Lecture 11: Welfare and Allocation
Advanced Microeconomics I
Yosuke YASUDA
National Graduate Institute for Policy Studies
November 18, 2013
How to Measure Welfare Change
When the economic environment or market outcome changes, a consumer may be made better off or worse off. Economists often want to measure how consumers are affected by these changes, and have developed several tools for the assessment of welfare. The obvious measure of the welfare change involved in moving from (p0, ω0) to (p1, ω1) is just the difference in indirect utility:
v(p1, ω1) − v(p0, ω0).
◮ If the utility difference is positive, then the policy change is worth doing, at least as far as this consumer is concerned.
◮ If it is negative, the policy change is not worth doing.
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Q Is there any monetary measure that quantifies welfare changes?✆
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Consumers’ Surplus
Suppose that the price of some good moves from p0 to p1 while the prices of other goods and initial wealth remain unchanged.
Def The classical measure of welfare change is consumers’ surplus (CS), which is the area below the Marshallian demand curve and above market price. The change of CS is defined as
∆CS := CS(p0, ω) − CS(p1, ω) = Z p1
p0
x(p, ω)dp.
◮ This is simply the area to the left of the Marshallian demand curve between p0 and p1.
◮ Although CS is intuitive and simple, it is an exact measure of welfare change only in special circumstances.
Beyond Consumers’ Surplus
Depending on how to quantify utility changes, we have two different measures which are better than CS.
Def The compensating variation (CV) and equivalent variation (EV) are defined as follows:
v(p1, ω+ CV ) = v(p0, ω), v(p0, ω− EV ) = v(p1, ω).
◮ EV uses the current prices as the base and asks what income change at the current prices would be equivalent to the proposed change in terms of its impact on utility.
◮ CV uses the new prices as the base and asks what income change would be necessary to compensate the consumer.
◮ That is, EV (resp. CV ) requires to keep a consumer’s utility constant before (resp. as a result of) a price change.
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Fg Figure 4.5 (see JR, pp.181)✆
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Compensating Variation
Using the definitions of CV and expenditure function, CV can be written by
e(p1, v(p0, ω)) = e(p1, v(p1, ω+ CV ))
= ω + CV
⇒ CV = e(p1, v(p0, ω)) − e(p0, v(p0, ω)) By Shepard’s lemma, we obtain
CV = e(p1, v(p0, ω)) − e(p0, v(p0, ω))
= Z p1
p0
∂e(p, v(p0, ω))
∂p dp=
Z p1 p0
qh(p, v(p0, ω))dp.
This is simply the area to the left of the Hicksian demand curve between p0 and p1, when the target utility level is v(p0, ω).
Equivalent Variation
Similarly, EV can be expressed by
e(p0, v(p1, ω)) = e(p0, v(p0, ω− EV )) = ω − EV
⇒ EV = e(p1, v(p1, ω)) − e(p0, v(p1, ω))
= Z p1
p0
qh(p, v(p1, ω))dp.
This is simply the area to the left of the Hicksian demand curve between p0 and p1, when the target utility level is v(p1, ω). The absolute value of ∆CS is always between that of CV and EV. These three measures coincide if and only if there is no income effect, for instance, when the utility function is quasi-linear: u(x1, x2) = f (x1) + x2.
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Fg Figure 5.9 (see SN, pp.160)✆
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General Environmental Changes
Consider the move from (p0, ω0) to (p1, ω1). Then, Def The compensating variation (CV) and equivalent variation (EV) are defined as follows:
v(p1, ω1+ CV ) = v(p0, ω0), v(p0, ω0− EV ) = v(p1, ω1).
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Rm CV and EV can be expressed as follows:✆
e(p1, v(p0, ω0)) = e(p1, v(p1, ω1+ CV )) = ω1+ CV
⇒ CV = e(p1, v(p0, ω0)) − ω1, e(p0, v(p1, ω1)) = e(p0, v(p0, ω0− EV )) = ω0− EV
⇒ EV = ω0− e(p0, v(p1, ω1)).
Exchange Economy
Next, we introduce the welfare measures that do not rely on any quantitative assessment.
Consider an exchange economy with I people and n goods where all of the economic agents are consumers and production is absent.
◮ Let e = (e1, ..., eI) denote the economy’s initial endowment vector, where ei = (ei1, ..., ein) denotes i’s initial endowment.
◮ Define an allocation as a vector, x = (x1, ..., xI), where xi = (xi1, ..., xin) denotes i’s consumption bundle according to the allocation.
◮ The set of feasible allocations is this economy is given by F(e) = {x |X
i∈I
xi =X
i∈I
ei}.
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Pareto Efficiency (1)
A situation is called Pareto efficient if there is no way to make someone better off without making someone else worse off.
◮ That is, there is no way to make all agents better off.
◮ To put it differently, each agent is as well off as possible, given the utilities of the other agents.
This is a central welfare notion in Economics, which is formally defined as follows.
Def A feasible allocation, x ∈ F (e), is (strongly) Pareto
efficient if there is no other feasible allocation, y ∈ F (e) such that ui(yi) ≥ ui(xi) for all i ∈ I, and
ui(yi) > ui(xi) for at least one i ∈ I.
Pareto Efficiency (2)
Def A feasible allocation, x ∈ F (e), is weakly Pareto efficient if there is no other feasible allocation, y ∈ F (e) such that
ui(yi) > ui(xi) for all i ∈ I
It is straightforward that an allocation that is (strongly) Pareto efficient is also weakly Pareto efficient.
In general, the reverse is not true.However, under some additional weak assumptions, the reverse implication is true.
Thm Suppose that preference relations are continuous and monotonic. Then an allocation is weakly Pareto efficient if and only if it is strongly Pareto efficient.
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Pareto Efficiency and Calculus
Thm A feasible allocation x∗ is Pareto efficient if and only if x∗ solves the following maximization problems for i = 1, ..., I:
max
x
ui(xi) s.t. XI h=1
xkh≤ ek k= 1, ..., n uj(x∗j) ≤ uj(xj) for all j 6= i.
Proof (⇐) Suppose x∗ solves all maximization problems but x∗ is not Pareto efficient. This means that there is some allocation x′ where someone i is strictly better off. But then x∗ cannot solve the problem for i, a contradiction.
(⇒) Suppose x∗ is Pareto efficient, but it does not solve one of the problems. Instead, let x′ solve that particular problem. Then x′ makes one of the agents strictly better off without hurting any other agents, a contradiction.
Pareto Efficiency and Social Welfare Function (1)
Def A social welfare function W : RI → R is a hypothetical scheme for ranking potential allocations of resources based on the private utilities they provide to individuals:
Social Welfare = W (u1(x1), · · · , uI(xI))
Assume that W is increasing in each of its arguments. Then we immediately obtain the following theorem.
Thm If x∗ solves the following (social welfare) maximization problem, then x∗ is Pareto efficient.
max
x
W(u1(x1), · · · , uI(xI)) s.t.
XI h=1
xkh ≤ ek for k = 1, ..., n.
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Pareto Efficiency and Social Welfare Function (2)
Imposing additional assumptions, we can completely characterize Pareto efficient allocations by the maximization problem of weighted average of individual utilities.
Thm If ui is an increasing and concave function for all i ∈ I. Then, x∗ is Pareto efficient if and only if x∗ solves the following maximization problem for some (λ1,· · · , λI) ∈ RI+\ {0}.
max
x
XI i=1
λiui(xi)
s.t. XI h=1
xkh ≤ ek for k = 1, ..., n.
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Q What happens if a consumer’s utility function is not concave?✆
Block and Core
Let S ⊂ I denote a coalition (subset) of consumers.
Def A coalition S blocks x ∈ F (e), if there is an allocation (among S) y such that
X
i∈S
yi=X
i∈S
ei,
ui(yi) ≥ ui(xi) for all i ∈ S, and ui(yi) > ui(xi) for at least one i ∈ S.
That is, an allocation x is blocked whenever some group can do better than they do under x by simply going it alone.
Def The core of an exchange economy C(e) is the set of all feasible allocations which cannot be blocked by any coalition.
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Rm The core must be Pareto efficient, since the core cannot be✆ blocked any allocation including the grand coalition, i.e., S = I.
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Edgeworth Box
The most useful example of an exchange economy is one in which there are two people and two goods. This economy’s set of allocations can be illustrated in an Edgeworth box diagram.
◮ The length of the horizontal axis measures the total amount of good 1.
◮ The height of the vertical axis measures the total amount of good 2.
◮ Each point in this box is a feasible allocation.
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Fg Figures 5.1 and 5.2 (see JR, pp.196-197)✆
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Q How will agents trade their goods in voluntary exchange?✆
⇒ If they trade both efficiently and in mutually beneficial way, then the allocation must be on the contract curve.
