3. consumer choice2

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Consumer Choice 2

Chapters 5, 6, 8

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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” <0, 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.

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Suppose a consumer’s 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-5y=0

ii. L(x, y, λ) = y +100 ln(x) + λ (600 – l0x - 5y)

iii. L1(x, y, λ) = (100/x) + λ (-10)

L2(x, y, λ) = 1 + λ (-5)

L3(x, y, λ) = 600 -10x -5y

so the three equations are

(100/x)+ λ (-10)=0 (1)

1+ λ(-5)=0 (2)

600-10x-5y=0. (3)

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(100/x)+ λ (-10)=0 (1)

1+ λ(-5)=0 (2)

600-10x-5y=0. (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

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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

y*=0 and x* = m/p.

(i.e, all of consumer’s income is spent on the first good).

The outcome would be a corner solution.

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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, λ) = y + 100 ln(x) + λ(m – px - qy)

iii. L1(x, y, λ) = (100/x) + λ (-p)

L2(x,y, λ) = 1 + λ (-q)

L3(x,y, λ) = m – px – qy

so the three equations are

(100/x) + λ (-p) = 0 (1)

1 + λ (-q) = 0 (2)

m – px – qy = 0.(3)

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(100/x) + λ (-p) = 0 (1)

1 + λ (-q) = 0 (2)

m – px – qy = 0.(3)

iv. By (2), λ =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 consumer’s income is too small to afford 100q/p units of the first good (i.e., if m <100q), then the optimal solution is really

x = m/p and y = 0.

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Thus the demand functions take two forms, depending on whether m>100q.

For m> 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.

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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.

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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.

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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 < 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?

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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 student’s 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.

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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.

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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* > m - A - Bx**).

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.

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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.

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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.

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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.

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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).

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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.

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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 * *, , .

bm

m ax y

b bp q p q

a a

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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.

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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.

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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 < 12 (since p > 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.

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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).

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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 <a/b

the optimal bundle is (m/p, 0), and the ICC is a line along the horizontal axis.

Whenever

p/q > a/b,

the optimal bundle is (0, m/q) and the ICC is a line along the vertical axis.

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Example: If

u(x, y) =2x + y

and p = 4, q = 7,

then p/q = 4/7 <2/1 = a/b.

The optimal bundle is (m/4, 0), and the ICC

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

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

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Allowing the first price to change, with

m = 140 and q = 7,

the optimal bundle is (140/p, 0) if p < 14 (i.e., if p/q <a/b)

and (0, 20) if p> 14 (i.e., if p/q > 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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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).

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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.

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

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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.]

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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.)

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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).

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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.

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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 m> 600.

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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 > 600.

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Allowing the first price to change, with

m = 900 and q = 6, the optimal bundle is (600/p, 50).


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Again allowing the first price to change, but with

m = 500 and q = 6,

the optimal bundle is (500/p, 0).


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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 > 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

,u x y y x

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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 > 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.

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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.

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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.

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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* <4. (Why <4?)

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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.

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.

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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).

Plan B increases the consumer’s 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.

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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.

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.

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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.

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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.

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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.

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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.

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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.

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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 consumer’s 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.

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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.]

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.

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).

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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.

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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.