Econ 3010: Intermediate Price Theory (microeconomics)
Professor David L. Dickinson
PROBLEM SET #1…….ANSWERS
BACKGROUND MATERIAL
1)
S
P
Price Support at $8
P
10
10
8
8
6
6
4
S
4
p*=3.50
p*= 3.50
2
D
2
4
6
q*=8
10
Q
2
D
2
4
6
q*=8
10
Q
Equilibrium is at P*=3.50, Q*=8. CS is the single-shaded area above the price, below the
demand curve, out to q, which means total CS=$19.50. PS is the double-shaded area
below p* but above the supply curve, out to q*, which means total PS=$12. All the
buyers purchase wheat in equilibrium, but Pat only purchases one bushel of the total three
that she would have been willing to buy. Only the low cost wheat farmers sell wheat in
equilibrium.
At a price support of $8 per bushel. All farmers will want to supply their wheat, but only
2 bushels will be traded (Qd=2<10=Qs). So, there is a surplus of 8 bushels of wheat.
Now, CS=$3, and PS=$12. (An alternative policy might have the price support with a
gov’t guarantee to buy up all the surplus bushels. While apparently a good deal for the
farmers, someone (i.e., the taxpayer) has to pay for buying up all the surplus units.
Needless to say, from the standpoint of society as a whole, the best outcome is the free
market equilibrium.
1
2)
First, note that we can also write the D-curve as P=25-Qd, and the S-curve as
P=10+(1/2)Qs
CDs
P
25
S
Solving simultaneously
yields P*=15, Q*=10. At this
equilibrium, CS =50, PS=25.
P*
10
D
25 Q
Q*
CDs Price ceiling
P
25
S
21
P*
ceiling
12
10
4
P
D
25 Q
Q*
CDs (tax on buyers)
25
S
Pd
P*
Ps
10
Dtax
Qt Q*
The price ceiling will lead to Qs<Qd,
which implies a shortage. The actual
amount traded in the market will be the
lower Qs amount, and so the new
outcome is at P=$12, and Q=Qs=4. In
this situation, CS (shaded)=8+36=44,
and PS (double-shaded)=4. So total
surplus is now 48, down fro 75. The
difference is the deadweight loss (the
inefficiency) of the price controlled
outcome. The DWL=27 area is speckleshaded.
D
25 Q
In This case, Demand shifts down by the size of the tax
(a vertical shift of $1). This leads to a new equilibrium
quantity at Q=9.33. At this quantity, suppliers receive a
price of 14.67, but demands are paying 15.67. Get this
by starting the problem with Pd=Ps+1 (since now there
are two prices to consider). Given the original
equilibrium at P=$15, Q=10, you can see the tax
reduced quantity traded, increases the price paid by
buyers but reduced the price received by sellers. In
other words, the tax is “paid” by both parties (though a
bit more in this example by the buyers, who have more
inelastic demand overall). The outcome would be
identical if the $1 tax were place on the sellers.
2
3)
CDs
P
25
Now the supply curve shifts to S’ on the
graph—the improved technology
increases the quantity supplied at all
prices—and the new equilibrium is at
Q=16 and price=$9.
S
P*
S’
10
9
5
Q*
16
D
25 Q
32  40
1
36
4   1  8  2 . Often the absolute value of d is reported
4)  d 

17  15
1
4 1
8
16
(since we all know it will always be negative due to the law of demand), and so we would
say in this case that the elasticity of demand for Bob’s shoes is 2, which is elastic
demand.
(Note: to calculate d above, I’ve used the mid-point method, so that the numerator of
each percentage change calculation is the average of the two quantities (or the average of
the two prices).
5)
Here, we can use simple calculus given that we have the full functional form of
this linear demand curve.
dQ P  1 P
. So, at a P=4 (where Qd=9), d= -1/9, which is inelastic
d  d 


dP Qd
4 Qd
demand. At a P=20 (where Qd=5), d= -1, which is unitary elastic demand. At a P=36
(where Qd=1) d= -9, which is elastic demand. So, elasticity of demand changes along a
linear demand curve, with elastic demand for high prices, inelastic demand for low prices
(although “high” and “low” are relative to the exact specification of the demand curve)
and a point of unitary elastic demand dividing the two. (Note: in order to have a demand
curve with a constant d at all points, it has to be a curve (and even at that it has to be a
special curve)).
3
BUDGET CONSTRAINTS
1)
Beverage
(liters)
36
12
Pizzas
An appropriate budget line would be 360=30x1+10x2, where x1=pizzas and x2=beverages
(i.e., income is $360, and p1=$30 and p2=$10). There are, however, an infinite number of
budget equations that would produce the same budget line and set—just multiple all
prices and income by any number (e.g., 180=15x1+5x2, 3600=300x1+100x2, etc.)
The slope of the budget line is -3. Note that it matters which good is on which axis to
determine the slope. With x1 on the vertical axis and x2 on the horizontal axis, the slope
of the budget constraint is always –p1/p2.
If you want to buy an extra pizza, you will have to give up 3 liters of beverages. Not
surprisingly, this is just the slope of the budget constraint (i.e., the market trade-off rate
of x1 for x2).
2) Pat’s budget equation is $500=2x1+x2+3x3+15x4+8x5. Pat would not be able to
consume the bundle x1=50, x2=100, x3=50, x4=5, and x5=10 because it would cost Pat
more than $500 (it would cost $505 to be precise.)
3)
Newspaper
Ads
town folk
100
10,000
5
TV ads
15,000
City Slickers
4
In the first case, Charlie should spend his budget entirely on TV ads, because he can
reach the most people with those ads. However, if town folk are known to each twice as
much of Charlie’s chicken after seeing an ad, then Charlie should focus on newspaper
ads. Though he’ll reach less total people, the number he’ll reach will consume more of
his chicken.
5
4)
The slope of Betty’s first budget line is -1/2. After the tax, the slope is -5/12. If
the tax is only for x2>10, then the budget kinks (in the bad way for the size of the budget
set) at x2=10. That is, it has slope =-1/2 for x2 less than or equal to 10, and slope=-5/12
for x2>10, which makes it trickier to find the vertical axis intercept. If we also do not
allow consumption of x1>50, then this “chops” off the budget constraint at x1=50, as in
the graph below). The exact kink point is real tough on this one. Find it by starting at the
vertical intercept where you spend all your income in x2 (keeping in mind that the first 10
can be bought at $10 each, then $12 thereafter). Then calculate how much less x2 it will
take to free up the $250 to buy the max 50 units of x1. Since subtraction x2 units saves
$12 each (as long as we’re still at x2>10), this is found by calculating 250/12=20.83. So,
we need to consume 20.80 units of x2 less than the max in order to have enough income
to buy x1=50. This leaves us at x2=14.17. Note also that giving up more x2 to purchase
additional x1 is wasteful because x1 is rationed at 50 units. As we’ll see after our material
on preferences and utility, a rational person would therefore never choose a point on that
vertical segment of the budget set.
x2
x2
Betty
40
Betty
$2 tax on x2
40
33.3
80
x1
Betty
($2 tax on x2>10)
x2
New budget line
slope= -5/12
80
Betty
($2 tax on x2>10 plus x1
rationed beyond 50 units
x2
40
35
x1
40
35
kink
kink
14.17
10
10
60
80
x1
50
60
80
x1
6
5)
income
income
Jasmine
1394
1344
Jasmine
(leisure on horiz. axis)
1344
budget constraint in
bold includes $50
nonwage income
50
112
labor
0
112
112
0
leisure
labor
PREFERENCES AND UTILITY
1)
Jack’s utility function could be U=min{(1/2)x1, x2}. The indifference curves are
drawn below. If the price of pizza were to go up, this would do nothing to Jack’s
preferences or indifference curves (it would alter the budget constraint, however, and so
Jack’s choice in equilibrium will be altered as we’ll see in the next chapter of material).
Jack
Beer=x2
U=3
U=2
U=1
x2=(1/2)x1
3
2
1
2
4
6
pizza
slices=x1
2)
Some indifference curves for U=min{3x1,7x2} are shown below. Note the levels
of utility associated with each I-curve.
These are perfect complement preferences, where the consumer likes to consume
the two goods in the ratio 7 units of x1 with each 3 units of x2. Notice that the kink will
always occur where 3x1=7x2, which is just the line x2=(3/7)x1 (i.e., the line with intercept
0 and slope of (3/7). You will note with these preferences that the MRS is not well
defined at all points (you can tell by looking at the I-curves. Depending on where you lie
on an I-curve, the MRS could be 0 (the horizontal segment),  (the vertical segment), or
undefined (the kink)). In general, indifference curves with kinks or sharp points will
suffer from this problem.
7
x2
U=63
U=42
U=21
x2=(3/7)x1
9
6
3
7
14
21
x1
3)
Joanne’s Husbands indifference curves are shown below. Ties are a “neutral”,
and so adding more ties does not make him any better or worse off. Preferences increase
as we go up from U1 to U2 to U3 (note: the I-curves would be vertical instead of
horizontal if the “neutral” ties were placed as x2 on the vertical axis).
Joanne’s Husband
x2=BB
tickets
U3
U2
U1
x1=ties
4)
Note that these I-curves slope upward because x1 (risk of injury) is a “bad” for
Bill. Utility is highest for U3, then U2, and the U1 on the graph. These preferences would
still be convex if Bill’s preferences had diminishing marginal utility for wages and
increasing marginal disutility of injury risk. You can think of the MRS here as the wage
“bribe” necessary to get Bill to accept an extra amount of on-the-job injury risk.
8
Bill
x2=wages
U3
U2
U1
x1=injury
risk
Here, MU1= 8 x1 x 25 , and MU2= 20 x12 x 24 . The MRS is therefore given by
5)
MRS=
 MU 1  8 x1 x25  8 x 2   2  x 2




MU 2
20 x1  5  x1
20 x12 x 24
  5  x2
If the utility function were U  4 x15 x22 , the we would have MRS= 

 2  x1
If you were to accurately graph out the indifference curves, they would look something
like this graph below. You can see from these indifference curves that when the
exponent on x1 is greater in the Cobb-Douglas utility function, the I-curves are steeper.
You will see after our next chapter that these steeper I-curves will give rise to the
consumer choosing more of x1 at his optimal choice, ceteris paribus. As such, the relative
sizes of the exponents on the goods in the C-D utility function tell us something about
which good the consumer weights more heavily in his preferences (and is therefore more
likely to choose).
x2
x2
Indifference
curves for
x1
Indifference
curves for
x1
9
CHOICE
1)
Note that the optimal choice is going to be where one of the kinks in Johnny’s Lshaped indifference curves hits the budget constraint. The budget line is 20=x1+(1/10)x2,
which can also be written as x2=200-10x1 (easier to graph this way). The kinks on the Icurves will all lie on the line x2=4x1. We can simultaneously solve to get x1*=14.29 and
x2*=57.16.
x2=coffee
x2=coffee
Johnny
x2=4x1
200
I-curve*
57.2
Johnny
x2=4x1
200
I-curve*
57.2
New I-curve*
42
budget line
rotate
14.3
20
x1=whiskey
13.3
10.5
20
x1=whiskey
After the tax, we can take the same approach to find the new x1*=10.5 and x2*=42. This
is graphed above along with the old optimum choice point. The tax on whiskey reduces
Johnny’s weekly consumption by about 4 ounces. This reduction could also be
accomplished by taxing coffee (since the are consumed together by Johnny). The
reduction could also be accomplished by taxing income, which would shift the budget
line towards the origin, while not altering its slope).
First, we need to find MU1 and MU2. They are MU 1  1 and MU 2  12x22 . So,
1
1
2
Beatrice’s MRS=
. Setting MRS=-p1/p2 yields
, which reduces to x2=2.

2
2
96
12 x 2
12 x 2
So, no matter, what, Beatrice’s optimal choice will involve 2 units of x2 (as long as she
can afford this), but not more than that. Here, we can see that with income of 192,
Beatrice’s optimal choice is on the U1* indifference curve where she consumes no x1 and
2 units of x2. However, when income rises to 384, Beatrice’s optimal choice is on the U2*
indifference curve where she spends all that extra income on x1 (and so x2*=2 still, but
now x1*=96). This is a particular property of quasi-linear preferences, although I’m not
sure my graph looks the best.
2)
10
x2
Beatrice
4
x2=2
2
U2*
U1*
96
192
x1
3)
An appropriate utility function to represent Todd’s preferences would be
U=2x1+x2. We can see in the diagram to the left-below that Todd will choose a corner
solution, where he consumes only gallons of milk (x1*=5 to be exact). This should not be
surprising because 2 half-gallons, which are a perfect substitute for 1 gallon for Todd,
cost more than a gallon. So, it would make sense never to buy half gallons (in case
you’re wondering why anyone would ever buy half gallons then, keep in mind that not
everyone consumes enough milk to justify a whole gallon, or people may have limited
space in their refrigerators, or they think the big gallons are extra ugly, etc).
The next graph to the right shows what happens when p2=$1.00. Now it is
cheaper to buy two half-gallons than a single gallon. In other words, the absolute value
of Todd’s MRS=2 is less than the absolute value of the budget line slope=1. Again, we
do not have MRS=p1/p2, and so he optimizes at a corner solution again. If the p1=2.50,
then the market trade-off rate is identical to Todd’s preference trade off rate (i.e., the
MRS). So, the budget line will have a slope equal to the slope of Todd’s linear
indifference curves, and an optimal solution would lie anywhere along the budget line
(trying drawing this situation).
11
x2=halfgallons milk
20
Todd
(budget line in
bold)
optimal
choice
x2=halfgallons milk
10
Todd
(budget line in
bold)
10
8
optimal
choice
4 5
8
x1=gallons
milk
5
10
x1=gallons
milk
4)
The budget constraint can be written as x2=2-(2/5)x1. This is graphed below
along with Joanne’s Husband’s preferences (according to his wife). Given that ties are a
neutral, and given that Joanne has now taken intermediate microeconomics, she realizes
now that she should spend all the birthday money on BB tickets (and she would buy two).
This will maximize her husband’s satisfaction.
(from this day on, Joanne never bought ties for her husband anymore).
Joanne’s Husband
x2=BB
tickets
optimal
choice
2
U3
1
U2
U1
5
x1=ties
12
5)
For Tommy, we have MU 1  3x23 and MU 2  9 x1 x22 and so Tommy’s
 3x23   1  x2
   . Setting MRS=p1/p2 yields x2=6x1. By substituting this into
9 x1 x22  3  x1
the budget equation, we have 16=x1+(1/2)(6x1), which solves as x1*=4. This means
x2*=24. You will notice at this optimum that Tommy spends a total of $4 on x1 and a
total of $12 on x2. In other words, he spends ¼ of his budget on x1 and ¾ of his budget
on x2 at his optimal choice.
If p2=2, Tommy will still spend ¼ of his budget on x1. So x1*=4 still. Now, when
he spends the remaining ¾ of his budget on x2, he will be only able to purchase x2*=6
(you can go through and solve this the long way, and you should get this exact same
answer).
If Tommy’s utility function is now U  3x13 x2 , and p1=$1, p2=$.50, and
income=$16, it should be the case that Tommy spends ¾ of his income on x1 now, and ¼
on x2. This would yield x1*=12, x2*=8 (again, try this the long way to double-check how
the Cobb-Douglas utility function result plays itself out).
MRS 
DEMAND
1)
With U=min{2x1,x2}, the consumer will always choose optimally at a point where
2x1=x2 (i.e., choice will be on the “kink” of one of the L-shaped indifference curves).
The exact indifference is determined by the budget constraint, m=p1x1+p2x2. So, we
must simultaneously solve these two equations to find the optimal choice of x1 and x2 as a
function of income, m. We can substitute the first equation 2x1=x2 into the budget
constraint to get m=p1x1+p2(2x1). This can be written as m=x1(p1+2p2). So writing the
m
optimal choice x1* as a function of income gives us x1* 
. Similarly, we can
p1  2 p 2
m
solve for x 2* 
. To graph these, it is more convenient to solve for income,
1 p1  p 2
2
because we place income in the vertical axis. So we can rewrite these as
 p1  2 p2   x1*  m and 1 2 p1  p2  x2*  m . You will note that the Engle curves are
both linear and the Engle curve for x1 is twice as steep as the one for x2 (see below).
 
 
income

Engle Curve for x1
income
slope =
p1 +2p2
x1*
Engle Curve for x2
slope =
(1/2)p1 +p2
x2*
13
2)
Here, we go through the same steps as in question (1) to solve for the optimal
m
m
demanded bundles. In other words, we have x1* 
and x 2* 
. How
1 p1  p 2
p1  2 p 2
2
can it be that the equation for the Engle curves and the demand functions are the same.
Well, remember that for the Engle curves we hold p1 and p2 constant while varying
income. For the demand curves, we typically focus on changing the price of the good,
while holding the price of the other good constant. So, we just focus on altering a
different variable really.
 
3)
For this, because Cobb-Douglas preferences are convex, we plan on a solution
where MRS=p1/p2, and we are on the budget constraint. To find the MRS, we need to
find MU1 and MU2. We solve to find MU 1  2 x1 x25 and MU 2  5x12 x24 . So, MRS=p1/p2
5 p1
2 x1 x 25
p
2 x2
p
 1 or
 1 which can be written as x 2 
 x1 . Now,
2 4
p2
p2
2 p2
5 x1 x 2
5 x1
substitute this into the budget constraint to get
 5p


5 
7 
m  p1 x1  p2  1  x1   x1  p1    p1   m  x1  p1 
2 
2 

 2 p2

implies
2 m 
 .
Solving for x1 we get x1*  
7
p
1 

5 m 
 .
Likewise, we would get x2*  
 7 p2 
In other words, the consumer will spend 2/7 of his budget on x1 and 5/7 of his budget on
5m
x2 at the optimum. The demand curve for x2 is graphed below (write as p 2  * to
7 x2
graph. This is the equation for a rectangular hyperbole (like xy=1, which can be written
as y=1/x)). So the demand curve is not linear, but if you draw it carefully, you’ll get
something like
Demand curve for x2
p2
D
x2
4)
For perfect substitute preferences, the demand curve will always be composed of
three parts. For Tiffany, if the p2 is low enough relative to p1, then she will only consume
x2 and thus the optimal choice will be however much x2 she can purchase with her
income (which is m/p2). If p2 is high relative to p1, then she will consume only x1 (and so
14
x2*). If the relative price of the two goods p1/p2 is equal the Tiffany’s MRS, then the
highest indifference curve she can reach lies on top of the budget line, and any amount
between zero and spending all her income on x2 would be optimal. So, we have
m
p1

if
 3
2
p2
 p2

p1
x 2*  0 if
 3
2
p
2

  m
p1
 0,  if
 3
2
p2
  p 2 
 
 
 
For this set of convex preferences, MU1=1, and MU 2 
5)
x2  x12/ 2 , and so the derivative of
MU2 by recalling that
   1 / 2x
x
x2
1/ 2
2
1 / 2
2

1
2 x2
(Hint: you get
x2 can be written as
1
1
). So MRS=p1/p2 implies 2 x2 =p1/p2. This can be

1/ 2
2 x2
2 x2
2
 p 
solved for x2 to get x   1  . You will note here that x2* is not a function of
 2 p2 
income. This implies that the demand for x2 is a fixed amount (as long as we have
enough income to purchase that amount, more income will not lead us to purchase any
more x2). This is actually a bit more complicated still, but for our purposes, it suffices to
realize that x2* is fixed, and our demand for x1 is whatever amount can be purchased with
the leftover income (after purchasing x2*).
*
*
2
SLUTSKY EQUATION
1)
For these preferences x1*=10, and x1’=20 (x1’ is the new optimal choice after the
price change). The question is, what portion of this change is due to substitution effect
and which is due to the income effect? If you consider new prices, p1=$1, and then wish
to compensate income so that x1*=10 just affordable, you would have to take away
10*($1-$2)=$10 dollars (i.e., the change in prices times the original consumption
amount). So, if we take away $10, we now have income of $20, and x1*=13.3 when
income=$20 and p1=$1. So, the amount 13.3-10, or + 3.3 units of x1 is due to the
substitution effect (consume more x1 because it is now relatively cheaper), and 20-13.3=
+6.7 is due to the income effect (consume more x1 because real income goes up when p1
falls).
(The trick in the middle to find the income compensation amount is not that hard.
Assume that original income is m, and prices are p1 and p2. Then say that the price of
good 1 changes to p1’, and m’ is the amount of income that makes the original
consumption bundle (x1,x2) just affordable. Because (x1,x2) is affordable in both cases,
we have m=p1x1+p2x2 and m’=p1’x1+p2x2. This implies (m’-m)=(p1’-p1)x1, which is what
we used above).
15
2)
Suppose that U=3x+x2, and p1=p2=$1, and income=$10. Given this, the optimal
choice of the consumer would be x1*=10, and x2*=0 (i.e., x1 is more important to utility,
but it costs the same as x2, so the utility maximizing consumer would consume only x1).
If we then assume the price of x1 changes to p1=$2 (notice, that this is not a large enough
change to induce switching the consumption choice to x2…..this is a critical point). Now,
the optimal choice is x1*=5, and x2*=0, and the change is all income effect (see graph
below).
x2
Total effect=income effect
UC
10
UA=B
A=B
C
5
10
x1
3)
This is a creative story to illustrate a simple point. If Betty is compensated the
amount necessary for her to be able to purchase her original consumption bundle, then
this compensation will actually increase her utility. You can see this in the graph below
to the left. The appropriate amount of compensation to make Betty “whole” in the sense
of restoring her original level of satisfaction would actually be a smaller compensation, as
can be seen in the graph to the right.
x2=all other
goods
x2=all other
goods
Betty
Betty
A
A
U3
U2
U2
U1
x1=gas
x1=gas
(Betty’s problem shows the principal point behind economists’ argument that the CPI
contains a “substitution” bias. In other words, it overstates the amount of income needed
to account for price changes because it does not take into account the fact that consumers
substitute towards different goods when prices change (and the CPI bases its calculations
on a fixed basket of market goods)).
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