University of Victoria
Economics 312
Urban Land Economics
Martin Farnham
Problem Set #0 (optional micro review)
SOLUTIONS
1) Practice calculating individual welfare with an individual demand curve.
Consider the individual demand equation Q=120-P.
a) Draw this demand curve.
b) At a price of $100, how much does this individual choose to consume? Why do they
choose that quantity?
You can solve this graphically or algebraically (usually it’s easiest to draw the picture,
then let the picture guide your algebra). The individual chooses to consume where MB=P
(we’ll try to develop an intuition for why this is the case, later in this problem). Since the
demand curve is a marginal benefit curve, we just find the point where P=100 intersects
the demand curve.
Note that we can write the demand equation as an equation for marginal benefit:
Demand equation: Q=120-P
Rearranging, we get P=120-Q (this is the usual slope-intercept form that allows you to
draw the equation easily—remember y=mx+b from high school algebra?).
Since MB and P are both measured on the vertical axis, we can express this as
MB=120-Q.
Now just set P=MB, or 100=120-Q to solve for where the price line and marginal benefit
curves intersect. This gives us Q=20.
c) How much is the consumer willing to pay (per unit) for a tiny bit more of the good, at
this quantity?
Plugging back into the equation for marginal benefit (or by inspection of the graph)
MB=100 when Q=20. The consumer is willing to pay at a rate of $100 per unit to get a
tiny bit more of the good when they’re consuming 20 units.
d) How much is the consumer willing to pay (in total) for the quantity they choose to
consume at a price of $100?
We know they buy 20 units. To get their maximum willingness to pay for 20 units, we take
the area under the demand curve between Q=0 and Q=20. Graphically, it’s
To calculate this area, we can either use our formula for the area of a trapezoid:
area of trapezoid=(average height)x(width)=110*20=2200,
or we can calculate the area of the lower rectangle (100*20)=2000 and the area of the
upper triangle (20*20/2)=200, and add these together to get 2200.
So the maximum willingness to pay for those 20 units is $2200.
e) What are the net benefits the consumer gets at this quantity?
The net benefits, or consumer surplus, the consumer gets when consuming 20 units at a
price of $100 per unit is what she values those goods at, minus what she has to pay for
them. The consumer values those goods at $2200. She has to pay P*Q or
100*20=$2000 for those goods. So she ends up with $2200-$2000=$200 in consumer
surplus from consuming 20 units at a price of $100 per unit.
f) Suppose, instead, she chooses the quantity 10 at a price of $100. What are the net
benefits the consumer gets at this quantity? Show your answer numerically and
graphically. What is the relationship between marginal benefit and price at this quantity?
What should the consumer do to make herself better off? Why?
If the consumer purchases 10 units at a price of $100 each, she spends $1000. The total
benefit (maximum willingness to pay) for those 10 units is the entire shaded trapezoid
above. That has an average height of 115 and a width of 10, so the total benefit from
consuming those 10 units is 1150. Net benefits are, therefore $1150-$1000=$150.
Alternatively, we call this consumer surplus. The little shaded trapezoid above P=100 is
the consumer surplus. Note that she is worse off consuming 10 units than she was when
she was consuming 20 units (her CS in that situation was $200). The small unshaded
triangle to the right of CS is the lost CS as a result of consuming too little. We can think
of that unshaded triangle as the lost consumer surplus (compared to the optimum at
Q=20) from consuming the wrong amount.
We can figure out marginal benefit at this quantity by plugging Q=10 into the MB
function above. MB=120-Q=110. This means that when she’s consuming 10 units of the
good, she’s willing to pay for more of the good at a rate of $110 per unit, yet it only costs
her $100 per unit to buy more. Therefore, at Q=10, she’d be better off buying more of the
good (her consumer surplus would rise).
g) Now suppose she chooses the quantity 30 at a price of $100. What are the net benefits
the consumer gets at this quantity? Show your answer numerically and graphically. What
is the relationship between marginal benefit and price at this quantity? What should the
consumer do to make herself better off? Why?
The diagram above shows the total benefits (maximum willingness to pay) that the
consumer gets from consuming 30 units. This is a trapezoid with average height of
(120+90)/2 = 105 and a width of 30 (to find the 90, just plug Q=30 into the MB
function). So the area of the trapezoid is $3150.
The consumer has to pay $100 per unit times 30 units, or $3000 (this rectangle is visible
above—I didn’t both to shade it). Subtracting expenditure from maximum willingness to
pay, we get $3150-$3000=$150. Thus, the net benefits (consumer surplus) the consumer
gets when consuming 30 units at a price of $100 per unit is $150. Graphically, consumer
surplus is equal to the rectangle of height 100 and base 30, minus the shaded trapezoid.
Another way to graphically represent consumer surplus at Q=30 is that it is the
consumer surplus triangle from when Q=20 (see above) minus the small unshaded
triangle in the upper right of the diagram. We can think of that unshaded triangle as the
lost consumer surplus (compared to the optimum at Q=20) from consuming the wrong
amount.
At Q=30, marginal benefit equals $90. This means that when she’s consuming 30 units of
the good, she’s willing to pay for more of the good at a rate of $90 per unit, yet it costs
her $100 per unit to buy more. This means she’d have been better off not buying the last
little bit that brought her up to Q=30. She’d be better off reducing her consumption of
the good (her consumer surplus would rise).
Note that when she’s consuming where P<MB, she could be better off by increasing her
consumption; when she’s consuming where P>MB, she could be better off by decreasing
her consumption; and, therefore, when she’s consuming where P=MB, she’s as well off
as she can be (because she needs to neither increase or decrease her consumption).
2) Practice calculating individual firm welfare with an individual supply curve.
Consider the individual supply equation Q=P-2.
a) Draw this supply curve.
b) At a price of $100, how much does this firm choose to produce? Why do they choose
that quantity?
You can solve this graphically or algebraically (usually it’s easiest to draw the picture,
then let the picture guide your algebra). The firm chooses to produce where MC=P (we’ll
try to develop an intuition for why this is the case, later in this problem). Since the supply
curve is a marginal cost curve, we just find the point where P=100 intersects the supply
curve.
Note that we can write the supply equation as an equation for marginal cost:
Supply equation: Q=P-2
Rearranging, we get P=2+Q.
Since MC and P are both measured on the vertical axis, we can express this as
MC=2+Q
Now just set P=MC, or 100=2+Q to solve for where the price line and marginal cost
curves intersect. This gives us Q=98.
c) How much does it cost the firm (per unit) to produce a tiny bit more of the good, at this
quantity?
Plugging back into the equation for marginal cost (or by inspection of the graph)
MC=100 when Q=98. Increasing output costs the firm at a rate of $100 per unit when
it’s producing 98 units.
d) What are the firm’s variable costs of producing the quantity they choose to produce at
a price of $100?
We know it produces 98 units. To get the variable cost of producing 98 units, we take the
area under the supply curve between Q=0 and Q=98. Graphically, it’s
This trapezoid has an average height of 51 and a width of 98. Thus its variable costs are
$4998.
e) What are the net benefits the producer gets at this quantity? Now assume the firm faces
fixed costs of $10. What are the firm’s profits at this quantity?
Net benefits, or producer surplus, are total revenues minus variable costs. If the firm
sells 98 units at $100 each, their total revenues are $9800. So producer surplus is 98004998=$4802. Profits are TR-TC. We know TR. TC is variable costs plus fixed costs.
TC=4998+10=5008. TR-TC=9800-5008=4792. Profits are $4792 (we can’t show this
graphically without an average total cost curve).
You could also get this by recalling from lecture that profits=PS-FC.
f) Suppose, instead, the firm chooses the quantity 90 at that price. What are the net
benefits the firm gets at this quantity? What are the profits they get at this quantity? Show
your answer numerically and graphically. What is the relationship between marginal cost
and price at this quantity? What should the firm do to make itself better off? Why?
To calculate net benefits at Q=90 we need to calculate variable costs at 90 and total
revenues at 90. TR=100*90=$9000.
The trapezoid above represents variable costs of the firm from producing 90 units. We
need to find the height of the tall end of this trapezoid. To do so, plug Q=90 into the MC
function. This yields MC=92 when Q=90. Variable costs are [(92+2)/2]*90=$4230. So
producer surplus is 9000-4230=$4770. Graphically this is the rectangle with height 100
and base 90 minus the shaded trapezoid. Another way to see producer surplus
graphically is that it is the producer surplus triangle from when the firm produces Q=98,
minus the small unshaded triangle at the upper right of the diagram. In fact if you
calculate the area of that triangle, you’ll see that it equals the difference between
producer surplus at Q=98 and producer surplus at Q=90. You can think of that triangle
as representing the loss of producer surplus from the firm choosing the wrong quantity to
produce (rather than the optimal quantity).
Profits are $4760 (we can’t show this graphically without an average total cost curve).
MC<P (92<100) at Q=90. This means that if it produced a bit more, it would bring in
more extra revenue than it would spend producing the extra output. This tells us that the
firm could increase its producer surplus and profits by producing extra units.
g) Now suppose the firm chooses the quantity 110 at that price. What are the net benefits
the producer gets at this quantity? What are the profits they get at this quantity? Show
your answer numerically and graphically. What is the relationship between marginal cost
and price at this quantity? What should the producer do to make itself better off? Why?
The diagram above shows variable costs associated with producing 110 units. TR from
110 units is 110*100, or $11,000. Variable costs are [(112+2)/2)*110]=$6270. So
producer surplus is 11,000-6270=$4730. Graphically, producer surplus is the area of the
rectangle with height 100 and base 110 minus the shaded area of the trapezoid. Another
way of describing producer surplus graphically is to say that it’s the producer surplus
triangle from when the firm produces Q=98, minus the little shaded triangle in the upper
right portion of the diagram. In fact, if you calculate the area of that little triangle, you’ll
find it matches the difference in producer surplus at Q=98 versus Q=110. You can think
of that triangle as representing the loss of producer surplus from the firm choosing the
wrong quantity to produce (rather than the optimal quantity).
Profits are $4720 (we can’t show these graphically).
At Q=110, the firm’s MC is greater than P (112>100). This means that the firm lost
money on the last little bit produced. It would be made better off by reducing output.
Note that when the firm is producing where P<MC, it could be better off by decreasing
production; when it’s producing where P>MC, it could be better off by increasing
production; and, therefore, when it’s producing where P=MC, it’s as well off as it can be
(because it needs to neither increase or decrease production).
3) Practice Calculating Social Welfare (aggregate net benefits) with Supply-Demand
diagram.
Consider the market demand equation Q=1200-2P.
a) Draw this demand curve.
b) What is the total benefit to consumers associated with consuming 200 units of the
good? How much would total benefit go up by if they switched from consuming 200
units of the good to 300 units of the good? Shade this area in a diagram.
The dark shaded trapezoid has an average height of 550. Its base has a width of 200. So
the area is 110,000. So the total benefit to consumers of consuming 200 units of the good
is $110,000. Total benefit rises by the area of the light shaded trapezoid if they switch
from consuming 200 to 300 units of the good. This trapezoid has a height of 500 on the
left side and a height of 450 on the right size. To find the amount 450, plug Q=300 into
the equation for the MB curve, MB=600-(1/2)Q. Plugging 300 in gives us MB=600150=450.
c) Suppose the price of this good is $100. What is the consumer surplus associated with
the quantity of goods that will be consumed at this price? Shade this area in a diagram.
If P=100, then Q=1200-2(100)=1000. The area under the demand curve between Q=0
and Q=1000 gives the total benefit to consumers. This is $350,000. Total benefit minus
total expenditure gives consumer surplus. Total expenditure on 1000 units at a price of
$100 per unit is $100,000. So consumer surplus is $250,000.
d) Now assume the market supply curve is given by Q=P. Add this supply curve to your
diagram.
e) What is the equilibrium quantity of goods produced and consumed in this market?
The intersection of the two curves will give the equilibrium price and quantity.
(1) Q=1200-2P
(2) Q=P
Substituting (2) into (1) we get Q=1200-2Q, or 3Q=1200, or Q=400. If Q=400 then
(substituting into (2)) P=400.
f) What is the producer surplus at this equilibrium? Shade this area in a diagram.
Producer surplus is total revenue minus total variable costs. Total revenue is $400x400
or $160,000. Total variable cost is the triangle below the supply curve up to Q=400.
This is $80,000. So producer surplus is $80,000.
g) What is social welfare at this equilibrium? Assume there are no externalities or other
market failures.
Social welfare (or net social benefits) is consumer surplus plus producer surplus.
Consumer surplus at this equilibrium is $200*400/2=$40,000. So social welfare is
$40,000+$80,000=$120,000.
h) Why is this equilibrium efficient?
Because MB=MC. If we were in a situation where MB>MC, this would tell us that
society could be made better off by producing and consuming a little bit more. If we were
in a situation where MB<MC, this would tell us that society could be made better off by
producing and consuming a little bit less. Because MB=MC, society can’t be made any
better off by changing the quantity. So the equilibrium, in this case, is efficient (it
maximizes social welfare).
i) By how much would social welfare change if the government imposed a binding quota
at Q=300? Shade total social welfare (net social benefits) in a diagram.
Social welfare falls by $2500+$5000=$7500. So social welfare with the quota is
$112,500.
j) What is the deadweight loss resulting from implementing the quota?
Notice that aggregate net benefits (social welfare) before the quota was implemented was
the triangle bounded by the supply curve, the demand curve, Q=0 and Q=400. Now it’s
that triangle, but with its tip cut off. So it’s the shaded area above. The tip that was cut
off (the two small unshaded triangles) are welfare that we could have, if we weren’t at
this inefficient allocation at Q=300. Those two small unshaded triangles represent the
deadweight loss. They have a combined area of $7500.
Problems 4-6 give a partial tour of equilibrium, market success, and market failure
4) Consider the market for slices of pizza. Suppose that the market is perfectly
competitive. There are 4 consumers and 2 producers (but each acts as price taker).
Consumers are identical and producers are identical. Assume that partial slices of pizza
may be produced and consumed.
individual demand curve: q=6-P
individual supply curve: q=P
a) Write an equation for the market demand curve. Why does the demand curve slope
down?
Pizza is a private good. Therefore to obtain market demand, we horizontally aggregate
the individual demands. Thus Q=24-4P. To convince yourself, draw the four individual
demand curves so they are lined up horizontally. Then draw a fifth set of axes. How
much is the sum of individual demands at P=8? (plot the point) at P=6? (plot the point);
at P=4? (plot the point); etc. down to P=0.
The demand curve slopes down because as consumers eat more pizza, their desire for
more (on the margin) declines. Therefore their willingness to pay declines as they
consume more. If you’ve been stranded in a blizzard for 3 days without food, your
marginal WTP for the first slice will be huge. Once you’ve eaten 20 slices, your
marginal WTP will probably be quite small, or even negative.
b) Write an equation for the market supply curve. Why does the supply curve slope up?
Again, we horizontally aggregate the individual supply curves. This gives market supply
of Q=2P. Complete the same exercise as in (a) if you’re not entirely convinced.
The supply curve slopes up (in the short run) due to some input into production being
fixed (typically we assume this is capital). As more labour is combined with the fixed
amount of capital, labour productivity (marginal product of labour) falls. This causes
the cost of an extra unit to rise, as more stuff is produced (and more labour is used).
Hence MC is thought to slope up. MC above average variable cost is an individual
firm’s supply curve.
c) Find equilibrium price and quantity in the market for pizza slices. How many slices
does each person consume? How many does each producer make?
D: Q=24-4P
S: Q=2P
Set this equal to find equilibrium P.
24-4P=2P
6P=24
P=4 (plug this into either D or S to get equilibrium Q)
Q=8
Each consumer faces P=4; plugging this into individual demand, q=6-P, yields q=2, so
each consumer buys 2 slices.
Each producer faces P=4; plugging this into individual supply, q=P, yields q=4, so each
producer supplies 4 slices.
d) Calculate the total benefit derived from the equilibrium consumption of pizza. Draw a
picture and shade in the area representing total benefit.
TB is given by area under demand curve up to Q=8. This is $40. TB=
e) Calculate the total variable cost of the equilibrium quantity of slices produced. Draw a
picture and shade in the area representing total variable cost.
TVC is given by area under supply curve up to Q=8. This is $16. TVC=
f) Calculate consumer surplus. Draw a picture and shade in the area representing CS.
CS is TB-expenditure, where expenditure is P*Q. In this case CS=40-32=$8.
CS=
g) Calculate producer surplus. Draw a picture and shade in the area representing PS.
PS is Total Revenue-TVC. In this case, PS=32-16=$16. PS=
P
S
6
4
D
8
24
Q
Note: for problems like d-g, you should draw the whole picture and shade in the relevant
area for each question. This is time-consuming on a computer, so I’ve only drawn the
market picture once (directly above). When you do this for practice or on exams, you
should draw the picture and shade in the area.
h) Is the equilibrium quantity efficient? Why?
In the absence of any externality, S=MSC and D=MSB. Therefore, in equilibrium
MSC=MSB, which is the condition for efficiency. Therefore, the equilibrium quantity
happens to be efficient. This means social welfare is maximized at Q=8. If output were
to go up a bit, marginal social cost would exceed marginal social benefit, thus lowering
social welfare. If output were to go down a bit, total social benefit would decline by
more than total social cost would decline, thus lowering social welfare. So the point
where MSB=MSC maximizes social welfare.
i) Using the efficiency criterion, could the government do any better than the market in
allocating goods in this case?
Nope. If the government knew the MSB and MSC curves, it could do as well as the
market, but not better. And if it were wrong, it would likely make things worse (from an
efficiency perspective)
5) (Using the setup from problem 4) Now suppose there is an externality associated with
pizza slices. Consumers of the pizza have a nasty habit of dropping their paper plate on
the sidewalk after eating. This presents a form of visual pollution to everyone else in the
area.
Assume that, on average, ½ plate is dropped on the sidewalk per slice of pizza consumed.
Each plate on the sidewalk causes $2 of collective unhappiness to society.
a) Is this a positive or a negative externality?
Negative, because it imposes costs on others.
b) Is it a consumption externality or a production externality?
Consumption, because it is the actions of consumers that have effects on others.
c) What is the marginal external cost of a slice of pizza?
On average, 1 slice of pizza is associated with ½ plate dropped. Each plate dropped
causes $2 of external damage. Therefore the marginal external cost of each slice of pizza
is (1/2)*2=$1.
d) Draw the marginal social benefit curve.
To get this, shift down the MPB curve by $1 to reflect the constant MEC=1.
P
S
6
5
4
D=MPB
8
MSB
24
Q
e) Does equilibrium production of pizza change in the face of the externality.
No. Private markets ignore externalities. This should change neither consumer nor
producer behavior. The equilibrium will be at (Q,P)=(8,4)
f) Calculate the total external cost of pizza consumption in equilibrium.
TEC is $1 per slice for 8 slices consumed. So TEC=$8. It’s the area between MSB and
MPB from Q=0 to Q=8.
g) Calculate the total social benefit of pizza consumption in equilibrium.
TSB=TPB-TEC. Graphically it is the area under the MSB curve up to Q=8. A trapezoid
with base 8 and average height 4. So TSB=$32.
h) Calculate total welfare in equilibrium.
SW=TSB-TVC =32-16=$16
i) What is the efficient level of pizza production, in light of the externality?
Efficient Q is where MSC=MSB.
MSB=5-(1/4)Q
MSC=(1/2)Q
Setting these equal gives 5-(1/4)Q=(1/2)Q, or 5=(3/4)Q, so Q=20/3
j) How much would total social benefit decline by moving to that efficient level?
Oops. Thought I’d picked numbers that worked out well. Doesn’t look that way
now…my apologies! Still, if you’ve got the graphical intuition, this isn’t too hard.
Social benefit declines by a trapezoid of width 8-(20/3), or 4/3, and average height of
19/6. To get the average height, look at MSB where Q=8: MSB=3 there. Then look at
MSB where Q=20/3 (the efficient quantity). MSB=10/3 there. The average of 3 and 10/3
is 19/6.
So social benefit declines by (4/3)*(19/6)=38/9
k) How much would total social cost decline by moving to that efficient level?
Social cost falls by a slightly larger trapezoid, bounded by Q=8, Q=20/3 and the MSC
curve. Again, this has width 4/3. The average height is 11/3 (average of 4 and 10/3). So
this trapezoid has area 44/9.
Social cost declines by 44/9.
l) What would total social welfare be at the efficient level? Compare that with total
social welfare in equilibrium.
Social welfare in equilibrium is $16 or $48/3. At the efficient level of pizza consumption,
SW is $50/3.
SW=TSB-TVC
TVC=20/3*10/3*1/2 = 200/18
TSB=20/3*[(15/3+10/3)*(1/2)] = (20/3)*(25/6) =500/18
SW=300/18=$50/3
m) What is the deadweight loss associated with the pizza market?
DWL=$(2/3)
n) What are some policy instruments the government could use to eliminate the
deadweight loss?
The government could tax pizza, it could pay people a reward to use trashcans, it could
impose a quota on pizza consumption, or set a price control that would induce the
efficient quantity. Try drawing each of these cases for practice.
6) Now suppose the externality occurs in the following form. The consumers don’t dump
their plates on the ground. Instead, the pizza parlors dump their trash (tomato cans,
cheese wrappers, etc.) on the street at the end of the day. How would your analysis differ
from in problem 2? Which curve would you alter now, to reflect the externality.
Supposing the marginal external cost is the same per slice as above, does the efficient
equilibrium differ?
Now we’re talking about a production externality. If we assume the same MEC
associated with littering by the owner, we’ll now shift the MPC curve up by $1 to reflect
the externality. Notice that if you work through this, shifting the MPC up vertically by $1
yields the same efficient quantity as shifting the MPB down vertically by $1. This points
to the fact that if you’re not sure about whether to count something as a production or a
consumption externality, just pick one and move forward with the analysis. It will yield
the same efficient quantity.
7) (Using setup from problem 5) Suppose the Victoria town council wants to apply a tax
on pizza, in order to induce the efficient level of pizza consumption.
a) Suppose the tax will be imposed on consumers. That is, each consumer must pay
amount $t per slice of pizza they buy. Draw a picture showing how the tax affects the
marginal private benefit curve.
See below. A $t per slice tax on consumers shifts the demand curve down vertically by $t.
You can think of the tax as reducing consumers’ willingness to pay for each slice by the
$t they have to hand over to the government.
b) What is the level of tax (per slice) that should be chosen in order to induce an efficient
equilibrium?
A tax of $1 per slice will induce the efficient equilibrium. By setting the tax equal to the
MEC, consumers are forced to internalize the externality. They still don’t care about the
damage their litter causes, but they now take into account the extra $1 cost associated
with a slice of pizza (due to the tax).
c) What is the new equilibrium quantity? Price paid by consumers? Price received by
producers?
New equilibrium occurs where MPB(w/tax) = MSC. MPB(w/tax)=5-(1/4)Q.
MSC=(1/2)Q. So the equilibrium Q (w/tax) is 20/3. Consumers pay producers Ps=$10/3
per slice in equilibrium. Consumers also pay the government $1, so Pc=$13/3.
P
S=MPC
6
4
D=MPB
8
D’=MPB(w/tax) 24
Q
Now suppose the tax will be imposed on producers of pizza instead of on consumers.
That is, for each slice produced, the pizza parlor must pay $t.
d) What is the level of tax (per slice) that will induce efficiency?
It doesn’t matter who pays the tax in the administrative sense of who gives the money to
the government. Result will be the same. Redo the analysis by shifting up the MPC curve
by $1, to reflect higher marginal cost to suppliers. Here, the tax induces the suppliers to
take into account the extra dollar cost associated with each slice (they may not care
about the littering, but they care about the tax and so act as if they care about the
littering). A $1 per slice tax levied on producers will have exactly the same result as a $1
per slice tax levied on consumers. Producer and consumer prices will be the same, as
will equilibrium quantity under the tax (which will be efficient).
e) What is the new equilibrium quantity? Price paid by consumers? Price received by
producers?
Q’=20/3
Pc=$13/3
Ps=$10/3
f) Calculate total social welfare (consumer surplus+producer surplus+government
revenue-total external cost).
SW=CS+PS+GR-TEC
CS=(5/3)*(20/3)*(1/2)=100/18
PS=(10/3)*(20/3)*(1/2)=200/18
GR=t*Q’=1*20/3=20/3=120/18
TEC=(20/3)*1=20/3=120/18
SW=100/18 + 200/18 + 120/18 – 120/18=300/18=50/3
Note: This is what we calculated SW to be at the efficient outcome earlier.