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Advanced Topics in Consumer Theory

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Advanced Topics in Consumer Theory Topics Aggregation Evaluating Welfare Choice Under Uncertainty Advanced Topics in Consumer Theory Juan Manuel Puerta November 9, 2009 Topics Aggregation Evaluating Welfare Choice Under Uncertainty Introduction In this section, we will focus on some selected advanced topics in consumer theory. These are: 1 Aggregation: Could we construct aggregate demand functions out of individual maximization? 2 Evaluating Welfare: How do we assess the welfare effect on changes in prices? 3 Choice under Uncertainty: How does utility maximization looks like when outcomes are uncertain? Topics Aggregation Evaluating Welfare Choice Under Uncertainty The problem of aggregation Assume we have I consumers with Walrasian demands xi(p; mi) for i = 1; 2; :::; I. The aggregate demand of the I consumers could be written as, PI x(p; m1; m2; :::; mI) = i=1 xi(p; mi) The main question is under which conditions we can write the aggregate demand as, PI x(p; m1; m2; :::; mI) = x(p; i=1 mi) In a word, under which conditions can we write the aggregate demand as a function of prices and aggregate income? Topics Aggregation Evaluating Welfare Choice Under Uncertainty Conditions for Aggregation It is apparent that in order for the aggregate demand to be written as a function of aggregate income alone, it is necessary that for 0 0 0 any two income vectors (m1; m2; :::; mI) and (m1; m2; :::; mI) that PI PI 0 yield equal aggregate income (i.e. i=1 mi = i=1 mi ), aggregate P P 0 demands are equal (i.e. i x(p; mi) = i x(p; mi ) P Assume a disturbance vector dm such that i dmi = 0. Consider the effect of this change in the aggregate demand for a given good `, PI @x`i(p;mi) dm = 0 i=1 @mi i for every ` Topics Aggregation Evaluating Welfare Choice Under Uncertainty Conditions for Aggregation (cont.) Recall that this has to be true for any initial m with dm satisfying P i di = 0. Then, the only way in which the equation above can be fulfilled is if, @x`i(p;mi) = @x`j(p;mj) @mi @mj for every ` and every two individuals i and j, and income distributions (m1; m2; :::; mI) y. Implication. This condition just means that the effect of income on demand has to be equal for every individual and income level. Geometrically, this implies that the IEP of the consumers are parallel linear paths. This property holds in particular with homothetic and quasilinear demands. But there is even a more general result applying (next slide) Topics Aggregation Evaluating Welfare Choice Under Uncertainty Conditions for Aggregation (cont.) Proposition: A necessary and sufficient condition for the set of consumers to exhibit parallel, straight income expansion paths (IEP) at any price vector (p) is that preferences admit indirect utility functions of the Gorman form with the coefficient on mi the same for every consumer i. That is, vi(p; mi) = ai(p) + b(p)mi Proof: Omitted (Straightforward application of Roy’s identity). Topics Aggregation Evaluating Welfare Choice Under Uncertainty Final Considerations on Aggregation Note that there are a few properties that carry over from individual demands to aggregate demands regardless of the income effects. Notably, continuity and homogeneity hold for the aggregate demand. Notice though that even when continuity of the individual demands is sufficient for continuity of the aggregate demand, it is not necessary (y). That is, non-continuous demand functions may yield a continuous aggregate demand function. Topics Aggregation Evaluating Welfare Choice Under Uncertainty Welfare Motivation: We often want to measure how certain policies affect consumer welfare. A first question would be, how do we measure welfare. Assume that we moved from (p0; m0) to (p1; m1) as a consequence of a new policy. An intuitive way of checking how the welfare changed would be to look at the indirect utility function and check whether υ(p1; m1) − υ(p0; m0) is positive/negative, so that welfare increased/decreased. Note that the result of such calculation is a “utility” difference that may be hard to interpret. In particular, it is difficult to answer to the following question: How much better/worse is the consumer as a consequence of the new policy? Topics Aggregation Evaluating Welfare Choice Under Uncertainty Money Metric Indirect Utility Functions A way about this problem would be to use a money metric indirect utility function µ(q; p; m) Recall that µ(q; p; m) measures how much income the consumer needs at prices q in order to attain the same utility he had when prices were p and income was m. More precisely, µ(q; p; m) = e(q; υ(p; m)) So letting again the superscript (1) denote “after” and (0) “before”, the utility difference above can be rewritten as, µ(q; p1; m1) − µ(q; p0; m0) Now the question is, at which prices q should we evaluate this expression? Topics Aggregation Evaluating Welfare Choice Under Uncertainty Equivalent and Compensating Variations Two obvious choices are: p0 and p1, EV = µ(p0; p1; m1) − µ(p0; p0; m0) = µ(p0; p1; m1) − m0 CV = µ(p1; p1; m1) − µ(p1; p0; m0) = m1 − µ(p1; p0; m0) Under the Equivalent Variation, everything is measured with respect to the initial prices. This just asks how much money would be equivalent, at current prices, to the proposed change in terms of utility. Under the Compensating Variation, everything is measured with respect to the new prices. In this case, the second term asks in which amount you have to be compensated in order for you to accept the price change. ( Draw figure y) Topics Aggregation Evaluating Welfare Choice Under Uncertainty Which Measure is more appropriate? If you are trying to arrange a compensation for a welfare change, it seems appropriate to use the new prices, and consequently, CV. If instead you want to measure willingness to pay for a new policy, it would seem more appropriate to use EV. If there are more than 2 policies, it would seem reasonable to compare them all with respect to the same benchmark prices. In this case, also, EV seems more reasonable. Topics Aggregation Evaluating Welfare Choice Under Uncertainty Empirical Issues and the “Integrability Problem” How can we measure µ in practice? This is related to the integrability problem Assume that we observe demand functions x(p; m). Can we recover the underlying preferences from the observed behavior? This is the so-called “integrability problem”. We have established that if the expenditure function exists and is differentiable, it should fulfill the following conditions: @e(p;u¯) = h (p; u¯) = x (p; e(p; u¯)) @pi i i for i = 1; 2; ::; k We could rewrite these conditions using the identityu ¯ = υ(q; m), @e(p,υ(q;m)) = x (p; e(p; υ(q; m))) @pi i for i = 1; 2; ::; k Topics Aggregation Evaluating Welfare Choice Under Uncertainty Empirical Issues and the “Integrability Problem” Using the definition of the Money metric indirect utility function, we can write these conditions as, ∂µ(q;p;m) = x (p; µ(q; p; m)) for i = 1; 2; ::; k @pi i And the boundary condition, µ(q; q; m) = m These are the integrability equations. The solution to this system of partial differential equations allows you to find µ which you can then evaluate to compute EV and CV Topics Aggregation Evaluating Welfare Choice Under Uncertainty Empirical Issues and the “Integrability Problem” The classic tool for measuring welfare changes is the consumer’s surplus. The consumer surplus associated to a change in prices between p0 and p1 is, R p1 CS = p0 x(t)dt: This is simply the area to the left of the demand curve between p0 and p1 It turns out that for some particular preferences, these measures coincide. In particular, if preferences are quasilinear (U(x0; x1; :::; xk) = x0 + u(x1; :::; xk)), then CS=CV=EV. But let’s talk some more about quasilinear utility functions. Topics Aggregation Evaluating Welfare Choice Under Uncertainty Quasilinear Utility As stated before, a utility function is quasilinear if it can be written as U(x0; x1; :::; xk) = x0 + u(x1; :::; xk). For the rest of the discussion, we will consider the case of 2 goods (without loss of generality). We usually assume that u(:) is strictly concave. The maximization problem is: maxx0;x1 x0 + u(x1) such that x0 + p1x1 = m FOC for the interior solution yield: u(x1) = p1 and from substitution into the budget constraint x0 = m − p1x1(p1). Note that x1 does not depend on m.The indirect utility function can be written as, V(p1; m) = u(x1(p1)) + m − p1x1(p1) = υ(p1) + m where υ(p1) = u(x1(p1)) − p1x1(p1) Topics Aggregation Evaluating Welfare Choice Under Uncertainty Quasilinear Utility Note that x1 cannot possibly be independent of income for every income level. If income is low enough, the “implicit” constraint x0 ≥ 0 may be binding. In that case x1 = m=p1 and V(p1; m) = u(m=p1). In what follows we will assume out this possibility. Note that in the standard case, demand depends only on prices and there are no income effects to worry about (How would IEP look like?). This property makes Quasilinear preferences particularly convenient for welfare analysis. However, are QL preferences a correct description of consumer behavior? It turns out that in cases in which the demand of the good is relatively independent from income, they are. Imagine the demand for paper or pencils, beyond the minimum income, it is unlikely that further increases in income would trigger increases in demands. Topics Aggregation Evaluating Welfare Choice Under Uncertainty Integrability under Quasilinear Utility It turns out that the integrability problem is greatly simplified under quasilinear preferences.
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