticities: the income elasticity of demand (using
dates 1 &2) and the cross-price elasticity of demand (using dates 1&3). (b) The income elasticity of
demand = (5/12.50)/(100/150) = .4/.67 = .60. The cross-price elasticity of demand = (2/9)/(.50/.75) =
.22/.67 = .33. (c) Oranges are a necessity and oranges and apples are substitutes.
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