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Minimisation Expenditure Problem
Simon Board This Version: November 3, 2008 First Version: October 27, 2008.
The expenditure minimisation problem (EMP) looks at the reverse side of the utility maximisation problem (UMP). The UMP considers an agent who wishes to attain the maximum utility from a limited income. The EMP considers an agent who wishes to find the cheapest way to attain a target utility. This approach complements the UMP and has several rewards: It enables us to analyse the effect of a price change, holding the utility of the agent constant. It enables us to decompose the effect of a price change on an agent's Marshallian demand into a substitution effect and an income effect. This decomposition is called the Slutsky equation. It enables us to calculate how much we need to compensate a consumer in response to a price change if we wish to keep her utility constant.
1
Model
We make several assumptions: 1. There are N goods. For much of the analysis we assume N = 2 but nothing depends on this. 2. The agent takes prices as exogenous. We normally assume prices are linear and denote them by {p1 , . . . , pN }. 1
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3. Preferences satisfy completeness, transitivity and continuity. As a result, a utility function exists. We normally assume preferences also satisfy monotonicity (so indifference curves are well behaved) and convexity (so the optima can be characterised by tangency conditions).
The expenditure minimisation problem is
N x1 ,...,xN
min
pi xi
i=1
subject to
u(x1 , . . . , xN ) u xi 0 for all i
(1.1)
The idea is that the agent is trying to find the cheapest way to attain her target utility, u. The solution to this problem is called the Hicksian demand or compensated demand. It is denoted by hi (p1 , . . . , pN , u) The money the agent must spend in order to attain her target utility is called her expenditure. The expenditure function is therefore given by
N
e(p1 , . . . , pN , u) = min
x1 ,...,xN
pi xi
i=1
subject to
u(x1 , . . . , xN ) u xi 0 for all i
Equivalently, the expenditure function equals the amount the agent spends on her optimal bundle,
N
e(p1 , . . . , pN , u) =
i=1
pi hi (p1 , . . . , pN , u)
1.1
Example
Suppose there are two goods, x1 and x2 . Table 1 shows how the agent's utility (the numbers in the boxes) varies with the number of x1 and x2 consumed. To keep things simple, suppose the agent faces prices p1 = 1 and p2 = 1 and wishes to attain utility u = 12. The agent can attain this utility by consuming (x1 , x2 ) = (6, 2), (x1 , x2 ) = (4, 3), (x1 , x2 ) = (3, 4) or (x1 , x2 ) = (2, 6). Of these, the cheapest is either (x1 , x2 ) = (4, 3) or (x1 , x2 ) = (3, 4). In either case, her expenditure is 4 + 3 = 7.
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x1 \x2 1 2 3 4 5 6
1 1 2 3 4 5 6
2 2 4 6 8 10 12
3 3 6 9 12 15 18
4 4 8 12 16 20 24
5 5 10 15 20 25 30
6 6 12 18 24 30 36
Table 1: Utilities from different bundles. Now suppose the agent faces prices p1 = 1 and p2 = 3 and still wishes to attain utility u = 12. The combinations of (x1 , x2 ) that attain this utility remain unchanged, however the price of these bundles is different. Now the cheapest is (x1 , x2 ) = (6, 2), and the agent's expenditure is 6 + 2 3 = 12. While this "table approach" can be used to illustrate the basic idea, one can see that it quickly becomes hard to solve even simple problems. Fortunately, calculus comes to our rescue.
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2.1
Solving the Expenditure Minimisation Problem
Graphical Solution
We can solve the problem graphically, as with the UMP. The components are also similar to that problem. First, we need to understand the constraint set. The agent can choose any bundle where (a) the agent attains her target utility, u(x1 , x2 ) u; and (b) the quantities are positive, x1 0 and x2 0. If preferences are monotone, then the bundles that meet these conditions are exactly the ones that lie above the indifference curve with utility u. See figure 1. Second, we need to understand the objective. The agent wishes to pick the bundle in the constraint set that minimises her expenditure. Just like with the UMP, we can draw the level curves of this objective function. Define an isoexpenditure curve by the bundles of x1 and x2 that deliver constant expenditure: {(x1 , x2 ) : p1 x1 + p2 x2 = const}
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Figure 1: Constraint Set. The shaded area shows the bundles that yield utility u or more.
These isoexpenditure curves are just like budget curves and so have slope -p1 /p2 . See figure 2 The aim of the agent is to choose the bundle (x1 , x2 ) in the constraint set that is on the lowest isoexpenditure curve and hence minimises her expenditure. Ignoring boundary problems and kinks, the solution has the feature that the isoexpenditure curve is tangent to the target indifference curve. As a result, their slopes are identical. The tangency condition can thus be written as M RS = This is illustrated in figure 3. The intuition behind (2.1) is as follows. Using the fact that M RS = M U1 /M U2 ,1 equation (2.1) implies that M U1 p1 = M U2 p1
1
p1 p1
(2.1)
(2.2)
Recall: M Ui = U/xi is the marginal utility from good i.
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Figure 2: Iso-Expenditure Curve. This figure shows the bundles that induce constant expenditure.
Figure 3: Optimal Bundle. This figure shows how the cheapest bundle that attains the target utility satisfies the tangency condition.
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Rewriting (2.2) we find p1 p2 = M U1 M U1 The ratio pi /M Ui measures the cost of increasing utility by one util, or the "costperbang". At the optimum the agent equates the costperbang of the two goods. Intuitively, if good 1 has a higher costperbang than good 2, then the agent should spend less on good 1 and more on good 2. In doing so, she could attain the same utility at a lower cost. If preferences are monotone, then the constraint will bind, u(x1 , x2 ) = u, (2.3)
The tangency equation (2.2) and constraint equation (2.3) can then be used to solve for the two Hicksian demands. If there are N goods, the agent will equalise the costperbang from each good, giving us N - 1 equations. Using the constraint equation (2.3), we can solve for the agent's Hicksian demands. The tangency condition (2.2) is the same as that under the UMP. This is no coincidence. We discuss the formal equivalence in Section 4.2.
2.2
Example: Symmetric Cobb Douglas
Suppose u(x1 , x2 ) = x1 x2 . The tangency condition yields: x2 p1 = x1 p2 Rearranging, p1 x1 = p2 x2 . The constraint states that u = x1 x2 . Substituting (2.4) into this yields, u= p1 2 x p2 1 (2.4)
Solving for x1 , the Hicksian demand is given by h1 (p1 , p2 , u) = p2 u p1
1/2
(2.5)
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Similarly, we can solve for the Hicksian demand for good 2, h2 (p1 , p2 , u) = We can now calculate the agent's expenditure e(p1 , p2 , u) = p1 h1 (p1 , p2 , u) + p2 h2 (p1 , p2 , u) = 2(up1 p2 )1/2 (2.6) p1 u p2
1/2
2.3
Lagrangian Solution
Using a Lagrangian, we can encode the tangency conditions into one formula. As before, let us ignore boundary problems. The EMP can be expressed as minimising the Lagrangian L = p1 x1 + p2 x2 + [u - u(x1 , x2 )] As with the UMP, the term in brackets can be thought as the penalty for violating the constraint. That is, the agent is punished for falling short of the target utility. The FOCs with respect to x1 and x2 are u L = p1 - =0 x1 x1 L u = p2 - =0 x2 x2 If preferences are monotone then the constraint will bind, u(x1 , x2 ) = u These three equations can then be used to solve for the three unknowns: x1 , x2 and . Several remarks are in order. First, this approach is identical to the graphical approach. Dividing (2.7) by (2.8) yields u/x1 p1 = u/x2 p2 which is the same as (2.2). Second, if preferences are not monotone, the constraint (2.9) may not bind. If it does not bind, 7 (2.9) (2.7) (2.8)
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the Lagrange multiplier in the FOCs will be zero. Third, the approach is easy to extend to N goods. In this case, one obtains N first order conditions and the constraint equation (2.9).
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3.1
General Results
Properties of Expenditure Function
The expenditure function exhibits four important properties. 1. The expenditure function is homogenous of degree one in prices. That is, e(p1 , p2 , u) = e(p1 , p2 , u) for > 0. Intuitively, if the prices of x1 and x2 double, then the cheapest way to attain the target utility does not change. However, the cost of attaining this utility doubles. 2. The expenditure function is increasing in (p1 , p2 , u). If we increase the target utility u, then the constraint becomes harder to satisfy and the cost of attaining the target increases. If we increase p1 then it costs more to buy any bundle of goods and it costs more to attain the target utility. 3. The expenditure function is concave in prices (p1 , p2 ). Fix the target utility u and prices (p1 , p2 ) = (p1 , p2 ). Solving the EMP we obtain Hicksian demands h1 = h1 (p1 , p2 , u) and h2 = h2 (p1 , p2 , u). Now suppose we fix demands and change p1 , the price of good 1. This gives us a pseudoexpenditure function h1 ,h2 (p1 ) = p1 h1 + p2 h2 This pseudoexpenditure function is linear in p1 which means that, if we keep demands constant, then expenditure rises linearly with p1 . Of course, as p1 rises the agent can reduce her expenditure by rebalancing her demand towards the good that is cheaper. This means that real expenditure function lies below the pseudoexpenditure function and is therefore concave. See figure 4. More formally, the expenditure function is given by the lower envelope of the pseudo-expenditure 8
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Figure 4: Expenditure Function. This figure shows how the expenditure function lies under the
pseudoexpenditure function.
functions. That is, for any bundle (x1 , x2 ), the cost of this bundle at prices (p1 , p2 ) is given by x1 ,x2 (p1 , p2 ) = p1 x1 + p2 x2 The expenditure function is then the minimum of these pseudoexpenditure functions given the bundle (x1 , x2 ) attains the target utility. Mathematically, e(p1 , p2 , u) = min{p1 x1 + p2 x2 : u(x1 , x2 ) = u} (3.1)
Thus the expenditure function is the lower minimum of a collection of linear functions, and is therefore concave.2 See figure 5. 4. Sheppard's Lemma: The derivative of the expenditure function equals the Hicksian demand. That is, e(p1 , p2 , u) = h1 (p1 , p2 , u) p1 (3.2)
The idea behind this result can be seen from figure 4. At p1 = p1 the expenditure function is tangential to the pseudoexpenditure function. The pseudoexpenditure is linear in p1 with slope h1 (p1 , p2 , u). Hence the expenditure function also has slope h1 (p1 , p2 , u).
2
Exercise: Show that the minimum of two concave functions is concave.
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Figure 5: Envelope Property of Expenditure Function. This figure shows the expenditure function equals the lower envelope of the pseudo expenditure functions.
The intuition behind Sheppard's Lemma is as follows. Suppose an agent wishes to attain target utility u = 25 and faces prices p1 = $1 and p2 = $1. Furthermore, suppose that the cheapest way to attain the target utility is by consuming h1 = 5 and h2 = 5. Next, consider an increase in p1 of 1. This change has a direct and indirect effect. The direct effect is that, holding demand constant, the agent's spending rises by h1 1 = 5; the indirect effect is that the agent will change her demands. However, since p1 has only changed a little, this indirect effect will be small. We thus conclude that e = h1 p1 , Rewriting, e = h1 p1 This is the discrete version of equation (3.2). Here is a formal proof of Sheppard's Lemma. By definition of the expenditure function, e(p1 , p2 , u) = p1 h1 (p1 , p2 , u) + p2 h2 (p1 , p2 , u) Differentiating with respect to p1 yields h1 (p1 , p2 , u) h2 (p1 , p2 , u) e(p1 , p2 , u) = h1 (p1 , p2 , u) + p1 + p2 p1 p1 p1 (3.3)
As discussed above, we have decomposed the effect of the price change into a direct effect (the
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first term) and an indirect effect (the second and third terms). We now wish to show the indirect effect is zero. From the agent's minimisation problem in Section 2.3, the FOCs are pi = u(h1 , h2 ) xi
We also know that the agent's constraint binds: u(h1 (p1 , p2 , u), h2 (p1 , p2 , u)) = u Substituting the FOCs into (3.3) u(h1 , h2 ) h1 (p1 , p2 , u) u(h1 , h2 ) h2 (p1 , p2 , u) e(p1 , p2 , u) = h1 (p1 , p2 , u) + + p1 x1 p1 x2 p1 Differentiating (3.4) with respect to p1 yields u(h1 , h2 ) h1 (p1 , p2 , u) u(h1 , h2 ) h2 (p1 , p2 , u) + =0 x1 p1 x2 p1 Substituting (3.6) into (3.5) yields Sheppard's Lemma. (3.6) (3.5) (3.4)
3.2
Properties of Hicksian Demand
Hicksian demand has three important properties. These follow from the properties of the expenditure function derived above. 1. Hicksian demand is homogenous of degree zero in prices. That is, h1 (p1 , p2 , u) = h1 (p1 , p2 , u) for > 0. Intuitively, doubling both prices does not alter the cheapest way to obtain the target utility u. 2. The Law of Hicksian Demand: The Hicksian demand for good i is decreasing in pi . That is, hi (p1 , p2 , u) 0 pi Intuitively, when p1 rises the relative prices become tilted in favour of good 2. The cheapest way to attain the target utility then consists of less of good 1 and more of good 2. Graphically
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Figure 6: Hicksian Demand and Own Price Effects. This figure shows the effect of an increase in p1 , from p1 to p1 . The optimal bundle moves from A to B.
this can be seen from figure 6. As p1 rises to p1 , the isoexpenditure function becomes steeper and the optimal bundle involves less of good 1 and more of good 2.3 A formal proof of this result uses the properties of the expenditure function: 2 h1 (p1 , p2 , u) = 2 e(p1 , p2 , u) 0 p1 p1 where the equality comes from Sheppard's Lemma and the inequality follows from the concavity of the expenditure function. This result highlights a big difference between Hicksian demand and Marshallian demand. An increase in p1 always reduces the Hicksian demand for good 1 but may, in the case of a Giffen good, increase the Marshallian demand. This is because the effect of a price change on Marshallian demand has two effects: a substitution effect (a change in relative prices) and an income effect (a change in the consumer's purchasing power). In comparison, the change in Hicksian demand isolates the substitution effect. 3. Hicksian demand has symmetric cross derivatives. That is, h1 (p1 , p2 , u) = h2 (p1 , p2 , u) p2 p1
3
The fact that the demand for good 2 always rises is an artifact of there only being 2 goods.
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The proof of this result also uses the properties of the expenditure function. h1 (p1 , p2 , u) = e(p1 , p2 , u) = e(p1 , p2 , u) = h2 (p1 , p2 , u) p2 p2 p1 p1 p2 p1 The first and third equalities come from Sheppard's Lemma and the second from Young's theorem. We say goods x1 and x2 are net substitutes if h1 (p1 , p2 , u) > 0 p2 and h2 (p1 , p2 , u) > 0 p1
We say goods x1 and x2 are net complements if h1 (p1 , p2 , u) < 0 p2 and h2 (p1 , p2 , u) < 0 p1
The symmetry of the cross derivatives means that we cannot have one crossderivative positive negative and the opposite crossderivative negative, as with gross substitutes and complements.4
4
Income and Substitution Effects
We are often interested in how price changes affect Marshallian demand. This matters to firms when choosing prices, to government when choosing tax rates and to economists when making forecasts. For example: how much will demand for ethanol increase if we lower the price by $10? We saw with the UMP that an increase in p1 may lead to a large decrease in demand (if demand is elastic), may lead to a small decrease in demand (if demand is inelastic) or may lead to an increase in demand (in the case of a Giffen good). One major issue is that an increase in the price of good 1 has two effects: it both makes good 1 relatively more expensive (the substitution effect) and reduces the agent's purchasing power (the income effect). This section will separate these effects. In Section 4.1 we do this graphically. In Section 4.3 we do this mathematically.
4
Exercise: Suppose there are two goods. Show they must be net substitutes.
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Figure 7: Substitution and Income Effects with Normal Good. With a normal good, both
substitution effect (SE) and income effect (IE) are negative.
4.1
Pictures
Suppose we start at point A in figures 7 and 8. When p1 increases, the budget line pivots around it's left end and demand falls from A to C. We can decompose this change into two effects.
1. A change in relative prices, keeping utility constant. This is the shift from A to B, and is called the substitution effect. This equals the change in Hicksian demand and, appealing to the Law of Hicksian Demand, is negative. 2. A change in income, keeping relative prices constant. This is the shift from B to C, and is called the income effect. This effect is positive if the good is normal, and negative if the good is inferior.
Exercise: draw the equivalent picture for a Giffen good.
4.2
Relation between the UMP and EMP
The EMP and UMP are closely related. To illustrate, suppose the agent has $10 to spend on two goods. Suppose her utility is maximised when (x1 , x2 ) = (5, 5) and she can attain 25 utils.5
5
These numbers come from assuming p1 = 1, p2 = 1 and u(x1 , x2 ) = x1 x2 .
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Figure 8: Substitution and Income Effects with Normal Good. With an inferior good, substitution effect (SE) is negative while the income effect (IE) is positive.
What is the cheapest way for the agent to attain 25 utils? Given this information, the answer must be (x1 , x2 ) = (5, 5). Moreover, her expenditure is $10. The reason is as follows. First, we know that the agent can obtain 25 utils from $10, so the cheapest way to obtain 25 utils is at most $10. That is, e $10. Now suppose, by contradiction, that the agent can obtain 25 utils for, say, $8. Then, if preferences are monotone, she will be able to obtain strictly more than 25 utils with $10, contradicting our initial assumptions. We can state this result formally. Fix prices (p1 , p2 ) and income m. Marshallian demand is given by x (p1 , p2 , m) and indirect utility is v(p1 , p2 , m). Consider the EMP: i
x1 ,x2
min p1 x1 + p2 x2
subject to
u(x1 , x2 ) v(p1 , p2 , m)
The induced Hicksian demand is given by hi (p1 , p2 , v(p1 , p2 , m)) while the expenditure function is e(p1 , p2 , v(p1 , p2 , m)). Then using the reasoning above, one can show that e(p1 , p2 , v(p1 , p2 , m)) = m hi (p1 , p2 , v(p1 , p2 , m)) = x (p1 , p2 , m) i (4.1) (4.2)
Suppose we start with income m. Equation (4.1) says that the minimum expenditure required to reach v(p1 , p2 , m), the most utility from m, is just m. Equation (4.2) says that an agent who wishes to maximise her utility from m and one who wishes to find the cheapest way to attain v(p1 , p2 , m) will buy the same goods. Intuitively, in both cases, they will spend m and will do so by equating the bangperbuck from each good.
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Equation (4.1) is practically useful. Fixing prices and omitting them from the arguments, it says that e(v(m)) = m. Since the expenditure function is increasing in u, we can invert it and obtain: v(m) = e-1 (m) (4.3)
Hence the indirect utility function equals the inverse of the expenditure function. To illustrate this result, suppose u(x1 , x2 ) = x1 x2 . From equation (2.6), we know that e(u) = 2 up1 p2 We invert this equation by letting m = e(u) and v(m) = u, and solving for v(m). This yields v(m) = m2 4p1 p2
One can verify that this indeed the indirect utility function. We can also state a second, closely related, result. Fix prices (p1 , p2 ) and target utility u. Hicksian demand is given by hi (p1 , p2 , u) and the expenditure function is e(p1 , p2 , u). Consider the UMP: u(x1 max , x2 )
x1 ,x2
subject to
p1 x1 + p2 x2 e(p1 , p2 , u)
The induced Marshallian demand is given by x (p1 , p2 , e(p1 , p2 , u)) while the indirect utility is i v(p1 , p2 , e(p1 , p2 , u)). One can show that v(p1 , p2 , e(p1 , p2 , u)) = u x (p1 , p2 , e(p1 , p2 , u)) = hi (p1 , p2 , u) i (4.4) (4.5)
Suppose we start with target utility u. Equation (4.4) says that the most utility the agent can get from e(p1 , p2 , u), the money required to reach u, is just u. Equation (4.5) says that an agent who wishes to find the cheapest way to attain u and one who wishes to maximise her utility from e(p1 , p2 , u) will buy the same goods. Intuitively, in both cases, they will attain utility u and will do so by equating the bangperbuck from each good. Fixing prices and omitting them from the arguments, equation (4.4) says that v(e(u)) = u. Since the indirect function is increasing in m, we can invert it and obtain: e(u) = v -1 (u) 16 (4.6)
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Hence the expenditure function equals the inverse of the indirect utility function. Together, equations (4.3) and (4.6) mean we can move back and forwards between the expenditure function and indirect utility function.
4.3
Slutsky Equation: Own Price Effects
Suppose p1 increases by p1 . There are two effects: 1. Fixing the agent's utility, relative prices change causing demand to rise by
h1 p1 h1 p1 p1 .
Since
< 0, this effect causes demand to fall. This is the substitution effect.
x
2. Fixing relative prices, the agent's income falls by x p1 . As a result, her demand falls 1
1 by x m p1 . This is the income effect. 1
Putting these effects together, we have x = 1 h1 x p1 - x 1 p1 1 p1 m
Dividing by p1 yields the Slutsky equation. Theorem 1 (OwnPrice Slutsky Equation). Fix prices (p1 , p2 ) and income m, and let u = v(p1 , p2 , m) be the indirect utility. Then x (p1 , p2 , m) = h1 (p1 , p2 , u) - x (p1 , p2 , m) x (p1 , p2 , m) 1 p1 1 p1 m 1 A formal proof is reasonably straightforward. Using equation (4.5), hi (p1 , p2 , u) = x (p1 , p2 , e(p1 , p2 , u)) i Differentiating with respect to p1 yields x (p1 , p2 , e(p1 , p2 , u)) x (p1 , p2 , e(p1 , p2 , u)) e(p1 , p2 , u) 1 h1 (p1 , p2 , u) = + 1 p1 p1 m p1 x (p1 , p2 , e(p1 , p2 , u)) x (p1 , p2 , e(p1 , p2 , u)) 1 = + 1 x1 (p1 , p2 , e(p1 , p2 , u)) p1 m (4.7)
(4.8)
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where the second line comes from Sheppard's Lemma. Using the definition of u and equation (4.1), e(p1 , p2 , u) = e(p1 , p2 , v(p1 , p2 , m)) = m Substituting (4.9) into (4.8) and rearranging yields (4.7), as required. (4.9)
4.4
Slutsky Equation: Cross Price Effects
Equation (4.7) analyses the effect of a change in p1 on the demand for good 1. We can use the same approach to analyse the effect of a change in p2 on the demand for good 1. Suppose p2 increases by p2 . As before, there are two effects: 1. Fixing the agent's utility, relative prices change causing demand to rise by that
h1 p2 h1 p2 p2 .
Recall
> 0 if the goods are net substitutes and
h1 p2
< 0 are net complements.
2. Fixing relative prices, the agent's income falls by x p2 . As a result, her demand falls 2
1 by x m p2 . 2
x
Putting these effects together, we have x = 1 h1 x p2 - x 1 p2 2 p2 m
Dividing by p2 yields the Slutsky equation for crossprice effects. Theorem 2 (CrossPrice Slutsky Equation). Fix prices (p1 , p2 ) and income m, and let u = v(p1 , p2 , m) be the indirect utility. Then x1 (p1 , p2 , m) = h1 (p1 , p2 , u) - x (p1 , p2 , m) x (p1 , p2 , m) 2 p2 p2 m 1 The proof is almost identical to that of (4.7). Using equation (4.5), hi (p1 , p2 , u) = x (p1 , p2 , e(p1 , p2 , u)) i (4.10)
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Differentiating with respect to p2 yields x (p1 , p2 , e(p1 , p2 , u)) x (p1 , p2 , e(p1 , p2 , u)) e(p1 , p2 , u) 1 h1 (p1 , p2 , u) = + 1 p2 p2 m p2 x (p1 , p2 , e(p1 , p2 , u)) x (p1 , p2 , e(p1 , p2 , u)) 1 = + 1 x2 (p1 , p2 , e(p1 , p2 , u)) (4.11) p2 m where the second line comes from Sheppard's Lemma. Using the definition of u and equation (4.1), e(p1 , p2 , u) = e(p1 , p2 , v(p1 , p2 , m)) = m Substituting (4.12) into (4.11) and rearranging yields (4.10). (4.12)
4.5
Slutsky Equation: Example
We illustrate the Slutsky equation with our running example. Let u(x1 , x2 ) = x1 x2 . From the UMP we know that x (p1 , p2 , m) = 1 m 2p1 m2 v(p1 , p2 , m) = 4p1 p2
From the EMP (see Section 2.2) we know that h1 (p1 , p2 , u) = u p2 p1
1/2
e(p1 , p2 , u) = 2(up1 p2 )1/2
The left hand side of the Slutsky equation states 1 x1 (p1 , p2 , m) = - mp-2 p1 2 1 The right hand side is x 1 1 h1 -1/2 1/2 - x 1 = - u1/2 p1 p2 - mp-2 1 p1 m 2 4 1 1 1 = - mp-2 - mp-2 1 4 4 1 19 (4.13)
(4.14)
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where the second line uses u = v(p1 , p2 , m). Observe that (4.13) equals (4.14) as we would hope. Moreover, the two terms in equation (4.14) are identical. This means that the substitution and income effects are of equal size: both account for 50% of the fall in demand.
5
Consumer Surplus
It is often important to put a monetary value on the effect of a price change on an agent's utility. For example, the government may wish to evaluate the impact of a tax change; or a court may wish to evaluate the negative effect of collusion on consumers. To gain some intuition, suppose the consumer has monetary valuations for each unit of the good. In particular suppose their valuations are given by table 2. Unit 1 2 3 4 5 Valuation $ 10 8 6 4 2
Table 2: Agent's Valuations Suppose the price of the good is initially p1 = 3. Since the agent buys a unit if and only if her valuation exceeds the price, she will buy 4 units. Her consumer surplus, the difference between her willingness to pay and the price she pays, equals CS = (10 - 3) + (8 - 3) + (6 - 3) + (4 - 3) = $16 Suppose the price rises to p1 = 7. The agent then consumes 2 units and her consumer surplus is CS = (10 - 7) + (8 - 7) = $4. Hence the agent would need to be compensated $12 for this price increase. This is shown in figure 9. This exercise is familiar from introductory economics courses: consumer surplus is the area under the agent's Marshallian demand curve. However this approach assumes the agent has 20
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Figure 9: Consumer Surplus with Quasilinear Demand. The figure shows the agent's demand
curve. The shaded area is the loss in CS due to the price increase.
quasilinear utility, allowing us to associate a monetary value to each unit demanded by the agent. In Section 5.1 we show that the welfare effect of a price change is determined by the area under the Hicksian demand curve rather than the Marshallian demand. In Section 5.2 we see that, when utility is quasilinear then Hicksian demand and Marshallian demand coincide, justifying the approach taken above.
5.1
Compensating Variation
Suppose prices and income are initially (p1 , p2 , m), and that p1 increases to p1 . The compensating variation is defined by CV = e(p1 , p2 , u) - e(p1 , p2 , u) The CV is thus the extra spending needed to keep the agent at their original utility level. That is, an increase in income of CV completely compensates the agent for the price increase.6 This is shown in figure 10. The compensating variation can be related to the Hicksian demand curve. Applying the fun6
There is a closely related measure of welfare called the equivalent variation. We will not discuss this here.
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Figure 10: Compensating Variation and Indifference Curves. This figure shows the effect on
an increase in p1 . The Marshallian demand falls from A to C. The Hicksian demand moves from A to B. The compensating variation equals the difference between the consumer's original income and the income she would need to attain u.
damental theorem of calculus,7 CV = =
p1 p1 p1 p1
e(~1 , p2 , u) d~1 p p p1 h1 (~1 , p2 , u) d~1 p p (5.1)
where the second equation follows from Sheppard's Lemma. Equation (5.1) says that the lost welfare from the price change equals the area under the Hicksian demand curve. See figure 11.
5.2
Quasilinear Utilities
While we may wish to calculate the area under the Hicksian demand, it is often easier to calculate the area under the Marshallian demand curve. For example, in empirical applications, it is easy to estimate the Marshallian demand by looking at how much people buy at different prices.
7 b a
The fundamental theorem of calculus says that f (b) - f (a) =
f (x)dx.
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Figure 11: Compensating Variation and Hicksian Demand. This figure shows that CV equals
the area under the demand curve.
Suppose utility is quasilinear in that it can be represented by a utility function of the form u(x1 , x2 ) = v(x1 ) + x2 where we assume v() is increasing and concave. Under this specification, the marginal utility of the second good is constant. For example, x2 could be a general aggregate good or cash. When utility is quasilinear we can think of an agent's utility in terms of dollar valuations, as at the start of this section. The argument is as follows. The agent's problem is to maximise her utility subject to her budget constraint, p1 x1 + p2 x2 m. Since utility is monotone, the budget constraint will bind. Using the substitution method, the budget constraint becomes x2 = m p1 - x1 p2 p2
Substituting this into the utility function, the agent maximises v(x1 ) - m p1 x1 + p2 p2 (5.2)
Notice the last term is a constant and can be ignored. If x2 is interpreted as cash, we can normalise p2 = 1. The agent then chooses x1 to maximise v(x1 ) - p1 x1 23
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The agent's choice is independent of m, so she acts as if she values x1 units of good 1 at v(x1 ), independent of the units of x2 being consumed. We can then think of v (x1 ) as her valuation of the marginal unit of x1 . Under quasilinear utility, the Hicksian and Marshallian demands coincide. Ignoring boundary problems, the Marshallian demand is derived by maximising (5.2). The firstorder condition implies that Marshallian demand is implicitly given by v (x (p1 , p2 , m)) = 1 Turning to the EMP, the agent minimises L = p1 x1 + p2 x2 + [u - v(x1 ) - x2 ] The first firstorder conditions are p1 = v (x1 ) p2 = Looking at the ratio of these two equations, Hicksian demand is implicitly given by v (h1 (p1 , p2 , u)) = p1 p2 (5.4) p1 p2 (5.3)
From equations (5.3) and (5.4) we see that Marshallian demand and Hicksian demand coincide. Hence the compensating variation is given by CV = =
p1 p1 p1 p1
p h1 (~1 , p2 , u) d~1 p x (~1 , p2 , m) d~1 p 1 p
This result provides a foundation for the classical measure of consumer surplus.
6
Endowments of Goods
In the UMP we assume that agents are endowed with income m and use it to maximise their utility. While this is a useful model to address demand for retail products, it is sometimes more accurate to assume agents are endowed with goods which they can sell on the open market. 24
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Figure 12: Marshallian Demand with Endowments This figure shows the optimal choice when
the agent has endowments of the two goods.
There are two reasons for analysing this model:
The model is important for understanding practical problems such as a worker's choice of labour supply (Section 6.1), and an agent's decision to smooth consumption over time (Section 6.2). When we analyse the entire economy, we will want to close the model. Hence we wish the agents who demand goods to also work for firms that make goods.
Suppose there are N goods and the agent starts with endowments {1 , . . . , N }, where i 0 for all i. The consumer can sell these goods at market prices {p1 , . . . , pN }. For example, an agent may own a farm which produces vegetables and may sell the produce to buy meat. The agent has income
N
m=
i=1
pi i
(6.1)
Given equation (6.1) the agent's problem is the same as that studied so far. We can derive her Marshallian demand and indirect utility (see figure 12). We can also derive her Hicksian demand and expenditure function (since these are independent of income)
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The one major difference from the model with exogenous income is that a price change now affects the agent's income as well as the goods she buys. We study this in Section 6.3. We first consider two applications.
6.1
Labour Supply
Suppose an agent has utility u(x1 , x2 ) = x1 x2 over leisure x1 and a general consumption good x2 . The agent has exogenous income m and can also work at wage w. She has T hours which she can allocate to either work or leisure. We normalise the price of x2 to p2 = 1. The agent's budget constraint is x2 = w(T - x1 ) + m The left hand side equals the agent's spending on consumption; the right hand side equals her income. As a thought experiment, one can imagine the agent selling all T units of her labour and then buying x1 units of it back at price w to be consumed as leisure. We can thus rewrite the budget constraint as wx1 + x2 = wT + m The left hand side is the goods consumed (including leisure, consumed at price w). The right hand side is the agent's endowment income, as in equation (6.1). Ignoring boundary constraints,8 her problem is max L = x1 x2 + [m + w(T - x1 ) - x2 ]
x1 ,x2
The FOCs are x2 = w x1 = Taking the ratio of these FOCs, we see that x2 =w x1 As before, the left hand side is the MRS, while the right hand side is the price ratio. Using the
With this problem the boundary constraints are slightly different to normal since the agent cannot consume more that T units of leisure. We thus have T x1 0 and x2 0.
8
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budget constraint the agent's demands are given by 1 [wT + m] 2w 1 x = [wT + m] 2 2 x = 1 (6.2) (6.3)
Equations (6.2) and (6.3) show that the consumer splits her endowment income of wT + m equally between leisure and consumption. This is just like the solution to the Cobb Douglas problem without endowments (see UMP notes), where we found that x = 1 1 m 2p1 and x = 2 1 m 2p2 (6.4)
We can now evaluate an effect of a change in wages. Differentiating (6.2) and (6.3), 1 x 1 1 1 = wT - [wT + m] = - 2 m 2 2 w 2w 2w 2w 1 x 2 = T w 2 (6.5) (6.6)
From (6.5), we see an increase in the wage reduces the amount of leisure the agent consumes. There are two effects here: an increase in the wage raises the relative price of leisure and reduces demand (the substitution effect); it also makes the agent richer and increases the demand for leisure (the income effect). In this case the substitution effect dominates the income effect: we analyse this formally in Section 6.4. From (6.6), we see an increase in the wage increase the amount of x2 the agent consumes. This is because an increase in wages increase the value of the agent's endowment; in comparison, without endowments, equation (6.4) shows that x is independent of p1 . 2
6.2
Intertemporal Optimisation
Suppose an agent allocates consumption (e.g. money) across two periods. Let the consumption in period 1 and 2 be x1 and x2 respectively. The agent's utility is u(x1 , x2 ) = ln(x1 ) + (1 + )-1 ln(x2 ) where 0 is the agent's discount rate. (6.7)
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In periods 1 and 2 the agent is endowed with income m1 and m2 , respectively. The agent can save at interest rate r 0, so that $1 in period 1 is worth $(1+r) in period 2. As a result, the agent's budget constraint is m1 + (1 + r)-1 m2 = x1 + (1 + r)-1 x2 (6.8)
The left hand side of (6.8) is the agent's lifetime income in terms of period 1 dollars. The right and side is the agent's lifetime spending. We say they are borrowing when x1 > m1 and saving when x1 < m1 . We can solve this problem just as we would solve a regular utility maximisation problem, where p1 = 1 and p2 = (1 + r)-1 . See figure 13. Using (6.7) and (6.8) the tangency condition, M RS = p1 /p2 , becomes (1 + ) Rearranging, x = 1 1+ x 1+r 2 (6.9) 1/x1 = (1 + r) 1/x2
Equation (6.2) immediately implies that if r = then the agent consumes the same in each period, x = x . Intuitively, since the agent's perperiod utility is concave, she wishes to smooth 1 2 her consumption across time. If r = , then the agent is just as impatient as the market, so she will perfectly smooth her consumption across the two periods. If > r then the agent is more impatient than the market and she consumes more in the first period, x > x . 1 2 Using the budget constraint, demand is given by x = 1 1+ [m1 + (1 + r)-1 m2 ] 2+ and x = 2 1+r [m1 + (1 + r)-1 m2 ] 2+
6.3
Own Price Effects
Suppose there is an increase in p1 . As with a fixed income m, the budget line becomes steeper. However, since the value of the endowment changes, it is no longer true that the budget set shrinks. Rather, the budget line pivots around the endowment: see figure 14.
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Figure 13: Intertemporal Optimisation This figure shows an agent who has a high income in period
1 and a low income in period 2. At the optimum, she saves in period 1.
Figure 14: OwnPrice Effects with Endowments. This figure shows the effect on a decrease in p1
when the agent is endowed with {1 , 2 }. Note the income looks like it goes down, even though prices fall. This is because, at point A, the agent owns more of good 1 than she buys, 1 > x . Hence a 1 decrease in p1 reduces her purchasing power.
29
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As in Section 4, we can decompose the price change into a substitution and income effect. However, the income effect has to be adjusted for the change in the value of the endowment. Suppose p1 increases by p1 . Then there are two effects: 1. Fixing the agent's utility, relative prices change causing demand to rise by
h1 p1 h1 p1 p1 .
Since
< 0, this effect causes demand to fall. This is the substitution effect.
2. Fixing relative prices, the agent's income rises by (1 - x )p1 . This means that the 1 agent's income rises if she is a net seller of the good (as in the labour example), and falls
1 if she is a net buyer of the good. As a result, her demand rises by (1 - x ) m p1 . This 1
x
is the income effect.
Putting these effects together, we have x = 1 x h1 p1 + (1 - x ) 1 p1 1 p1 m
Dividing by p1 yields the Slutsky equation. Theorem 3 (OwnPrice Slutsky Equation with Endowments). Fix prices (p1 , p2 ), income m and endowments (1 , 2 ), and let u = v(p1 , p2 , m) be the indirect utility. Then x1 (p1 , p2 , m) = h1 (p1 , p2 , u) + (1 - x (p1 , p2 , m)) x (p1 , p2 , m) 1 p1 p1 m 1 (6.10)
This result follows from the regular Slutsky equation (4.7). All we need to do is define net
demand for good 1 by z1 (p1 , p2 , m) = x (p1 , p2 , m) - 1 . We can then apply the regular Slutsky 1
equation to the agent's net demand: z (p1 , p2 , m) = h1 (p1 , p2 , u) - z1 (p1 , p2 , m) z (p1 , p2 , m) p1 1 p1 m 1 x (p1 , p2 , m), yielding equation (6.10). i (6.11)
Since z1 and x differ by a constant term, we can put equation (6.11) back in terms of 1
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6.4
Labour Supply and the Slutsky Equation
We now apply the Slutsky equation (6.10) to the labour supply problem in Section 6.1. From equation (6.2), the Marshallian demand is x (p1 , p2 , m) = 1 Using the (6.2) and (6.3) the indirect utility is v(p1 , p2 , m) = x x = 1 2 1 [wT + m]2 4w (6.13) 1 [wT + m] 2w (6.12)
From (2.5) and using p1 = w and p2 = 1, the Hicksian demand is h1 (p1 , p2 , u) = u1/2 w-1/2 We now have all the elements we need. The left hand side of the Slutsky equation is 1 x1 = - 2 m p1 2w The right hand side of the Slutsky equation is h1 x 1 1 1 + (1 - x ) 1 = - w-3/2 u1/2 + T - [wT + m] 1 p1 m 2 2w 2w 1 1 1 = - w-3/2 w-1/2 [wT + m] + [wT - m] 2 2 4w2 1 1 = - 2 [wT + m] + [wT - m] 4w 4w2 1 = - 2m 2w where the first line uses (6.12) and (6.14), and the second uses u = v(p1 , p2 , m) and equation (6.13). We can therefore see that the substitution effect outweighs the income effect, and as m becomes smaller these two effects grow closer in magnitude. In the limit, as m 0, leisure demand (and hence labour supply) are independent of the wage. (6.14)
31

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