Economics 323H Professor Gronberg
Fall 2007 Problem Set 4 - Answer Key
a) If the income consumption curve is vertical the utility function has no income
effect. This will occur, for example, with a quasi-linear utility function. This
utility function will have the same marginal rate of substitution for any particular
value of tea regardless of the level of total utility. If the price of tea falls,
flattening the budget line, the consumer will reach a new optimum where the
marginal rate of substitution is equal to the slope of the new budget line. Since
the budget line has flattened, this cannot occur at the previous optimum amount of
tea. The substitution effect implies that this new optimum level of tea will be
greater than the previous level. Thus, when the price of tea falls, the quantity of
tea demanded increases, implying a downward sloping demand curve. This can
be seen in the following figure.
Level of tea
Price of tea falls
b) Yes, the values will be exactly $30. When the income consumption curve is
vertical, the consumer's utility function has no income effect. As stated in the
text, when there is no income effect, compensating and equivalent variation will
be identical and these will also equal the change in consumer surplus measured as
the change in the area under the demand curve.
400 Slope = -20
Kink in budget line
Slope = -10
0 5 10 15 20 25
Because the wage rate changes for any hours worked over eight (leisure less than
sixteen) the budget line has a kink at sixteen hours of leisure.
0 5 10 15 20 25
With this set of indifference curves, the consumer reaches an optimum at 12 hours
of leisure and 12 hours of labor, or $160 of income.
5.22) If Terry's wage rate is w, then the income he earns from working is (24 - L)w.
Since PY = 1, the number of units of other goods he purchases is Y = (24 - L)w.
Now at an optimal bundle, Terry's MRSL,Y must equal the price ratio w/PY = w.
Therefore, the tangency condition tells us that = w . The two conditions
imply w(1 + L) = (24 + L)w. This means that the optimal amount of leisure is
L = 11.5. You can see that this does not depend on the wage rate.
Assume the consumer's original income was M. His original budget constraint is a; after
the gas price increase, it is b; and after the money from his uncle it is c. At the very end,
he can afford the original bundle he bought. But his indifference curve at the original
optimum is below his new budget constraint. By decreasing his gas consumption and
increasing other goods consumption, he can reach a higher indifference curve. (see dotted
3. The loss in consumer's surplus is 5, the area of the shaded trapezoid in the diagram:
4 6 8
P= $4/movie, Q=8 movies/year, consumer surplus = $64. So maximum he will pay for
membership is $64.
5. a) The area under the curve from the 6th visit to the 10th is ($6 + $4 + $3 + $2 + $1) =
b) The total area under the curve adds ($20 + $16 + $12 + $10 + $8) to $16 for a
grand total of $82. Net of a $75 fee, benefits would be $7, so they would buy the
c) An $8 charge would support 5 trips. Total value is $66, but the total charge would
be $40 = $8 @ 5; consumer surplus would equal ($66 - $40) = $26. For a $4
charge, 7 visits would be enjoyed with total benefit of $66 + $6 + $4 = $76.
These 7 visits would therefore produce $76 - $28 = $48 in consumer surplus.
a) )CS = shaded area = $150
b) DWL = Cost of Subsidy – )CS
7. a) The demand curve is linear and downward sloping. The vertical intercept is 15
and the horizontal intercept is 30.
b) At a price of zero, the quantity demanded would be 30.
c) If the toll is $5 then the quantity demanded is 20. 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*20) + 0.5(5*10) = $125.
d) At a toll of $7, the quantity demanded would be 16. The initial toll revenue was
$5*20 = $100. The new toll revenue is $7*16 = $112. Since the revenue went up
when the toll was increased, demand is inelastic (the increase in price (40%)
outweighed the decline in quantity demanded (20%).
e) The lost consumer surplus is (7 - 5)*16 + 0.5(7 - 5)(20 - 16) = $36.