Consumer Choice 2

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Consumer Choice 2

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Title: Consumer Choice 2

1
Consumer Choice 2

Chapters 5, 6, 8

3d. Demand for quasi-linear utility (vertical
shift case)
Recall there are two classes of quasi-linear
utility functions
one corresponding to vertical shifts, the other
to horizontal shifts.
In the vertical shift case, utility is of the
form
u(x, y) f(x) y
where f is an increasing function with negative
second derivative,
f lt0, to guarantee strictly convex preferences.
Quasi-linear utility may lead to either interior
or corner solutions. We can use the Lagrange
multiplier method as long as we recognize that
negative answers mean the solution must be
adjusted.
We will see how to do the adjusting later in this
section. First we start with an example.

Suppose a consumers preferences can be
represented by the utility function
u(x, y) y 100 ln(x) and p 10, q 5, and m
600
where ln(x) is the natural logarithm function.
Using the Lagrange multiplier method,
i. 600-10x-5y0
ii. L(x, y, ?) y 100 ln(x) ? (600 l0x -
5y)
iii. L1(x, y, ?) (100/x) ? (-10)
L2(x, y, ?) 1 ? (-5)
L3(x, y, ?) 600 -10x -5y
so the three equations are
(100/x) ? (-10)0 (1)
1 ?(-5)0 (2)
600-10x-5y0. (3)

(100/x) ? (-10)0 (1)
1 ?(-5)0 (2)
600-10x-5y0. (3)
iv. By the second equation, lambda is 1/5.
Substituting this into the first equation,
(100/x)(-2)0 (1)
Or
x 50.
Then
y (600 - 500)/5 20
by the third equation

Notice that x was determined without using the
budget equation.
The budget equation was used to determine y
after x was found.
This is where the problem of negative solutions
arises.
If the consumer did not have enough money to
purchase 50 units of the first good (for example,
if income were 400 rather than 600), then the
solution for y would be negative.
In that case, the solution should be modified so
that
y0 and x m/p.
(i.e, all of consumers income is spent on the
first good).
The outcome would be a corner solution.

This same technique may be used when parameters
are left in the problem. For example, if we leave
p, q, and m as parameters in the previous
example, our problem becomes that of maximizing
y 100 ln(x) subject to px qy m.
The steps are
i. m px qy 0
ii. L(x,y, ?) y 100 ln(x) ?(m px - qy)
iii. L1(x, y, ?) (100/x) ? (-p)
L2(x,y, ?) 1 ? (-q)
L3(x,y, ?) m px qy
so the three equations are
(100/x) ? (-p) 0 (1)
1 ? (-q) 0 (2)
m px qy 0. (3)

(100/x) ? (-p) 0 (1)
1 ? (-q) 0 (2)
m px qy 0. (3)
iv. By (2), ? 1/q. Substituting this into (1),
(100/x) (1/q)(-p) 0 (1)
or x 100q/p.
Then from (3),
y (m - 100q)/q m/q - 100.
Once again, if the consumers income is too small
to afford 100q/p units of the first good (i.e.,
if m lt100q), then the optimal solution is really
x m/p and y 0.

Thus the demand functions take two forms,
depending on whether mgt100q.
For mgt 100q, there is an interior solution with
x(p, q, m) 100q/p
y(p, q, m) (m -100q)/q.
For m 100q, there is a corner solution with
x(p,q,m) m/p
y(p,q,m) 0.

The fact that the optimal amount of the first
good is independent of income (for incomes above
100q) should be no surprise. Because we are
dealing with the vertical case of parallel
preferences, the MRS depends on x alone. Thus the
MRS at bundles (200, y) is 1/2 for every y, and
if p/q 1/2, the tangency condition can hold
only if x 200.
The figure below shows several budgets with p/q
1/2 but different income levels, along with
indifference curves tangent to the budgets at x
200. For the lowest of the four budgets there is
no tangency. Instead, there is a corner solution
where the budget intersects the x-axis.

The horizontal shift case of quasi-linear utility
(u(x, y) x g(y))
works in similar manner, with the roles of x and
y reversed.

4. Income and Price Changes
4a. Simplified, Motivating Example
Bus trips in Freedonia are sold only on a one-way
basis, at price 1 each. The bus company is
considering the introduction of special fares for
students, that would work as follows.
Each month, each student could purchase a bus
card for A. During the month, the card could be
used by the student to reduce the fare on each
bus trip from 1 to B, where B lt 1. A new card
must be purchased each month.
As consultant to the bus company, you must
determine the effect of such a scheme on bus
ridership and on the total revenues received by
the bus company. If the special fares are
introduced, will the total number of bus trips in
Freedonia necessarily increase?

To answer this question, we will consider an
individual student.
Let x be the number of bus rides per month taken
by the student and
let y be a composite commodity, dollars spent on
all other goods per month at fixed prices.
Let m be the students income per month.
Then the student has two budget options.

Without the bus pass, she has a budget with slope
- 1 (since both goods have price 1 per unit) and
available funds m. With the bus pass, her
available funds have been reduced to (m A) and
the budget has slope - B. The budgets are
illustrated in the first figure.
13

There are three possibilities.
First, she might be strictly better off without
the pass. In this case, she will not buy the
pass, and there will be no effect on her bus
ridership. This is illustrated in the next
figure, where x is the optimal number of bus
trips per month.

14
The second possibility is that she is indifferent
between purchasing and not purchasing the pass
each month. Note that she will increase her
optimal number of bus trips per month if she
purchases a pass, as illustrated in the third
figure, where x is the optimal number of trips
without the pass and x is the optimal number
with the pass.

With diminishing marginal rate of substitution,
the flatter budget line must be tangent to the
indifference curve at a bundle with larger x.
Her optimal bundle with the pass has fewer
dollars spent on all other goods, so she would
be spending more money on bus trips than without
the pass
(m - x gt m - A - Bx).

The fact that with diminishing marginal rate of
substitution, the flatter budget line must be
tangent to the indifference curve at a bundle
with larger x, is an important observation that
will be used again.
It is an example of the Hicksian substitution
effect
If a price changes while income adjusts to
maintain the same level of preference at the new
optimal bundle as at the original one, then with
smooth indifference curves the individual buys
more of the good that has become relatively
cheaper than it originally was.
If the price of x goes down, it becomes
relatively cheaper than it was. If the price of x
goes up, the other good becomes relatively
cheaper than it was.
A crucial condition for the Hicksian substitution
effect is that both optimal bundles lie on the
same indifference curve.

16
The third possibility is that she is strictly
better off with the bus pass. This is the first
case in which we cannot unambiguously determine
the effect of the pass on the number of trips.
In the fourth figure, without knowing more about
the preferences we cannot say how she will change
the number of bus trips she takes each month. If
the price of a pass were C, then she would be
indifferent between having and not having a pass,
and we would know from the previous case that she
would take more rides with the pass than without.
17
Her actual budget with the pass has the same
slope but higher income than this fictional
budget. Thus in order to determine her response,
it would help to understand how she responds to
changes in income.
This motivates our next topic, Income Consumption
Paths (or Curves) and Engel Curves.
18

4b. Income Consumption Curve and Engel Curve
For fixed prices, p and q, the Income Consumption
Curve (or Path) is the collection of optimal
bundles corresponding to all the different
possible income levels.
The fixed prices determine the slope of the
budget line while the income determines the
location.
The Engel Curve is derived from this same
information but relates the quantity demanded for
one of the goods to the income level (i.e.,
income is on the horizontal axis and the quantity
demanded for the good of interest is on the
vertical axis).
A good is said to be
normal if the quantity demanded increases as
income increases (i.e., if the Engel curve is
upward sloping)
inferior if the quantity demanded decreases as
income increases (i.e., if the Engel curve is
downward sloping).

4c. Price Consumption Curve and Demand Curve
The Price Consumption Curve and Demand Curve are
analogs of the
ICC and Engel curve.
The difference is that income and one price are
fixed while the other price varies.

4d. Perfect complements
We have already seen the optimal bundle will be
at a kink. With
u(x, y) minimum x/a, y/b
the kinks lie along the line
y (b/a)x.

The optimal bundle (x, y) is
21

Example If
u(x, y) minimum x/2, y
(so a 2 and b 1) and
p 5, q 20,
then the optimal bundle is
(m/(5 10), (l/2)m(5 10)) (m/15, m/30).

In the graph of points (x, y), as m varies this
traces out the line y (1/2)x, which is the
Income Consumption Curve. Note the ICC coincides
with the kink points.
22

The corresponding Engel curve is the graph of the
optimal choice for one of the goods as a function
of income.
For the first good, x m/15, so the Engel curve
is

Note the axes are income and the quantity of the
good being considered.
23

Allowing the first price to change, with m 120
and q 20, the optimal bundle is
(120/(p 10), 60/(p 10)).
In the graph of points (x, y), as p varies this
traces out the line y (1/2)x, for x lt 12 (since
p gt 0), which is the Price Consumption Curve.
Perfect complements is a special case in which
the ratio of x to y in the optimal bundle does
not depend on the prices.

The demand curve for the first good is the graph
of x 120/(p 10).
Recall that compared to the graph of the
mathematical function x(p), with p on the
horizontal axis and x on the vertical axis, for
historical reasons economists graph the demand
curve x(p) with the axes flipped (i.e., with x
on the horizontal axis and p on the vertical
axis).

4e. Perfect substitutes
We have already seen the optimal bundle is
typically at a corner. With utility function
u(x, y) ax by,
whenever
p/q lta/b
the optimal bundle is (m/p, 0), and the ICC is a
line along the horizontal axis.
Whenever
p/q gt a/b,
the optimal bundle is (0, m/q) and the ICC is a
line along the vertical axis.

Example If
u(x, y) 2x y
and p 4, q 7,
then p/q 4/7 lt2/1 a/b.
The optimal bundle is (m/4, 0), and the ICC

The Engel curve for the first good is the graph
of x m/4,

while the Engel curve for the second good is the
graph of y 0,

Allowing the first price to change, with
m 140 and q 7,
the optimal bundle is (140/p, 0) if p lt 14 (i.e.,
if p/q lta/b)
and (0, 20) if pgt 14 (i.e., if p/q gt a/b).
At p 14 all bundles on the budget line are tied
for best (so demand is not a function in the
mathematical sense). The PCC and the demand curve
for the first good are

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31

4.f Cobb-Douglas
For Cobb-Douglas utility the optimal bundle is
interior and satisfies two conditions
MRS p/q
and
px qy m.
Since the MRS is constant along any ray from (0,
0),
the ICC will be the collection of all those
bundles at which the MRS is equal to the price
ratio.
This is always a ray through (0, 0).

Example For u(x, y) xy, and p 10, q 5,
the optimal bundle is (m/20, m/b).
The ICC can also be found as the set of bundles
at which MRS 2,
or y/x 2, or y 2x.

The Engel curve for the first good is the graph
of x m/20,

Allowing the first price to change, with m 100
and q 5, the optimal bundle is (50/p, 10).
The PCC and the demand curve for the first good
are

Cobb-Douglas preferences are a special case in
which the demand function x(p, q, m) does not
depend on q, and the demand function y(p, q, m)
does not depend on p. This yields the unusual
shape of the PCC a horizontal line when p
varies. If q varies, the PCC is a vertical line
in the Cobb-Douglas case.
35

In all three classes of preferences we have
considered so far, the ICC is always a straight
line through (0, 0).
For the perfect complements class, the slope of
the ICC depended on the preferences but not on
the price ratio, p/q.
For the perfect substitutes class, the slope of
the ICC depends on the preferences and on the
price ratio, p/q, but the ICC could only be
vertical or horizontal.
For the Cobb-Douglas class, the slope of the ICC
depended on the preferences and the price ratio,
and could take any positive value.
In the remaining class, the ICC is not a straight
line.
(Note. In the perfect substitutes case, when p/q
a/b the ICC is not a straight line. Recall the
optimal bundle is not unique in that case.
Instead, all bundles on the budget line are tied
for best.)

4g. Quasi-linear utility
For quasi-linear utility, the optimal bundle is a
corner solution for low income levels and an
interior solution for high income levels. For the
vertical shift case, with utility function
u(x, y) f(x) y,
the corner solution will be
(m/p. 0),
which will apply as long as m px, where x
satisfies
MRS p/q f (x) p/q.
For larger incomes, the optimal bundle will be
(x, (m - px)/q).

Example For f(x) 100 ln(x) and p 3, q 6,
the corresponding x satisfies
100/x 3/6 or x 200.
The ICC is horizontal between (0, 0) and (200, 0)
and then vertical.

The corresponding Engel curve for the first good
has two segments,
the first with x m/3 when m 600 and
the second with x 200 when mgt 600.

The corresponding Engel curve for the second good
has two segments,
the first with y 0 when m 600 and
the second with y (m - 600)/6 when m gt 600.

Allowing the first price to change, with
m 900 and q 6, the optimal bundle is (600/p,
50).
The PCC and the demand curve for the first good
are

Again allowing the first price to change, but
with
m 500 and q 6,
the optimal bundle is (500/p, 0).
The PCC and the demand curve for the first good
are

For our previous special classes of preferences,
the properties exhibited by the graphs for the
examples were general in a qualitative sense.
That is true for the ICC and Engel curves for the
quasi-linear case, but not for the PCC. To see
this, consider the quasi-linear utility function.
For m gt q2/4p, there is an interior solution and
the demand functions are
x(p, q, m) q2/4p2
y(p, q, m) m/q - q/4p
For m q2/4p, there is a corner solution and the
demand functions are
x(p,q,rn)m/p
(p,q, rn)0

For any p and q, the ICC and Engel curves have
shapes similar to those of the previous
quasi-linear example.
However, the PCC does not!
For example, for m 1 and q 2, when p 1,
x(p, 2, 1) 1/p and y(p, 2, 1) 0
while for p gt 1,
x(p, 2, 1) 1/p2 and y(p, 2, 1) 1/2 - 1/2p.
Thus the PCC has a shape that differs from that
of the previous quasi-linear example.

5. EXAMPLE Use of Demand Theory in Policy
Analysis
A government nutrition panel has determined that
low income individuals do not obtain an adequate
amount of calcium in their diets. As economic
consultant to the panel, your job is to evaluate
the cost effectiveness of various proposed
remedies. For simplicity
(i) assume that the only way to obtain an
adequate amount of calcium is to consume four
quarts of milk per week
(ii) consider only a single (representative) low
income consumer who purchases some milk at the
market price and who would purchase four quarts
per week if either the price were low enough or
income were high enough
(iii) assume all commodities are normal goods and
the consumer has the usual smooth, convex
indifference curves and
(iv) ignore any effects on the market price for
milk and any potential savings due to improved
health, etc.

The proposed remedies are
Plan A, milk price subsidies for low income
people (the subsidy must be large enough to raise
milk consumption to four quarts per week)
Plan B, direct cash payments to low income people
which can be used for any purpose (the payment
must be large enough to raise milk consumption to
four quarts per week)
Plan C, Government purchase and distribution to
low income individuals of four quarts of milk per
week (with no resale of the milk permitted).
Questions
a. Compare the cost to the government of plans A
and B.
b. Which of plans A and B does the consumer
prefer?
c Compare the cost to the government of plans A
and C.
d Compare the cost to the government of plans B
and C.

5a. Analysis of the Alternative Policies
Let the first good be quarts of milk per week and
let the second good be a composite good, dollars
spent on all other goods per week at fixed
prices. From the information provided in the
problem, the initial budget and optimal choice
must be as in the figure below. The budget line
includes (0, m) (since in dollars are available
for other goods if x 0) and has slope -p/q -
p, where p is the price of a quart of milk. The
optimal bundle is (x, m - px), where x lt4.
(Why lt4?)

47
Plan A lowers the price of milk. The budget line
still goes through (0, m), but it is now flatter.
As the subsidy is increased, the corresponding
optimal bundles trace out a portion of the Price
Consumption Curve. The subsidy is increased until
the optimal bundle contains 4 quarts of milk per
week. This occurs where the PCC crosses the
vertical line, x 4. Since vertical distances
are measured in dollars per week, the cost to the
government (per week) is the vertical distance
between the two budgets at x 4.

For example, if (4, 39) is the point on the old
budget and (4, 40) is the point on the new
budget, then the 1 per week difference must be
the total amount paid by the government as a
subsidy, with a subsidy of 0.25 per quart.

48
Plan B increases the consumers income. The slope
of the budget line is unchanged, but the line
shifts up. As the income is increased, the
corresponding optimal bundles trace out a portion
of the Income Consumption Curve. Since all goods
are normal, as income increases the quantity
demanded also increases for each good. Thus the
ICC must be an upward sloping curve.

The income is increased until the optimal bundle
contains 4 quarts of milk per week. This occurs
where the ICC crosses the vertical line, x 4.
Since vertical distances are measured in dollars
per week, the cost to the government (per week)
is again the vertical distance between the two
budgets at x 4 (or anywhere, since the budgets
are parallel).

49
5b. An Aside The Slutsky Substitution Effect To
compare the cost of these two plans, we need to
make one important observation. In the figure
below, the indifference curve is tangent to
budget 1 at (x, m - px). The other two budgets
also go through the point (x, m - px) but have
different slopes. Budgets 2 and 3 may have
different prices and different income levels than
budget 1.

The key point is that the optimal bundle for
budget 1 is just affordable with either of
budgets 2 and 3 (i.e., it is on budget line 2 and
on budget line 3). With a little thought, it is
easy to see how the optimal bundles for the
different budgets will compare.

Along budget 2, bundles with less than x units
of the first good are worse than (x, m -px)
while at least some of the bundles with more than
x units of the first good are strictly better.
Thus the optimal bundle will have more than x
units of the first good. Note that budget 2 is
flatter than budget 1, so x is relatively cheaper
under budget 2 than under budget 1.

Along budget 3, bundles with more than x units
of the first good are worse than (x, m -px)
while at least some of the bundles with less than
x units of the first good are strictly better.
Thus the optimal bundle will have less than x
units of the first good. Note that budget 3 is
steeper than budget I, so x is relatively more
expensive under budget 3 than under budget 1.

These are examples of substitution effects
(called Slutsky substitution effects for this
version as opposed to Hicksian substitution
effects for the version used in the bus example).
Starting from budget 1 and the optimal bundle
(x, m - px),
if the price of the first good changes and income
is adjusted so that the original optimal bundle
is still just affordable, then the price change
and quantity change for the first good are of
opposite sign.
In budget 2, the price of the first good
decreased while the optimal quantity of the first
good increased.
In budget 3, the price of the first good
increased while the optimal quantity of the first
good decreased.

The Hicksian and Slutsky substitution effects
come to the same conclusions
When income is adjusted appropriately, the
price change and quantity change for a good move
in opposite directions.
The Hicksian and Slutsky versions differ in terms
of what is used as the appropriate income
adjustment.
For the Slutsky version, income is adjusted so
that the optimal bundle for the original budget
is just affordable at the new prices.
For the Hicksian version, income is adjusted so
that the consumer is just as well off in
preference terms when she chooses the optimal
bundle under either budget.

Either version may be used to decompose a
response to a price change into income and
substitution effects. This is illustrated for the
Slutsky version using the following diagram. For
budget 1, x is the optimal amount of x, while
x is optimal for budget 2, in which x has
become cheaper. The change from x to x is
decomposed into income and (Slutsky) substitution
effects as follows. First, create an artificial
budget by using the prices from budget 2 but
adjusting the income so that it is just enough to
afford the bundle that was optimal for budget 1.

First, create an artificial budget by using the
prices from budget 2 but adjusting the income so
that it is just enough to afford the bundle that
was optimal for budget 1.
55

If we can find the optimal bundle for the
artificial budget, call it (x, y), then the
substitution effect is x - x (i.e., the
change in optimal x as we move from the original
budget to the artificial budget). The income
effect is x- x (i.e., the change in optimal
x as we move from the artificial budget to the
real new budget). It is called the income effect
because the artificial budget and budget 2 have
the same prices but differ in income.

To find x, we need to know something about the
consumers preferences. For example, if the
consumer had vertical-shift parallel preferences,
then the income change would not affect the
optimal x, and x x. In that case, the
entire change would be due to the substitution
effect, and the income effect would be zero.
56
5c. Back to the Policy Analysis We are now able
to compare the cost of plans A and B. Start with
the solution for plan A and calculate the total
cost to the government. Now consider giving this
same amount to the consumer as a direct cash
payment without subsidizing the price, and call
this plan B. The cost to the government is the
same under plans A and B, and the consumer has
just enough money at the unsubsidized prices of
plan B to afford the optimal bundle from plan A.

But at plan B prices, milk is more expensive
than under the subsidy of plan A, so by the
(Slutsky) substitution effect, the consumer will
purchase less than 4 quarts of milk per week.
Plan A corresponds to budget 1 and plan B
corresponds to budget 3 in the previous section.
Since exactly 4 quarts per week are consumed
under plan A (budget 1), the optimal bundle under
plan B (budget 3) includes less than 4 quarts
per week.

To get the consumer to purchase 4 quarts per week
under plan B, we would need an even larger cash
payment than that used in plan B. Thus plan B is
more expensive than plan A (and the consumer
would prefer plan B).
57

Plan C is clearly more expensive than plan A,
since the government pays the entire cost of the
milk as opposed to just a portion of the cost.

Plan C could be more or less expensive than plan
B, depending on the shape of the ICC. The cost of
plan C is 4p, which is the vertical distance at x
4 between the original budget and the
horizontal line y m. In the first figure below,
this is more than the cost of plan B. In the
second figure it is less than the cost of plan B.

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