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Chapter 4

Individual Demands

Having covered budgets and preferences in Chapters 2 and 3, we are now ready to focus on consumer choice behavior. A consumer’s demand for each good is found by maximizing her preferences over her budget, i.e., by find- ing a consumption bundle within her budget set which is strictly better or at least as good as any other affordable bundle. We find this preference- maximizing bundle graphically, deduce the necessary mathematical condi- tions, and apply these conditions to the preferences introduced in Chapter 3 to calculate demand functions.

4.1 Preference Maximization on Budgets

To maximize a consumer’s preferences over the bundles she can afford, bring together her budget and her preferences. This is illustrated in Figure 4.1 where her budget line is drawn in blue and her preferences are represented by the orange indifference curves.While bundle A in the interior of the bud- get set is certainly affordable, bundle B is also affordable and lies on a higher indifference curve than A. In fact, it is easy to verify that there is no other bundle in the budget set that lies on a higher indifference curve than ū2. Therefore, the quantities of the two goods (x̄1, x̄2) at B maximize this con- sumer’s preferences subject to the given budget.

The preference-maximizing bundle, B = (x̄1, x̄2), is said to be an interior solution to the consumer’s preference (or utility) maximization problem be- cause x̄1 > 0 and x̄2 > 0, i.e., an interior solution is one where both goods are consumed. Here the indifference curve that passes through B is tangent to the budget line, so the slope of the indifference curve (which is the negative

0 ? D?? 9 C 4 ? ? C ? 5C= I?=I I C=M , / :II 0 C C A / I =B 8I ? A? I7 ?M 3 IIE 1? B , ? IIE=? I ?M =I C ? C = CI . I=42-

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62 Chapter 4

0 u1-

x1-

x2-

u2-

B A

Figure 4.1 Interior preference maximization

of the MRS) at that point equals that of the budget constraint:

−MRS(x̄1, x̄2) = − p1 p2

or what amounts to the same thing,

MRS(x̄1, x̄2) = p1 p2

. (4.1)

The tangency condition given in equation (4.1) — the equality of the marginal rate of substitution to the ratio of the commodity prices — is the primary mathematical condition used to algebraically calculate individual demands in section 4.2.

Sometimes, however, the preference-maximizing bundle is a corner so- lution, meaning that either the quantity of x1 or that of x2 is zero. In other words, a corner solution is one where only one good is consumed. Several examples will be considered in more detail in sections 4.2.1 and 4.2.3 below in the context of linear and quasilinear preferences. As you will see, a corner solution along the horizontal axis (where x̄1 > 0 and x̄2 = 0) requires the indifference curve to be steeper than or the same slope as the budget line, i.e.,

−MRS(x̄1, 0) ≤ − p1 p2

or MRS(x̄1, 0) ≥

p1 p2

. (4.2)

0 ? D?? 9 C 4 ? ? C ? 5C= I?=I I C=M , / :II 0 C C A / I =B 8I ? A? I7 ?M 3 IIE 1? B , ? IIE=? I ?M =I C ? C = CI . I=42-

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Individual Demands 63

Similarly, for a corner solution along the vertical axis (where x̄1 = 0 and x̄2 > 0), the indifference curve must be as flat as or flatter than the budget line:

MRS(0, x̄2) ≤ p1 p2

. (4.3)

In summary, at an interior utility-maximizing bundle, equation (4.1) must hold; at a corner utility-maximizing bundle, either (4.2) or (4.3) must hold.

4.2 Calculating Individual Demands

4.2.1 Demands for linear preferences

Suppose a consumer’s utility is u(x1, x2) = 2x1 + x2 (so her indifference curves have a slope of −2) and her income is m = $60. There are three possibilities, each illustrated in the panels of Figure 4.2.

In the left panel, p1 = $15 and p2 = $6, so the blue budget line has a slope of −2.5 and is steeper than the orange indifference curve. In this case the utility-maximizing bundle is at A = (0, 10). Since the MRS = 2 and the price-ratio p1/p2 = 2.5, equation (4.3) holds with a strict inequality at A when the consumer buys no units of x1 and spends all her income on x2.

In the middle panel, p2 = $6 as before, but p1 = $12. The highest in- difference curve the consumer can reach coincides with the budget line, so there is no single bundle that maximizes her preferences: any point on the

0 x1

x2 A

E – 2.5

x2 B

– 2 – 1

Figure 4.2 Linear preferences maximization

0 ? D?? 9 C 4 ? ? C ? 5C= I?=I I C=M , / :II 0 C C A / I =B 8I ? A? I7 ?M 3 IIE 1? B , ? IIE=? I ?M =I C ? C = CI . I=42-

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64 Chapter 4

budget line from point B to point D inclusive maximizes her preferences. At an interior solution such as C, equation (4.1) holds. At the corner solution B where x̄1 = 0 and x̄2 > 0, (4.3) holds with an equality, while at the corner solution D where x̄1 > 0 and x̄2 = 0, (4.2) holds with an equality.

In the right panel, p1 = p2 = $12, and the blue budget line is flatter than the orange indifference curve. Preferences are maximized at point E = (5, 0) where x̄1 > 0 but x̄2 = 0. Since the MRS = 2 and the price-ratio p1/p2 = 1, equation (4.2) holds with a strict inequality.

Thus, which bundle is preference-maximizing depends on the slope of the budget relative to the slope of the indifference curves. Because we want to see how the preference-maximizing bundle changes with prices and in- comes, we will calculate demand functions, writing the demand for x1 as h1(p1, p2, m) and for x2 as h2(p1, p2, m). In other words,

x1 = h1(p1, p2, m) and x2 = h2(p1, p2, m),

signifying that the quantity demanded of each good depends in general on p1, p2, and m.1 For example, the preference-maximizing bundle in the left panel of Figure 4.2 is given by h1(15, 6, 60) = 0 and h2(15, 6, 60) = 10, while that in the right panel is h1(12, 12, 60) = 5 and h2(12, 12, 60) = 0. The de- mand functions for both goods together are written more compactly as

h(p1, p2, m) = (h1(p1, p2, m), h2(p1, p2, m)),

where h(p1, p2, m) refers to the pair of individual demands, the demands for good 1 and good 2 listed in order.

In general then, the demand functions corresponding to the left panel in Figure 4.2 are given by

h1(p1, p2, m) = 0 and h2(p1, p2, m) = m p2

when p1/p2 > 1. When p1/p2 = 1 as in the case of the middle panel, any bundle (x̄1, x̄2) that satisfies p1 x̄1 + p2 x̄2 = m is preference-maximizing, so

h1(p1, p2, m) = x̄1 and h2(p1, p2, m) = x̄2.

Finally, when p1/p2 < 1 as in the right panel,

h1(p1, p2, m) = m p1

and h2(p1, p2, m) = 0.

1Such demand functions are sometimes called Marshallian (after Alfred Marshall) or Wal- rasian (after Léon Walras) demand functions.

0 ? D?? 9 C 4 ? ? C ? 5C= I?=I I C=M , / :II 0 C C A / I =B 8I ? A? I7 ?M 3 IIE 1? B , ? IIE=? I ?M =I C ? C = CI . I=42-

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Individual Demands 65

Summarizing these derivations, the demand for the linear utility u(x1, x2) = ax1 + bx2 is given by

h(p1, p2, m) =

⎧ ⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

( 0,

m p2

) if p1/p2 > a/b

{(x̄1, x̄2) : p1 x̄1 + p2 x̄2 = m} if p1/p2 = a/b( m p1

, 0 )

if p1/p2 < a/b.

(4.4)

Note that even though demands depend on prices p1, p2, and income m in general, not all of these variables are necessarily present simultaneously on the right hand side; indeed, when the demand is zero, it is independent of all of these variables.

4.2.2 Demands for Leontief preferences

Suppose u(x1, x2) = min{x1, x2}. Then, all the kinks in the indifference curves lie along the dashed magenta 45◦ ray through the origin given by x2 = x1. The highest indifference curve attainable passes through the point where this line intersects the budget line p1x1 + p2x2 = m at point A in Fig- ure 4.3.

x2 = x1

45°

Figure 4.3 Leontief preference maximization

Replacing x2 with x1 in the budget and solving, we obtain the demand ¶ b function for good x1 to be

h1(p1, p2, m) = m

p1 + p2 .

0 ? D?? 9 C 4 ? ? C ? 5C= I?=I I C=M , / :II 0 C C A / I =B 8I ? A? I7 ?M 3 IIE 1? B , ? IIE=? I ?M =I C ? C = CI . I=42-

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66 Chapter 4

Since x2 = x1 at the demanded bundle, h2(p1, p2, m) is also given by the formula above. Then, the demand functions for this Leontief utility can be written as

h(p1, p2, m) = (

m p1 + p2

, m

p1 + p2

) . (4.5)

In general, when u(x1, x2) = min{ax1, bx2}, the kinks lie along the line x2 = ax1/b and the demand functions are given byb ·

h(p1, p2, m) = (

bm bp1 + ap2

, am

bp1 + ap2

) . (4.6)

Note that these demands are functions of all three variables, p1, p2, and m, and that the ratio of the demand for x2 to the demand for x1 is a : b.

4.2.3 Demands for quasilinear preferences

The solutions to preference maximization for quasilinear preferences may be interior or corner ones, depending on the prices and income. The following four steps provide an algorithm to find them.

1. Solve for an interior solution by using the equation MRS = p1/p2. This yields one demand function (either for good 1 or good 2 depending on the quasilinear utility function).

2. Solve for the demand for the other good by substituting the demand function obtained in step 1 into the budget equation, p1x1 + p2x2 = m.

3. For the demand derived in step 2, determine if it can be negative for certain values of p1, p2, and m and derive a condition for an interior solution.

4. If the demand derived in step 2 is for good 1, then the corner solution is (0, m/p2) when the condition for an interior solution does not hold. Conversely, if the demand derived in step 2 is for good 2, then the corner solution is (m/p1, 0).

Consider the case when the consumer’s quasilinear preferences are rep- resented by the utility function u(x1, x2) = 2

√ x1 + x2. To illustrate the steps

above, derive the marginal rate of substitution, MRS = 1/ √

x1, and equate

0 ? D?? 9 C 4 ? ? C ? 5C= I?=I I C=M , / :II 0 C C A / I =B 8I ? A? I7 ?M 3 IIE 1? B , ? IIE=? I ?M =I C ? C = CI . I=42-

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Individual Demands 67

it to the price ratio following step 1: 1/ √

x1 = p1/p2. Solving for x1 yieldsb · the demand for good 1:

h1(p1, p2, m) = p22 p21

. (4.7)

From step 2, substitute (4.7) in the budget equation p1x1 + p2x2 = m and solve for x2 to obtain the demand for good 2: ¶ b

h2(p1, p2, m) = m p2

− p2 p1

. (4.8)

Note that in order to have an interior solution, h2 must be positive. Setting the right hand side of the h2 equation to be greater than zero, we obtain the condition mp1 > p22 for an interior solution. Therefore, if mp1 ≤ p22, we ¶ b no longer have an interior solution. From step 4, the demand for good 2 vanishes at a corner solution and the entire income is spent on good 1.

In summary then, the demand for this quasilinear utility function is

h(p1, p2, m) =

⎧ ⎪⎪⎨

⎪⎪⎩

( p22 p21

, m p2

− p2 p1

) if mp1 > p22

( m p1

, 0 )

if mp1 ≤ p22, (4.9)

where the first case refers to the interior solution and the second, to the cor- ner solution.

To see a numerical example and its corresponding graph, suppose m = $8 and p1 = p2 = $1. In this case, the condition for an interior solution holds, so the utility-maximizing bundle is (1, 7) which is illustrated by point ¶ b A in the left panel of Figure 4.4. However, if m = $18, p1 = $2 and p2 = $9 instead, the condition for an interior solution no longer holds; we obtain a ¶ b corner solution, as shown by point B in the right panel of Figure 4.4.

A central feature of quasilinear preferences is that the demand for one of the goods does not depend on income at an interior solution. In this specific instance, ∂h1/∂m = 0 from equation (4.7), therefore good 1 has a zero income effect because the quasilinear indifference curves are vertically parallel (see Figure 3.8). Consequently, as long as the consumer can afford to purchase p22/p

2 1 units of good 1 with her initial income so as to be at an interior solu-

tion, the demand for good 1 will remain unchanged when m increases and all the additional income will be spent on buying more of good 2.

0 ? D?? 9 C 4 ? ? C ? 5C= I?=I I C=M , / :II 0 C C A / I =B 8I ? A? I7 ?M 3 IIE 1? B , ? IIE=? I ?M =I C ? C = CI . I=42-

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68 Chapter 4

0 1 8

7 8 A

9 B

u = 9-

u = 6-

Figure 4.4 Quasilinear interior and corner solutions

4.2.4 Demands for Cobb-Douglas preferences

We solve for the demand functions for Cobb-Douglas preferences by follow- ing these steps.

1. Set MRS = p1/p2 and obtain an expression for x2.

2. Substitute the expression from step 1 into the budget equation p1x1 + p2x2 = m and solve for x1 to derive the demand for good 1.

3. Substitute the demand derived in step 2 into the expression from step 1 (or the budget equation) to solve for the remaining demand.

To illustrate, consider the utility u(x1, x2) = x1x2. Then, from step 1,

x2 x1

= p1 p2

and we obtain the expression x2 = p1x1/p2. Following step 2, we obtainb · h1(p1, p2, m) = m/2p1. Finally, from step 3, we obtain h2(p1, p2, m) = m/2p2.b · Therefore, the demands for the Cobb-Douglas utility, u(x1, x2) = x1x2, can be written more compactly as

h(p1, p2, m) = (

m 2p1

, m

2p2

) . (4.10)

0 ? D?? 9 C 4 ? ? C ? 5C= I?=I I C=M , / :II 0 C C A / I =B 8I ? A? I7 ?M 3 IIE 1? B , ? IIE=? I ?M =I C ? C = CI . I=42-

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Individual Demands 69

A2p2 m

2p1 m

p2 m

p1 m

Figure 4.5 Cobb-Douglas preference maximization

Under these preferences, the consumer buys exactly half the total amount of good 1 that she could afford if she spent all her income on this good, and likewise for good 2, as shown by point A in Figure 4.5. In other words, for the Cobb-Douglas utility, u(x1, x2) = x1x2, the preference-maximizing quantities will always be at the midpoint of the budget line.

For the general Cobb-Douglas utility u(x1, x2) = Axa1x b 2, follow the same ¶ b

three steps to obtain the demand function

h(p1, p2, m) = (

am (a + b)p1

, bm

(a + b)p2

) . (4.11)

In general, the three-step process outlined above can be used to calculate the demand functions for any utility function that is differentiable2 so long as the preference-maximizing bundles are interior solutions (which is most often the case).

4.2.5 Demands for lexicographic preferences◦

We have been deriving demand functions from utility functions, but demand functions can be derived even when preferences cannot be represented by utility functions. As an example, consider the lexicographic preferences !L

2I.e., a utility function that has partial derivatives.

0 ? D?? 9 C 4 ? ? C ? 5C= I?=I I C=M , / :II 0 C C A / I =B 8I ? A? I7 ?M 3 IIE 1? B , ? IIE=? I ?M =I C ? C = CI . I=42-

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70 Chapter 4

we introduced in section 3.7.1 over combinations of food and shelter: two bundles are related by this binary relation ( f2, s2) !L ( f1, s1) if either

(a) f2 ≥ f1 regardless of the values of s1 and s2, or

(b) f2 = f1 and s2 ≥ s1.

Since this consumer cares about her consumption of food first, and her consumption of shelter second, this means that on a standard budget set, the bundle that maximizes her preferences is the one that has the greatest amount of food, namely, the bundle (m/p1, 0) where she spends all her in- come on good 1. This will be true no matter what the prices and income. Therefore, the demands for these lexicographic preferences are

h(p1, p2, m) = (

m p1

, 0 )

. (4.12)

Note that the linear utility function u(x1, x2) = x1 whose indifference curves are vertical (see the left panel of Figure 3.4) results in the same de- mand function as equation (4.12)! Thus it is not possible to tell from a con- sumer’s demand function whether the preferences she maximizes have a utility representation, or if they cannot be represented by a utility function.

4.3 Two Properties of Demand Functions

In this section, we look at two essential properties satisfied by all the demand functions you will encounter in this book.

4.3.1 Budget exhaustion

It is easy to check that when a consumer’s preferences are strictly monotonic, she will always maximize her preferences on the budget line and never inside the budget set. This is because at any bundle that is strictly inside the budget (i.e., not on the budget line), it is possible to move slightly northwest, remain inside the budget, and yet make the consumer better off. Since most of our preferences satisfy strict monotonicity (and even for some that do not), it will be the case that the preference-maximizing bundle is on the budget line and thus exhausts (i.e., uses up) the consumer’s entire income. We say that a consumer’s demand satisfies budget exhaustion so long as

p1h1(p1, p2, m) + p2h2(p1, p2, m) = m (4.13)

0 ? D?? 9 C 4 ? ? C ? 5C= I?=I I C=M , / :II 0 C C A / I =B 8I ? A? I7 ?M 3 IIE 1? B , ? IIE=? I ?M =I C ? C = CI . I=42-

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Individual Demands 71

for any positive values of p1, p2, and m. This property is satisfied by all the demands that have been derived so far. For instance, it may be checked that the demand functions in equation (4.10) that maximize the Cobb-Douglas utility u(x1, x2) = x1x2 satisfy budget exhaustion:

p1 · m

2p1 + p2 ·

m 2p2

= m 2 +

m 2

= m.

4.3.2 Homogeneity of degree zero in prices and income

Review the definition of homogeneous functions from section A.7.2 of the Mathematical appendix. Then, demand functions are homogeneous of de- gree zero in prices and income:

h(tp1, tp2, tm) = t0h(p1, p2, m) = h(p1, p2, m).

This means that if we were to scale all prices and income by the same factor (say, t = 4, and so we quadruple all prices and income), the demand remains unchanged. This is because scaling all prices and income by the same factor leaves the budget set unchanged: (tp1)x1 + (tp2)x2 = (tm) is identical to p1x1 + p2x2 = m for t > 0. Since the budget set remains unchanged, the preference-maximizing bundle must be the same.

Let us verify this for the demand function for the Leontief preferences derived in equation (4.5). In the case of the demand for good 1, note that

h1(tp1, tp2, tm) = tm

tp1 + tp2 =

m p1 + p2

= h1(p1, p2, m).

Verify that all the demand functions for goods 1 and 2 derived in this chapter ! " are homogeneous of degree zero in prices and income.

Exercises

4.1. For each of the following utility functions, calculate the demand func- tions for each good, h1(p1, p2, m) and h2(p1, p2, m), as functions of the prices and income.

(a) u(x1, x2) = 3x1 (b) u(x1, x2) = 2 ln x1 + 3 ln x2 (c) u(x1, x2) = x1(x2 − 1), where x2 > 1

0 ? D?? 9 C 4 ? ? C ? 5C= I?=I I C=M , / :II 0 C C A / I =B 8I ? A? I7 ?M 3 IIE 1? B , ? IIE=? I ?M =I C ? C = CI . I=42-

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72 Chapter 4

(d) u(x1, x2) = [min{x1, 2x2}]2

(e) u(x1, x2) = x1 + ln x2 (f) u(x1, x2) =

√ x1 +

√ x2

(g) u(x1, x2) = ln(x1 − 1)− 2 ln(x2 − 2) where x1 > 1 and 0 ≤ x2 ≤ 1.6

4.2. Verify that the demand functions calculated in 4.1 satisfy the budget exhaustion property. For each of the demand functions, show that it is homogeneous of degree zero in prices and income.

4.3. The following utility functions are defined over three goods, x, y, and z. Calculate the demand functions for each good, hj for j = x, y, z, as functions of the prices px, py, pz and income m.

(a) u(x, y, z) = xyz

(b) u(x, y, z) = x + ln y + ln z

Show that the demand functions satisfy the budget exhaustion prop- erty in each case, i.e., pxhx + pyhy + pzhz = m.

4.4. Ali spends his income of $64 on kerosene (x) and food (y) each week. The price of food is $8 per unit. The price of kerosene is $4/liter at a government-run store and he can purchase up to 8 liters there. On the market, kerosene costs $8/liter. His utility function is Cobb-Douglas: u(x, y) = xy3. How much kerosene and food does he buy given his budget constraint?

4.5. Consider Violet from problem 2.1. Suppose her utility function over pies (x) and champagne (y) (assumed to be divisible goods) is Cobb- Douglas and given by u(x, y) = xy.

(a) Given Violet’s budget in 2.1 part (a), calculate the quantities of pies and champagne she will consume when she maximizes her preferences.

(b) Given Violet’s budget in 2.1 part (b), calculate the quantities of pies and champagne she will consume when she maximizes her preferences.

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