Download Lab 17: Consumer and Producer Surplus

Lab 17: Consumer and Producer Surplus
Who benefits from rent controls? Who loses with price controls? How do taxes and subsidies
affect the economy? Some of these questions can be analyzed using the concepts of consumer
and producer surplus and the definite integral.
1. Demand and Supply Curves
Open the Surplus tool in the Integration Kit and look at examples of linear and non-linear
demand and supply curves. Keep checking the screen as you read about consumer and producer
surplus.
Economists assume that demand curves are never increasing, whereas supply curves are generally
assumed to be upward sloping. For the demand curve shown, we see that when the quantity of a
good is increased the unit price decreases, or conversely, when the price is lowered the amount
demanded increases. For the supply curve we see that when the quantity supplied to the market is
increased, the price to the supplier decreases. We see that when the price to the supplier is
increased, the quantity producers are willing to supply is increased.
The demand curve and supply curves intersect at the equilibrium point ( x eq , peq ). The price
peq is also called the market price and x eq is the corresponding market quantity. It is
generally assumed that the market will operate at the equilibrium point so that the quantity x eq
will be supplied and sold at the market price peq .
2. Consumer and Producer Surplus
When we assume that all consumers pay the equilibrium price peq , there are consumers who are
paying less that they were willing to pay. If each consumer bought the commodity at the
maximum price they were willing to pay, the total amount spent is represented by the area under
the demand curve from zero to x eq . The difference between this amount and the amount spent
at the equilibrium price is called the consumer surplus. It can be viewed as the consumers' gain
from the trade.
Consumer Surplus = area below the demand curve and above the line p = peq
xeq
=
∫ D( x)dx
0
− xeq peq
(1)
When the commodity is sold at the equilibrium price, there is a gain to the producers also.
Various producers were willing to supply the goods at the price given by the supply curve.
However the supply price is always less than or equal to the equilibrium price peq . The difference
between the amount received by the producers (i.e., x eq peq ) and the amount they would have
received at the price level determined by the supply curve (the area under the supply curve from
zero to x eq ) is called the producer surplus.
Producer surplus = area above the supply curve and below the line p = peq
xeq
= xeq peq − ∫ S( x)dx
(2)
0
The sum of the consumer surplus and the producer surplus is called the total trade gain, and
considered by economists to be a measure of the benefit to society of the transaction at the
equilibrium point.
2.1
In the supply-demand figure shown (in Figure 1), shade in and label the areas that represent
the consumer surplus and the producer surplus, respectively.
Figure 1
2.2
For the linear model, calculate the consumer surplus and the producer surplus by means of
a definite integral using the facts that D (x)= -0.5x +500 and S(x)=0.5x +20 in the
formulas given in the earlier part of this section. Find the x eq and peq values from the
tool.
2.3
Work through the following example:
Find the consumer surplus and the producer surplus for an item whose supply and demand
functions are given by:
S( x) = 4x + 2 and D( x) = 20 − x 2
for x thousands of units and prices in dollars/unit.
Solution:
First find the equilibrium point ( x eq , peq ) by equating S( x ) and D( x ) and
solve for x (which will be x eq ).
Start with:
3x + 2 = 20 − x 2
The positive solution to this equation is x eq =____(thousand units). The
market price is
peq = S( xeq ) = D( xeq ) = _________
Set up the integrals for the consumer and producer surplus and the evaluate
them. Show the intermediate steps.
The consumer surplus =
The producer surplus =
The total trade gain is ______________dollars.
3. The Effect of Price Controls on Consumer and Producer Surplus
Sometimes for the best motives, a government will interfere with the operation of the market
process. For example price controls above the market can be imposed to insure the survival of
marginal producers. The minimum wage and controls on cotton prices are two such examples.
Also, prices can be kept artificially low. Rent control is an example of this policy.
Select the box for price control to investigate these phenomena.
3.1 Use the price slider to select the controlled price pc to be higher than the market price
peq . Notice that the producer surplus grows as the consumer surplus shrinks. Explain.
3.2 There is a new area on the screen that grows as the controlled price increases beyond the
market price. This area (in yellow) represents the dead-weight loss (DWL) which is the
difference between the total trade gain for the equilibrium price and quantity and the
controlled price and quantity. The dead-weight loss is the loss in potential gains from all
those transactions that never took place. On the graph (Figure 2) identify each of these
areas.
Figure 2
3.3 Now look at the effect of rent controls when the price (the rent) is kept artificially low.
When the rent cannot be greater than pc (for pc lower than peq ), the quantity of rental
units supplied x c is less than x eq . The producer (landlord) surplus is the area above the
supply curve and below y = peq . Notice that it appears to be reduced. This effect is balanced
somewhat by the fact that those renters who have rental units have a greater consumer
surplus. What does the dead-weight-loss represent in terms of renters and landlords?
4. The Effect of a Tax per Unit on Producer and Consumer Surplus
Select the unit tax option at the top left of the tool.
Suppose the government wishes to tax the producers a set amount for each item produced. This
kind of tax is called a unit tax t. The unit tax adds a production cost equal to tx where x is the
quantity of items produced. We can see from this that there is an additional marginal cost t (since
dx
(t x) = t ). In the short run, the marginal cost curve is quite close to the supply curve (because
in the short run, suppliers are willing to produce up to the point where the price they will receive
for an additional item is equal to their marginal cost.) Consequently on the tool you can see that
as you move the tax slider t , the supply curve moves up or down by the amount t .
4.1 Select the unit tax t = $60/item. Label the areas in the graph shown in Figure 3. Identify
the areas that correspond to the consumer surplus, producer surplus, dead-weight loss and
the amount of money accrued by the government by means of the tax.
Figure 3
4.2 Experiment with various values of the tax t . Is it true that for positive t (that is, t is a
tax, not a subsidy), the greater the unit tax t, the greater the dead-weight loss?
Explain why tax increases might have this effect.
4.3 Now try negative values for the unit tax t . A negative unit tax is called a subsidy. The
government is paying the producer for each item producer. Note that the big rectangle that
contains the equilibrium point is the amount of money that the government pays out. Also
see that part of this rectangle is filled in with yellow when the tax slider is released. That
area represents the dead-weight loss (as before). What does the dead-weight loss mean in
this context?
5. The Effect of a Sales Tax on Producer and Consumer Surplus
A sales tax affects the demand curve as seen by the producers. Note that the price per item that
the consumers see is p consumer = p producer + t p producer which lowers their demand according to their
demand curve p = D( x ) and gives a net price to the producer of
p producer =
p consumer D( x )
=
1+ t
1+ t
(3)
In effect the producers see a lower demand curve.
5.1
Select the sales tax option on the tool and increase the tax t to verify this phenomenon
for positive values of t . Select the option for the linear supply and demand curves. Note
that the original equations are D(x)= -0.5x+500 and S(x)=0.5x+20. Determine the
following slopes:
Slope of the demand curve for 0% sales tax. (Recall that x is on thousands of units.)
Slope of the demand curve (as seen by the producers) for a 50% sales tax. Recall that 50% = .5.
5.2
Now select the nonlinear case and set t =50%. Label the areas in Figure 4 that correspond
to the producer and consumer surplus, the tax received by the government and the dead
weight loss.
Figure 4