Prerequisites Almost essential Firm: Optimisation Consumption: Basics Frank Cowell: Microeconomics October 2006 Consumer: Welfare MICROECONOMICS Principles and Analysis Frank Cowell
Using consumer theory Frank Cowell: Microeconomics n n n Consumer analysis is not just a matter of consumers' reactions to prices We pick up the effect of prices on incomes on attainable utility - consumer's welfare This is useful in the design of economic policy, for example u n The tax structure? We can use a number of tools that have become standard in applied microeconomics u price indices?
Overview. . . Consumer welfare Frank Cowell: Microeconomics Utility and income Interpreting the outcome of the optimisation in problem in welfare terms CV and EV Consumer’s surplus
How to measure a person's “welfare”? Frank Cowell: Microeconomics n n n We could use some concepts that we already have Assume that people know what's best for them. . . So that the preference map can be used as a guide We need to look more closely at the concept of “maximised utility”. . . the indirect utility function again
The two aspects of the problem Frank Cowell: Microeconomics C(p, u) V(p, y) x 2 DC DV l § Primal: Max utility subject to the budget constraint § Dual: Min cost subject to a utility constraint § What effect on max-utility of an increase in budget? § What effect on min-cost of an increase in target utility? x* l x* x 1 Interpretati on of Lagrange multipliers
Interpreting the Lagrange multiplier (1) Frank Cowell: Microeconomics Optimal value of demands Allofsummations value Second function Optimal for the are from 1 to n. Lagrange multiplier The solution primal: V(p, y) = U(x*) line follows because, at the optimum, either the constraint binds or the Lagrange multiplier is zero Vy(p, y) = Si. Ui(x*) Diy(p, y) We’ve just used the demand functions xi* = Di(p, y) ) n = U(x*) + m* [y – Si pixi* ] n Differentiate with respect to y: + m* [1 – n Rearrange: i because Sip. Vanishes y) ]of FOC i D y(p, Ui(x*) = m*pi Vy(p, y) = Si[Ui(x*)–m*pi]Diy(p, y)+m* Vy(p, y) = m* The Lagrange multiplier in the primal is just the marginal utility of money! And (with little surprise) we will find that the same trick can be worked with the solution to the dual…
Interpreting the Lagrange multiplier (2) Frank Cowell: Microeconomics n The solution function for the dual: Once again, at the optimum, C(p, u) = Sipi xi* – l* [U(x*) – u] n Differentiate with respect to u: Cu(p, u) = Sipi. Hiu(p, u) because of i (p, u – l* [Si Ui(x*) HVanishes ) – =1] u l*Ui(x*) FOC pi n either the constraint binds or the Lagrange multiplier is zero (Make use of the conditional demand functions xi* = Hi(p, u)) Rearrange: Cu(p, u) = Si [pi–l*Ui(x*)] Hiu(p, u)+l* Cu(p, u) = l* Lagrange multiplier in the dual is the marginal cost of utility Again we have an application of the general envelope theorem.
A useful connection Frank Cowell: Microeconomics Minimised budget in the dual Constraint incomesolution can be the underlying in the primal written this way. . . utility in y. Constraint = C(p, u)in. Maximised utility the primal the dual n the other solution this way. u = V(p, y) n n n Mapping utility into income Mapping income into utility We can get fundamental results on the person's welfare. . . Putting the two parts together. . . y = C(p, V(p, y)) marginal cost (in terms Differentiate ofwith utility)respect of a dollar of to y: 1 = Cu(p, u) Vy(p, y) the budget = l* 1 Cu(p, u) = ———— Vy(p, y). A relationship between the slopes of C and V. marginal cost of utility in terms of money = m*
Utility and income: summary Frank Cowell: Microeconomics n This gives us a framework for the evaluation of marginal changes of income… n …and an interpretation of the Lagrange multipliers n The Lagrange multiplier on the income constraint (primal problem) is the marginal utility of income n The Lagrange multiplier on the utility constraint (dual problem) is the marginal cost of utility n But does this give us all we need?
Utility and income: limitations Frank Cowell: Microeconomics n 1. This gives us some useful insights but is limited: We have focused only on marginal effects u 2. We have dealt only with income u n infinitesimal income changes. not the effect of changes in prices We need a general method of characterising the impact of budget changes: valid for arbitrary price changes u easily interpretable u n For the essence of the problem re-examine the basic diagram.
Overview. . . Consumer welfare Frank Cowell: Microeconomics Utility and income Exact money measures of welfare CV and EV Consumer’s surplus
The problem… Frank Cowell: Microeconomics x 2 u § Take the consumer's equilibrium § and allow a price to fall. . . u' § Obviously the person is better off. §. . . but how much better off? x** How do we quantify this gap? x 1
Approaches to valuing utility change Frank Cowell: Microeconomics n Three things Utility that are not much use: differences 1. u' – u. Utility ratios distance 2. u'some / u function 3. d(u', u) n depends on the units of the U function depends on the origin of the U function depends on the cardinalisation of the U function A more productive idea: 1. Use income not utility as a measuring rod 2. To do the transformation we use the V function 3. We can do this in (at least) two ways. . .
Story number 1 Frank Cowell: Microeconomics n n Suppose p is the original price vector and p' is vector after good 1 becomes cheaper. This causes utility to rise from u to u'. u u n Express this rise in money terms? u u n u = V (p , y ) u' = V(p', y) What hypothetical change in income would bring the person back to the starting point? (and is the right question to ask. . . ? ) Gives us a standard definition….
Frank Cowell: Microeconomics In this version of the story we get the Compensating Variation u = V(p, y) u = V(p', y – CV) the original utility level at prices p and income y the original utility level restored at new prices p' § The amount CV is just sufficient to “undo” the effect of going from p to p’.
The compensating variation Frank Cowell: Microeconomics § The fall in price of good 1 x 2 § The original utility level is the reference point. § CV measured in terms of good 2 u CV x* x** Original prices new price x 1
CV assessment Frank Cowell: Microeconomics n n n The CV gives us a clear and interpretable measure of welfare change. It values the change in terms of money (or goods). But the approach is based on one specific reference point. The assumption that the “right” thing to do is to use the original utility level. There alternative assumptions we might reasonably make. For instance. . .
Here’s story number 2 Frank Cowell: Microeconomics n n n Again suppose: u p is the original price vector u p' is the price vector after good 1 becomes cheaper. This again causes utility to rise from u to u'. But now, ask ourselves a different question: u u u Suppose the price fall had never happened What hypothetical change in income would have been needed … …to bring the person to the new utility level?
Frank Cowell: Microeconomics In this version of the story we get the Equivalent Variation u' = V(p', y) u' = V(p, y + EV) the utility level at new prices p' and income y the new utility level reached at original prices p § The amount EV is just sufficient to “mimic” the effect of going from p to p’.
The equivalent variation Frank Cowell: Microeconomics x 2 § Price fall is as before. u' § The new utility level is now the reference point § EV measured in terms of good 2 EV x* x** Original prices new price x 1
CV and EV. . . Frank Cowell: Microeconomics n Both definitions have used the indirect utility function. u u n n n But this may not be the most intuitive approach So look for another standard tool. . As we have seen there is a close relationship between the functions V and C. So we can reinterpret CV and EV using C. The result will be a welfare measure u the change in cost of hitting a welfare level. remember: cost decreases mean welfare increases.
Welfare change as – D(cost) Frank Cowell: Microeconomics n Prices before. Variation Compensating as Prices Reference –D(cost): after utility level CV(p®p') = C(p, u) – C(p', u) n Equivalent Variation as –D(cost): EV(p®p') = C(p, u') – C(p', u') Using the above definitions we also have n CV(p'®p) = C(p', u') – C(p, u') = – EV(p®p') (–) change in cost of hitting utility level u. If positive we have a welfare increase. (–) change in cost of hitting utility level u'. If positive we have a welfare increase. Looking at welfare change in the reverse direction, starting at p' and moving to p.
Welfare measures applied. . . Frank Cowell: Microeconomics n n The concepts we have developed are regularly put to work in practice. Applied to issues such as: u u u n Consumer welfare indices Price indices Cost-Benefit Analysis Often this is done using some (acceptable? ) approximations. . . Example of cost-ofliving index
Cost-of-living indices Frank Cowell: Microeconomics n An index based on CV: I C(p', u) = ——— C(p, u) All summations are. CV from 1 to n. n An approximation: IL = What's the change in cost of hitting the base welfare level u? ³ C(p', u) What's the change in cost of buying Si p'i xi = C(p, theu) base consumption bundle x? ——— This is the Laspeyres index – the S i pi x i basis for the Retail Price Index and other similar indices. ³I. CV n An index based on EV: C(p', u') IEV = ———— C(p, u') n An approximation: Si p'i x'i IP = ——— Si pi x'i £ IEV. What's the change in cost of hitting the new welfare level u' ? = C(p', u') ³ C(p, What's u') the change in cost of buying the new consumption bundle x'? This is the Paasche index
Overview. . . Consumer welfare Frank Cowell: Microeconomics Utility and income A simple, practical approach? CV and EV Consumer’s surplus
Another (equivalent) form for CV Prices before Reference utility level Prices after Frank Cowell: Microeconomics Use the cost-difference definition: CV(p®p') = C(p, u) – C(p', u) n Assume that the price of good 1 changes from p 1 to p 1' while other prices remain unchanged. Then we can rewrite the above as: n (–) change in cost of hitting utility level u. If positive we have a welfare increase. (Just using the definition of a definite integral) p 1 CV(p®p') = ò C 1(p, Hicksian u) dp 1(compensated) p 1' n demand for good 1 Further rewrite as: CV(p®p') = ò p 1' H 1(p, u) dp 1 You're right. It's using Shephard’s lemma again So CV can be seen as an area under the compensated demand curve Let’s see
Frank Cowell: Microeconomics Compensated demand the value of a price fall p 1 § The initial equilibrium § price fall: (welfare increase) compensated (Hicksian) demand curve original § value of price fall, relative to original utility level H 1(p, u) price fall initial price level §The CV provides an exact welfare measure. § But it’s not the only approach Compensating Variation x*1 x 1
Compensated demand the value of a price fall (2) § As before but use new utility level as a reference point § price fall: (welfare increase) compensated (Hicksian) demand curve § value of price fall, relative to new utility level H 1(p, u ) new utility level price fall Frank Cowell: Microeconomics p 1 Equivalent Variation §The EV provides another exact welfare measure. § But based on a different reference point §Other possibilities… x** 1 x 1
Ordinary demand the value of a price fall ordinary (Marshallian) demand curve § The initial equilibrium § price fall: (welfare increase) § An alternative method of valuing the price fall? D 1(p, y) price fall Frank Cowell: Microeconomics p 1 §CS provides an approximate welfare measure. Consumer's surplus x*1 x** 1 x 1
p 1 D 1(p, y) §Summary of the three approaches. H 1(p, u) H 1(p, u ) CV £ CS §Conditions for normal goods §So, for normal goods: CV £ CS £ EV price fall Frank Cowell: Microeconomics Three ways of measuring the benefits of a price fall x 1** § For inferior goods: CV >CS >EV x 1
Summary: key concepts Frank Cowell: Microeconomics Review n Review Interpretation of Lagrange multiplier Compensating variation Equivalent variation u u n CV and EV are measured in monetary units. In all cases: CV(p®p') = – EV(p'®p). Consumer’s surplus u u u The CS is a convenient approximation For normal goods: CV £ CS £ EV. For inferior goods: CV > CS > EV.