Economics 2010a Fall 2003 Edward L. Glaeser Lecture 4 4. Welfare Analysis and Other Issues a. Measuring Welfare b. First Order and Second Order Losses c. Taxes and Welfare d. Household Production (did last class) e. The Hedonic Approach f. The Theory of Equalizing Differentials g. Application: Land Prices and Consumption A few useful utility functions to think about: (1) Quasi-Linear Preferences, i.e. U?x 1 ,x 2 ,...x L ? ? x 1 ???x 2 ,...x L ? First order conditions are then: ?? ?x i ? p i for all i ? 1. This means that consumption of all goods, except for good one is independent of income and depends only on prices. Good one just takes up the residual income. (2) Homothetic Preferences (MWG Definition 3.B.6) A monotone preference relation on X ? ? ? L is homothetic if x ? y then ?x ? ?y for any ? ? 0. (Parallel indifference curves)– homothetic preferences can be represented by a utility function u?x? that is homogeneous of degree one, i.e. ?u?x? ? u??x? for all positive ?. Does that mean that utility functions that are not homogeneous of degree one can’t be homothetic? (3) Leontief/Fixed Proportions u?x 1 ,x 2 ,...x L ? ? min?? 1 x 1 ,? 2 x 2 ,...? 3 x L ? In this case, people always consume goods in exactly the same proportions and these proportions are independent of prices. (4) Cobb-Douglas (fixed budget shares) u?x 1 ,x 2 ,...x L ? ? ? i?1 L x i ? i or ? i?1 L ? i log?x i ? where ? i?1 L ? i ? 1 Maximization yields: ? i x i ? ?p i ,or ? i ? ? p i x i with ? i?1 L p i x i ? w, this means 1 ? ? w and ? i w ? p i x i this yields fixed budgets shares– what are income and price elasticities in this case? (5) Separable utility u?x 1 ,x 2 ,...x L ? ? ? i?1 L v i ?x i ? This yields first order conditions v i ? ?x i ? ? ?p i , so consumption depends only on the price for your own good and the marginal utility of income. There are cross-price elasticities but they only work through the marginal utility of income, or another way to think about this is if we normalize the price of good one to one, we have v 1 ? ?x 1 ? ? v i ? ?x i ? p i for all i. A particularly standard case of this function is u?x 1 ,x 2 ,...x L ? ? ? i?1 L ? i x i ? i which yields f.o.c. ? i ? i x i ? i ?1 ? ?p i Again if the price of good 1 is one, this tells us: ? i ? i p i x i ? i ?1 ? ? 1 ? 1 p 1 x 1 ? 1 ?1 for all i. (n.b. this is actually the same as the C.E.S. utility function) If ? i ? ?, then x i ? w? i 1 1?? p i ?1 1?? ? j?1 L p j ?? 1?? ? j 1 1?? Separability will particularly show up when thinking about time and uncertainty. In the temporal context, u?x 1 ,x 2 ,...x L ? ? ? i?1 L ? i?1 x i ? is a particularly usual formulation where the i subscripts generally refer to some time periods. and of course this means: ? i?1 p 1 p i x i ??1 ? x 1 ??1 or x i ? x 1 p 1 ? i?1 p i 1 1?? Consumption rises or falls depending on whether the change in prices is greater or less than the discount factor– the extent to which consumption rises depends on the elasticity term. If there is a fixed interest rate, then p i ? ?1 ?r? ??i?1? , which means x i ? x 1 ???1 ?r?? i?1 1?? . a. Measuring Welfare In general, "util" units is pretty hard to use in measuring welfare. Starting with utility maximization is helpful in showing whether utility rises or falls with a particular perturbation, but for quantifying changes, it’s a hard road. The expenditure function is a more natural means of approaching welfare calculations. It can put the losses/gains in dollar units. For example, for any change in prices (due to supply shifts or taxes) where w ? e?p 0 ,u?, then e?p 1 ,u? ? e?p 0 ,u? can be seen as the welfare loss from the change in prices. The quantity e?p 1 ,u? ? e?p 0 ,u? ? ? p 0 p 1 ?e?p,u? ?p dp ? ? p 0 p 1 h?p,u?dp If you happen to know the Hicksian demand function, you’re done. A few examples– if the price change is only for one commodity, and the Hicksian demand for that commodity is linear, you are in the usual triangle case, i.e. h?p,u? ? h 0 ? h 1 p, yields h 0 ?p 1 ? p 0 ? ? 1 2 h 1 ?p 1 2 ? p 0 2 ? or ?p 1 ? p 0 ? h 0 ? h 1 ?p 1 ?p 0 ? 2 or ?p 1 ? p 0 ??h 0 ? h 1 p 1 ? ?h 1 ?p 1 ?p 0 ? 2 2 This is our familiar graph. Technically, the graph really only makes sense for hicksian demands. Bounding Welfare Losses (from J. Green’s lecture notes) An example– what is the welfare loss of a price doubling, when a consumer spends 10 percent of his income on the commodity. The range is between 0 and 0.1w. Now assume that we know that the price elasticity of this commodity (Marshallian) is ? and the income elasticity lies between one and zero (i.e. it is a normal good) and the initial budget share is ?. Using the Slutsky equation, we know that: ? p x ?h?p,w? ?p ? ? p x ?x?p,w? ?p ? px w ?x?p,w? ?w w x where ? p h ?? p ? ?? w ? ? ? ?? w So ? ? ? ? ? p h ? ? and we let ? denote ? p h We know that h?p,u? ? ?w/p 0 If h?p,u? can be approximated with a constant price elasticity function in the neighborhood of p 0 , it equals: h?p,u? ? ?wp ?? /p 0 1?? Welfare loss is ?w p 0 1?? ? p 0 p 1 p ?? dp ? ?w ?1 ? ??p 0 1?? ?p 1 1?? ? p 0 1?? ? If we let g be the gross growth in prices, then the welfare loss is ?w ?1??? ?g 1?? ? 1? , which is always increasing with share, increasing with g, and falling with ? (prove this on your own– the intuition is that the more elastic is demand, the less the loss). So the biggest loss occurs when ? equals ? ? ? so the loss must be less than w times ? ?1????? ?g 1???? ? 1? So imagine the price doubles– and the budget share is .1, and ? equals 1, then we know that the welfare loss is less than .072 times W. Welfare Effects of Tax Changes Consider an increase in the tax rate from t to t??t. This could easily be a vector across commodities, but we will tend to focus on the two good world with one good taxed and the other not. What change in income would offset the price change to the consumers? What is the change in revenues to the government? Total welfare adds together these two quantities (triangles and rectangles). We assume that the supply price, p 0 ,is independent of taxes, so total price equals p 0 ?t with the first tax and p 0 ?t??t after the tax change. The expenditure function change is e?p 0 ?t??t,u? ? e?p 0 ?t?,u? To this quantity we need to add the change in tax revenues which includes both the change in tax rate and the change in quantity consumed. Consider now the two good case, and assume that the price of the untaxed commodity is one. The welfare change from the change in tax level equals: ?L ? e?p 0 ?t??t,u? ? e?p 0 ?t,u? ??th?p 0 ?t??t,u? ?t?h?p 0 ?t,u? ? h?p 0 ?t??t,u Total loss has three components: (1) loss to consumers, minus (2) gain to government from higher tax rate at new consumption level, plus (3) loss to government from less consumption at new tax level. We are going to handle this with a second order Taylor series expansion, and for that we need derivatives w.r.t ?t, evaluated at ?t ? 0: ?L ??t ? ?e ?p ? h? p?t,u? ? t ?h ?p ? ?t ?h ?p When the initial tax is zero– how big is this? ? 2 L ??t 2 ? ?t ? 2 h ?p 2 ? ?h ?p ?L ? ?t?t ?h ?p ? 1 2 ??t? 2 t ? 2 h ?p 2 ? ?h ?p ?... If h was linear and t was zero, this becomes ? 1 2 ??t? 2 ?h ?p , that should be familiar. If h was linear and there is an initial tax– we have a pre-existing wedge, so the loss is the welfare loss, plus the loss tax revenue due to lower consumption– ? t?t ?h ?p ? 1 2 ??t? 2 ?h ?p e. The Hedonic Approach (Rosen, 1974) u ? u?x,z 1 ,z 2 ,...z n ? x is serving as the composite commodity (at price 1) and the z’s (as in the case of household production) represent characteristics of a good, not prices. Start by assuming that consumers purchase only one good each of which has a vector of characteristics ?z 1 ,z 2 ,...z n ? and a price. Assuming that different models are being sold to the same type of consumer, we have: u ? u?w ? p?z 1 ,z 2 ,...z n ?,z 1 ,z 2 ,...z n ? Differentiating we have ?u ?x ?p ?z i ? ?u ?z i What happens with heterogeneous consumers? Assume that there are k different products each of which with a different level of z, and z 1 ? ? j?1 k q j z j 1 sums together all of the amount that you consume. u ? u w ? ? j?1 k q j p j , ? j?1 k q j z j 1 , ? j?1 k q j z j 2 ,... Then we have ?u ?x p j ? ? i?1 n ?u ?z i z j i If k ? n and for no model j is z j ? ?z i for any other commodity i (or if there exist at least n models for which z j ? ?z i ) then the existence of a price vector for models implies a price vector for each characteristic, so we get back to our more usual framework with linear prices for characteristics. The first order conditions can be written p ? Z? T , where p is the vector of model prices, Z is the matrix of z j i terms and ? is the vector with components ? i ? ?u ?z i / ?u ?x f. The Theory of Equalizing Differences or Compensating Differentials (Smith) Few things are as much at the heart of economics as the theory of equalizing differences, or compensating differentials. The concept is one of the purest expressions of our equilibrium ideas, that returns or costs must offset other things. Essentially this is just an application of the hedonic model to specific contexts. For example– each job has a set of characteristics, and we can write: u ? u?w?z 1 ,z 2 ,...z n ?,z 1 ,z 2 ,...z n ? ?u ?x ?w ?z i ? ? ?u ?z i This tells you that jobs that have unpleasant characteristics will have higher wages. Whywouldthisoftenbehardtofindin empirical work? The real estate market: Each location has different characteristics u ? u?w ? r?z 1 ,z 2 ,...z n ?,z 1 ,z 2 ,...z n ? ?u ?x ?r ?z i ? ?u ?z i Rents rise exactly to make you pay for characteristics. Of course, these prices need not be linear. In some cases, both prices and wages will differ over space: u ? u?w?z 1 ,..z n ? ? r?z 1 ,..z n ?,z 1 ,..z n ? ?u ?x ?w ?z i ? ?r ?z i ? ?u ?z i To close the models you need to think about both labor demand and housing supply. Also quantities of labor and housing. The Alonso-Muth-Mills Model Goal is to capture the relationship between distance from city center and housing prices and housing density. Hence we need at least three ingredients in the utility function: (1) housing consumption (assume we’re just talking about land quantity here), (2) the cost of commuting, and (3) some alternative use of money. Possible function # 1: u?A,T,C? where A is land area, T is commuting time and C is other forms of consumption. Where land is indexed by d, distance from city center, there is a price p?d? per unit of land as function of this distance, d also determines commuting time T?d?, and consumption equals W ? p?d?A,the residual. This yields the problem of maximizing u?A,T?d?,W ? p?d?A? over d and A,which yields: ?u ?A ? ?u ?C p?d?, and ?u ?T T ? ?d? ? ?u ?C p ? ?d?A This is pretty good, but we can do slightly better by being less general. Assume that commuting only enters through the budget constraint, i.e. again we have a time budget (normalized to one) and it can be spent either on earning money or commuting, so the problem becomes choosing d and A that solve max A,d u?A,W ? WT?d? ? p?d?A? This now yields first order conditions: ?u ?A ? ?u ?C p?d?, and WT ? ?d? ? ?p ? ?d?A If T?d? ? td then we have Wt A ? ?p ? ?d? In the case that the city is populated by homogeneous consumers, we can prove that lot size always decreases with distance. The first order condition is: u A ?A,W ? Wtd ? p?d?A? ? p?d?u C ?A,W ? Wtd ? Defining A ? ?d? as the values of A that solve this equation for different levels of d we get: ?A ? ?d ? ??Wt ? p ? ?d?A??u AC ? p?d?u CC ? ? p ? ?d?u C u AA ? 2p?d?u AC ?p?d? 2 u CC But we know we have Wt A ? ?p ? ?d? so ?A ? ?d ? ?p ? ?d?u C u AA ? 2p?d?u AC ?p?d? 2 u CC And that is positive– so lot sizes rise with distance from the city center– combining compensating differential with incentive effects to describe both prices and quantities. But we still don’t have anything that we can directly estimate. In all cases, we have p ? ?d? ? ?Wt/A Simplest, classic case: u?A,c? ? c??logA Then the first order condition for land consumption gives you ? A ? p?d?. Then you are left with the differential equation p ? ?d? p?d? ? ? Wt ? with the solution p?d? ? p?0?e ? Wtd ? . This also implies that density equals 1 ? times p?0?e ? Wtd ? This you can estimate. What if utility is Cobb-Douglas, then: ? A ? ?1 ? ??p?d? W ? Wtd ? p?d?A or p?d? ? ?W?1 ? td?/A, and using p ? ?d? ? ?Wt/A, we have p ? ?d? ? ?p?d? t ??1 ? td? This is another differential equation which solves to: p?d? ? p?0??1 ? td? 1 ? Density in this case (1/A) equals p?0??1 ? td? 1 ? ?1 /?W.