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University of Victoria
Economics 325
Public Economics
Martin Farnham
Problem Set #1
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 (compared to
the optimum at Q=20).
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.
The average height of the light shaded trapezoid is 475. The base is 100. So the added
total benefit moving from Q=200 to Q=300 is 47,500.
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.