ECN 100
Midterm
SSI 12
(16 pts) 1.
Consumer Surplus
Suppose you are in charge of a toll bridge that costs essentially nothing to operate. The demand
for bridge crossings Q is given by P = 30 - 0.5Q.
a. Draw the demand curve for bridge crossings. (2 pts)
b. How many people would cross the bridge if there were no toll? (2 pts)
c. What is the loss of consumer surplus associated with a bridge toll of $5? (4 pts)
d. The toll bridge operator is considering an increase in the toll to $6. At this new
higher price, how many people would cross the bridge? Would the toll bridge
revenue increase or decrease? What does your answer tell you about the elasticity of
demand? (4 pts)
e. Find the lost consumer surplus associated with the increase in the price of the toll
from $5 to $6. (4 pts)
Answer)
a. The demand curve is linear and downward sloping. The vertical intercept is 30 and
the horizontal intercept is 60.
P
30
5
50
60
Q
b. At a price of zero, the quantity demanded would be 60.
c. If the toll is $5 then the quantity demanded is 50. The lost consumer surplus is the
area below the price line of $5 and to the left of the demand curve. The lost consumer
surplus can be calculated as (5*50) + 0.5(5*10) = $275.
d. At a toll of $6, the quantity demanded would be 48. The initial toll revenue was
$5*50 = $250. The new toll revenue is $6*48 = $288. Since the revenue went up
when the toll was increased, demand is inelastic.
e. The lost consumer surplus is (6-5)*48+0.5(6-5)(50-48) = $49.
(9 pts) 2.
Tax, Price support, and Price Ceiling
The figure below shows the demand and supply curves in the market for milk. Currently
the market is in equilibrium.
a. If the government imposes a $2 per gallon tax to be collected from sellers,
estimate the change in p, Q, and social welfare.
b. If the government establishes a $4 per gallon price support, estimate the change in
p, Q, and social welfare.
c. If the government establishes a $2 per gallon price ceiling to ensure that children
are nourished, estimate the change in p, Q, and social welfare.
Answer:
a. The supply curve shifts vertically by $2. The price changes from $3 per gallon to
$4 per gallon. Quantity falls from 1,000 gallons to 500 gallons. For the 500
gallons no longer produced, consumer surplus was $250 and producer surplus was
$250. Producers and consumers also pay $1,000, but that represents a transfer to
the government.
b. Price rises to $4 per gallon. Consumers purchase only 500 gallons of milk. The
government purchases 1,000 gallons of milk to support the price at $4. Thus a
total of 1,500 gallons is produced. The loss in social welfare equals 1,000 gallons
of milk at $4/gallon (equals $4,000) less the producer surplus above the old
demand curve up to a price of $4 (which is $500). The loss in social welfare is
$3,500.
c. At a price of $2, only 500 gallons are produced. The deadweight loss equals
[(1000 - 500) ∗ (4 - 2)]/2 = $500.
(8 pts) 3. Tax
Incidence
What effect doe a $1 specific tax have on equilibrium price and quantity, and what is the
incidence on consumers, if the following is true:
a. The demand curve is perfectly inelastic.
b. The supply curve is perfectly elastic.
dP


. where  is the specific tax, η price
(Hint: Can use the following formula
d   
elasticity of supply and is price elasticity of demand)
Solution
a. If demand is perfectly inelastic, the demand curve is vertical. The supply curve
shifts up by $1, and all of the incidence falls on consumers. Price increases by $1,
and there is no change in quantity. Since   0, dP/d 
b. When supply is perfectly elastic, the supply curve is horizontal. The supply curve shifts
up by $1 increasing price by $1. The quantity falls and the incidence falls entirely on the
consumer. Since   , dP/d  1.
(4 pts) 4. Utility
Function
The following combinations of goods X and Y represent various market baskets.
Consumption is measured in pounds per month.
Market Basket
Units of X
Units of Y
A
4
6
B
16
7
C
15
3
D
3
2
Explain which market basket(s) is(are) preferred to other(s), and if there is any
uncertainty over which is preferable, point this out as well.
Solution:
Since more of each good is preferred to less, we can conclude that a market basket is
preferred to an alternative basket if it lies above and to the right of the alternative basket.
Combination B is preferred to all others. A and C cannot be compared without additional
information. A and C are both preferred over D.
(9 pts) 5. Food
Coupon vs Cash Transfer
Lisa's budget line and her satisfaction maximizing market basket, A, are shown in the
diagram below.
a. Suppose that Lisa is given $50 worth of coupons that must be spent on food.
How will the coupons alter Lisa's budget line?
b. Suppose that Lisa is given $50 in cash instead of $50 in coupons. How will this
alter Lisa's budget line?
c. Is Lisa indifferent between the food coupon and cash program, or does she prefer
one program over the other? Draw an indifference curve to illustrate your answer.
Solution:
Refer to the following diagram with the answers.
a. With the coupons, Lisa’s budget is abc.
b. With cash, Lisa’s budget line is dbc.
c. If Lisa’s preferences are as shown, she is indifferent between the two programs.
However, if her preferences were such that an indifference curve was tangent to
the db portion of dbc, she would prefer cash.
(28 pts) 6.
Consumer Problem and Marshallian Demand Function (each part 4 pts)
Nader has preferences over two goods x (measured in pounds/day) and y (=Other Goods
measured in terms of $/day) are presented by u = u(x, y). Nader faces market prices p and q (for
x and y respectively. Nader has m = $/day available to spend.
(1) Set up the consumer problem
(2) Show the solution to the consumer’s problem graphically (label your graph carefully)
(3) Characterize Nader’s the optimal choice mathematically (Hint: state the two conditions that
must hold at the optimum)
(4) Derive Nader’s Marshallian demand curve graphically
(5) Show Income effect and Substitution effect of a price increase using a graph.
3
4
1
4
(6) Assume the following Cobb-Douglas utility function: ux, y   x y .
a. Solve for the Marshallian demand functions of x and y.
b. If m = 12, p = 3, and q = 1, then show that Cobb-Douglas' optimal choice is x = 3 and y = 3.
Answer)
(1) Choose x and y that Maximizes u(x, y) subject to m = px + qy
(2) --MUx
p
(3) Tangency Condition: |slope of BL| =|slope of IC| or MRT= MRS or q = MU
y
Budget constraint: m = px + qy
(4), (5) ---
(6)
|slope of BL| = pq
=
3y
x

3  14 14
x y
MUx
4
= MRS = |slope of IC|
3
3 =
1 4  4 MUy
x y
4
px
(*)
3q
Use budget constraint together with (*) to get the demand functions for x and y:
px
Equation (*) can be rewritten qy 
3
which simplifies to
y 
Substituting this for qy in the Budget Equation gives: m

3m
m
Dd

(**) similarly we find y
4p
4q
which is the demand (Marshallian) functions for x and y
so
x Dd
px  qy 
px 
px
3

4px
3

b. Substituting these values for m, p and q into the demand formulas for x and y in a. gives
the specified values of x and y.
Givenm  12, p  3,and q  1,
3m 3 12
x

 3,
4 p 43
m 12
y

 3.
4q 4 1
(18 pts) 7.
Hicksian Demand Function (each part 3 pts)
Nader has preferences over two goods x (measured in pounds/day) and y (=Other Goods
measured in terms of $/day) are presented by u = u(x, y). Nader faces market prices p and q (for
x and y respectively. Nader has m = $/day available to spend.
(1) Set up Nader’s expenditure Minimization problem and conceptually write the solution to this
problem (Hint: minimum expenditure function)
(2) Derive Nader’s Hicksian demand curve graphically
(3) Assume the following Cobb-Douglas utility function: u(x,y) = x0.75y.25. The minimum
expenditure function for this utility function is: E=U_bar (p/0.75)0.75(q/0.25)0.25 or simply
E = 1.75U_bar p0.75q0.25.
a. Find Hicksian or compensated demand curve for x.
b. If m = 12, p = 3, and q = 1, Nader’s optimal choice is x = 3 and y = 3.
i.
Show that that Nader’s utility level at x = 3 and y = 3 is 3.
ii. How much is Nader worse off as a result of this price increase (calculate the CV).
iii. Show that for p=4 Nader’s Hicksian demand for x is 2.78.
(9 pts) 8.
Welfare Analysis
Rachel has an income of $10 which she spends on burritos and other goods (“other
goods” represents a composite of all other goods). The price of burritos is $1 as is the
price of other goods.
a. Suppose the government agrees to pay half of Rachel’s burrito bill, so burritos
now cost her $.50 apiece. She now chooses to buy eight burritos. On a graph
with Burritos on the x-axis and other goods on the y-axis, draw Rachel’s budget
constraints before and after the subsidy program. Also include an indifference
curve and the coordinates of Rachel’s optimal bundle with the subsidy.
b. Now, suppose the government ends the program in part a. and replaces it with a
new and simpler program: Rachel just gets a cash gift of $4. Show his new
budget line. Is Rachel still able to afford her optimal bundle that she chose with
the subsidy?
c. How much does each program cost the government? Which program does Rachel
prefer? Explain.
Answer:
a.
b. It goes through the point (6,8) so it is just affordable with the new policy.
c. They both cost the same. Before, Rachel was purchasing 4 burritos, so it cost $4 to
the government. Now, the government is paying $4 total.
Rachel is better because she can now reach a higher indifference curve.