Division of the Humanities and Social Sciences Monetary welfare measurement KC Border Fall 2008 Revised Fall 2014 One of the goals of consumer demand theory is to be able to measure welfare changes. The ideal is to measure welfare changes in monetary units, so that they can be added up across consumers to get an aggregate value. Unfortunately, there are problems with this approach that were pointed out by Kaldor [15], Scitovsky [23], and culminating in Arrow s famous impossibility theorem [1]. Chipman and Moore [2, 3, 4, 5, 6, 7] provide an extensive analysis of the difficulties involved. 1 Hicks s Compensating and Equivalent Variations Hicks [11, 12] assigns monetary values to budget changes. He defines the Compensating Variation [12, p. 126] to be the loss of income which would just offset a fall in price, leaving the consumer no better off than before. That is, starting from a budget (p 0, w), considering a price change to the new budget (p 1, w) he defines CV(p 0, p 1 ) by v(p 1, w CV) = v(p 0, w), where v is the indirect utility function. In terms of the expenditure function, letting υ 0 = v(p 0, w) and υ 1 = v(p 1, w), we have w CV = e(p 1, υ 0 ) and w = e(p 1, υ 1 ), so CV(p 0, p 1 ) = e(p 1, υ 1 ) e(p 1, υ 0 ). Similarly he defines the Equivalent Variation [12, p. 128] to be the change in income which, if it took place at [p 0 ] prices, would have the same effect on satisfactions as is produced by the change in prices from [p 0 ] to [p 1 ] as EV(p 0, p 1 ) = e(p 0, υ 1 ) e(p 0, υ 0 ). See Figure 1 for the case where the price of y is held fixed and and the price of x decreases. I hope it is apparent from the figure that the compensating and equivalent variations need not be equal. One difficulty with Hicks approach is that the expenditure function is not directly observable. Nevertheless, as we show in sections 3 and 9, there is a way to use ordinary demand functions to recover these welfare measures. 1
KC Border Monetary welfare measurement 2 y e(p 0, υ 1 ) u = υ 0 u = υ 1 EV(p 0, p 1 ) w CV(p 0, p 1 ) e(p 1, υ 0 ) ˆx(p 0, υ 1 ) x (p 0, w) x (p 1, w) ˆx(p 1, υ 0 ) (p 0, w) (p 1, w) x Figure 1. Compensating and equivalent variation for a decrease in p x. (The figure is drawn to scale for the case u(x, y) = x 2/3 y 1/3, w = 1, p 0 x = 1, p 1 x = 1/2, p 0 y = p 1 y = 1.)
KC Border Monetary welfare measurement 3 2 The quasi-linear case A utility function u(x 1,..., x n, y) is quasi-linear if it is of the form u(x 1,..., x n, y) = y + f(x 1,..., x n ), where f is concave. The indifference curves of a quasi-linear function are vertical shifts of one another, and so typically intersect the x-axis, so corner solutions with y = 0 are common. For interior points though, any additional income is spent on good y. In this case the CV and EV are just vertical shifts of each other and so are equal see Figure 2. But if corner solutions are involved, this is no longer true see Figure 3 for a particularly nasty example. y e(p 0, υ 1 ) EV(p 0, p 1 ) w CV(p 0, p 1 ) e(p 1, υ 0 ) ˆx(p 0, υ 1 ) x (p 0, w) x (p 1, w) ˆx(p 1, υ 0 ) (p 0, w) (p 1, w) x Figure 2. A nice quasi-linear case where CV = EV. (Here u(x, y) = y + (1/3) ln x, w = 1, p 0 x = 1, p 1 x = 1/2, p 0 y = p 1 y = 1.) 3 Money metric utility A more general approach to assigning monetary values to budgets is the following. Define µ(p ; p, w) = e ( p, v(p, w) ), ( ) where p is an arbitrary price vector. Since e is strictly increasing in v, this is an indirect utility or welfare measure. That is, v(p, w) v(p, w ) if and only if µ(p ; p, w) µ(p ; p, w ),
KC Border Monetary welfare measurement 4 y e(p 0, υ 1 ) EV(p 0, p 1 ) w CV(p 0, p 1 ) e(p 1, υ 0 ) (p 0, w) (p 1, w) ˆx(p 0, υ 1 ) x (p 0, w) = ˆx(p 1, υ 0 ) x (p 1, w) x Figure 3. A nasty quasi-linear case where CV EV. (Here u(x, y) = y + 0.6 ln x, w = 0.4, p 0 x = 1, p 1 x = 1/2, p 0 y = p 1 y = 1. It may not look like it, but the red indifference curve is a vertical shift of the blue one.)
KC Border Monetary welfare measurement 5 but the units of µ are in dollars (or euros, or whatever currency). The money values depend on the choice of p. This function is variously called a money metric (indirect) utility or an income-compensation function. This approach is taken by Varian [24] and Chipman and Moore [5, 7], who adapt ideas from McKenzie [18] 1 and Hurwicz and Uzawa [13]. 2 4 Compensating and equivalent variation in terms of money metrics Thus we may define the compensating and equivalent variations of arbitrary budget changes by CV(p 0, w 0 ; p 1, w 1 ) = e(p 1, υ 1 ) e(p 1, υ 0 ) = µ(p 1 ; p 1, w 1 ) µ(p 1 ; p 0, w 0 ) (1) EV(p 0, w 0 ; p 1, w 1 ) = e(p 0, υ 1 ) e(p 0, υ 0 ) = µ(p 0 ; p 1, w 1 ) µ(p 0 ; p 0, w 0 ). (2) 5 A single price change Now consider a decrease in the price of only good i. That is, and p 1 i < p 0 i, p 1 j = p 0 j = p j for j i w 0 = w 1 = w. Since the price of good i decreases, the new budget set includes the old one, so welfare will not decrease. To make the analysis non-vacuous, let us assume that x i (p1, w) > 0. By definition, CV(p 0, p 1 ) = µ(p 1 ; p 1, w 1 ) µ(p 1 ; p 0, w 0 ) }{{} w EV(p 0, p 1 ) = µ(p 0 ; p 1, w 1 ) µ(p 0 ; p 0, w 0 ) }{{} w We now use the following trick (which applies whenever w 0 = w 1 = w): µ(p 0 ; p 0, w 0 ) = w 0 = w = w 1 = µ(p 1 ; p 1, w 1 ). 1 McKenzie defines an expenditure function in terms of a preference relation by µ(p, x 0 ) = min{p x : x x 0 }. This enables him to dispense with a the utility function. 2 Varian attributes the money-metric function µ to Samuelson [22], but this seems a stretch to me. The closest Samuelson seems to come is found on page 1262 and again on page 1273, where he uses McKenzie s expenditure function. The discussion of how this can be used as an indirect utility is confusing at best.
KC Border Monetary welfare measurement 6 This enables us to rewrite the values as CV = µ(p 0 ; p 0, w 0 ) µ(p 1 ; p 0, w 0 ) }{{} w EV = µ(p 0 ; p 1, w 1 ) µ(p 1 ; p 1, w 1 ) }{{} w Or in terms of the expenditure function, we have: CV = e(p 0, υ 0 ) e(p 1, υ 0 ). EV = e(p 0, υ 1 ) e(p 1, υ 1 ) Compare these expression to the definitions (2) and (1). In terms of the expenditure function, definition (2) is equivalent to EV(p 0, w 0 ; p 1, w 1 ) = e(p 0, υ 1 ) e(p 0, υ 0 ), which by the trick is equal to e(p 0, υ 1 ) e(p 1, υ 1 ). In my opinion, using the trick obscures the true comparison, which is of the budgets (p 0, w 0 ) and (p 1, w 1 ) using the money metric defined by p 0. So why did Hicks rewrite things this way? So he could use the Fundamental Theorem of Calculus and the fact that e/ p i = ˆx i to get EV(p 0, p 1 ) = e(p 0, υ 1 ) e(p 1, υ 1 ) = p 0 i p 1 i ˆx i (p, p i, υ 1 ) dp (2 ) CV(p 0, p 1 ) = e(p 0, υ 0 ) e(p 1, υ 0 ) = p 0 i p 1 i ˆx i (p, p i, υ 0 ) dp, (1 ) where the notation p, p i refers to the price vector ( p 1,..., p i 1, p, p i+1,..., p n ). That is, for a change in the price of good i, the equivalent and the compensating variation are the areas under the Hicksian compensated demand curves for good i corresponding to utility levels υ 1 and υ 0 respectively. Also since p 0 i > p1 i, the integrals above are positive if ˆx i is positive. Assume now that good i is not inferior, that is, assume for all (p, w). Recall the Slutsky equation, x i w > 0 ( ) ˆx i p, v(p, w) = x i (p, w) + x p j p j(p, w) x i (p, w). (S) j w Under the assumption that good i is normal and that x j > 0, this implies that ( ) ˆx i p, v(p, w) > x i (p, w). p j p j
KC Border Monetary welfare measurement 7 We know that the Hicksian compensated demands for good i are downward-sloping as a function of p i (other prices held constant), that is, ˆx i / p i < 0, so we have for j = i 0 > ˆx ( ) i p, v(p, w) > x i (p, w). (3) p i p i Now by the equivalence of expenditure minimization and utility maximization we know that Then (3) tells us that x (p 1, w) = ˆx(p 1, υ 1 ) and x (p 0, w) = ˆx(p 0, υ 0 ). (4) 0 > ˆx i(p 1, υ 1 ) p i > x i (p1, w) and 0 > ˆx i(p 0, υ 0 ) > x i (p0, w) p i p i p i That is, as a function of p i. the ordinary demand x i is steeper (more negatively sloped) than the Hicksian demand ˆx i, where it crosses the Hicksian demand. Since x i is downward sloping and p 1 i < p 0 i it must be the case that the Hicksian demand ˆx i (, υ 1 ) lies above the Hicksian demand ˆx i (, υ 0 ). See Figure 4. Thus, for a price decrease, if good i is not inferior, then EV > CS > CV > 0. The inequalities are reversed for a price increase. Also note that if x i w demand curves coincide. = 0, then the three 6 Deadweight loss Consider a simple problem where the good 1 is subjected to an ad rem tax of t per unit, but income and other prices remain unchanged. The original price vector is p 0 and the new one is p 1 = (p 0 1 + t, p0 2,..., p0 n). Clearly the consumer is worse off under the price vector p 1. But how much worse off? We shall compare the tax revenue T = tx 1(p 1, w) (5) to the welfare cost measured by the money metric utility. Specifically, we shall compare tax revenue to the equivalent variation of the price change. The equivalent variation is negative, since the consumer is worse off. We shall show that EV > T. That is, the dollar magnitude of the welfare loss as measured by the equivalent variation exceeds the tax revenue raised. This excess of the welfare loss over the tax revenue is often referred to as the deadweight loss 3 from ad rem taxation. 3 I don t know why the term deadweight is used. Musgrave [19] uses the term excess burden in 1959, which dates back at least to Joseph [14] in 1939, who claims the concept was known to Marshall [16, 8th edition] in 1890. Harberger [10] uses the term deadweight loss in 1964, and claims the analysis of the concept goes back at least to Dupuit [9] in 1844.
KC Border Monetary welfare measurement 8 x i x i (p i; p i, w) CS EV ˆx i (p i ; p i, υ 1 ) CV ˆx i (p i ; p i, υ 0 ) p 1 i p 0 i p i Figure 4. Illustration of a single price decrease. (Graphs are for a Cobb Douglas utility.) N.B. The horizontal axis is the price axis and the vertical axis is quantity axis. The equivalent variation is the area under the Hicksian demand curve for utility level υ 0. The compensating variation is the area under the Hicksian demand curve for utility level υ 1. The consumer s surplus is the area under the ordinary demand curve.
KC Border Monetary welfare measurement 9 Let s abuse the notation slightly to get rid of some of the visual noise by setting x(p) = ˆx 1 (p, p 0 2,..., p 0 n, υ 1 ), where p is just a scalar that represents p 1. Then EV T = = = = p 0 1 +t p 0 1 p 0 1 +t p 0 1 p 0 1 +t p 0 1 p 0 1 +t p 0 1 x(p) dp T by (2 ) x(p) dp tx 1(p 1, w) by (5) x(p) dp tˆx 1 (p 1, υ 1 ) p 0 1 +t x(p) dp ˆx 1 (p 1, υ 1 ) dp. p 0 1 The last equality come from integrating the constant ˆx 1 (p 0 1 + t, p0 1, υ 1) over an interval of length t. But Hicksian compensated demands are downward sloping, so for p 0 1 p p1 1 = p0 1 +t, we have x(p) = ˆx 1 (p, p 0 2,..., p 0 n, υ 1 ) > ˆx 1 (p 1 1, p 1 2,..., p 1 n, υ 1 ), so the last expression is > 0. Therefore a lump-sum tax leaves the consumer better off than an ad rem tax that raises the same revenue. The amazing thing is not so much that the ad rem tax is inferior to the lump-sum tax, but that some taxes are worse than others at all, even when they collect the same amount of revenue! This would not be apparent without our theoretical apparatus. 7 Revealed preference and lump-sum taxation Recall that x is revealed preferred to y if there is some budget containing both x and y and x is chosen. If the choice function is generated by utility maximization, then if x is revealed preferred to y, we must have u(x) u(y). So consider an ad rem tax t on good 1 versus a lump-sum tax, as above. Assume both taxes raise the same revenue T. The ad rem tax leads to the budget (p 1, w), and the lump-sum to the budget (p 0, w T ), where p 1 1 = p 0 1 + t and p 1 j = p 0 j, j = 2,..., n. Let x 1 be demanded under the ad rem tax and x 0 be demanded under the lump-sum tax. Then x 0 is revealed preferred to x 1 : w p 1 x 1 = p 0 x 1 + tx 1 1 = p 0 x 1 + T,
KC Border Monetary welfare measurement 10 so p 0 x 1 w T, which says that x 1 is in the budget (p 0, w T ), from which x 0 is chosen. Thus u(x 0 ) u(x 1 ), so the lump-sum tax is at least as good as the ad rem tax. This argument is a lot simpler than the argument above, but we don t get a dollar value of the difference. Of course the previous argument gave us two or three different dollar values, depending on how we chose p for the money metric. 8 Money metrics and recovering utility from demand: A little motivation It is possible to solve differential equations to recover a utility function from a demand function. The general approach may be found in Samuelson [20, 21], but the following discussion is based on Hurwicz and Uzawa [13]. Consider the demand function x : R n ++ R ++ R n + derived by maximizing a locally nonsatiated utility function u. Let v be the indirect utility, that is, v(p, w) = u ( x (p, w) ). Since u is locally nonsatiated, the indirect utility function v is strictly increasing in w. The Hicksian expenditure function e is defined by e(p, υ) = min{p x : u(x) υ}. and the income compensation function µ is defined by µ(p; p 0, w 0 ) = e ( p, v(p 0, w 0 ) ), Set υ 0 = v(p 0, w 0 ). From the Envelope Theorem we know that e(p, υ 0 ) p i = ˆx i (p, υ 0 ) = x ( i p, e(p, υ 0 ) ). Suppressing υ 0, this becomes a total differential equation e (p) = x ( p, e(p) ). (6) What does it mean to solve such an equation, and what happened to υ 0?
KC Border Monetary welfare measurement 11 An aside on solutions of differential equations You may recall from your calculus classes that, in general, differential equations have many solutions, often indexed by constants of integration. For instance, take the simplest differential equation, y = a for some constant a. The general form of the solution is y(x) = ax + C, where C is an arbitrary constant of integration. What this means is that the differential equation y = a has infinitely many solutions, one for each value of C. The parameter υ in our problem can be likened to a constant of integration. You should also recall that we rarely specify C directly as a condition of the problem, since we don t know the function y in advance. Instead we usually specify an initial condition (x 0, y 0 ). That is, we specify that y(x 0 ) = y 0. In this simple case, the way to translate an initial condition into a constant of integration is to solve the equation y 0 = ax 0 + C = C = y 0 ax 0, and rewrite the solution as y(x) = ax + (y 0 ax 0 ) = y 0 + a(x x 0 ). In order to make it really explicit that the solution depends on the initial conditions, differential equations texts may go so far as to write the solution as y(x; x 0, y 0 ) = y 0 + a(x x 0 ). In our differential equation (6), an initial condition corresponding to the constant of integration υ is a pair (p 0, w 0 ) satisfying e(p 0, υ) = w 0. From the equivalence of expenditure minimization an utility maximization under a budget constraint, this gives us the relation υ = v(p 0, w 0 ) = u ( x (p 0, w 0 ) ). Following Hurwicz and Uzawa [13], define the income compensation function in terms of the Hicksian expenditure function e via µ(p; p 0, w 0 ) = e ( p, v(p 0, w 0 ) ). Observe that and µ(p 0 ; p 0, w 0 ) = w 0 µ(p; p 0, w 0 ) = e(p, υ0 ) = ˆx i (p, υ 0 ) = x ( i p, e(p, υ 0 ) ) = x ( i p, µ(p; p 0, w 0 ) ). p i p i
KC Border Monetary welfare measurement 12 In other words, the function e defined by e(p) = µ(p; p 0, w 0 ) solves the differential equation subject to the initial condition e (p) = x ( p, e(p) ). e(p 0 ) = w 0. We are now going to turn the income compensation function around and treat (p 0, w 0 ) as the variable of interest. Fix a price, any price, p R n ++ and define the function ˆv : R n ++ R ++ R by ˆv(p, w) = µ(p ; p, w) = e ( p, v(p, w) ). The function ˆv is another indirect utility. That is, ˆv(p, w) ˆv(p, w ) v(p, w) v(p, w ). We can use w to find a utility U, at least on the range of x by U(x) = µ(p ; p, w) where x = x (p, w). 9 Recovering utility from demand: The plan The discussion above leads us to the following approach. Given a demand function x : 1. Somehow solve the differential equation µ(p) p i = x ( ) i p, µ(p). Write the solution explicitly in terms of the intial condition µ(p 0 ) = w 0 as µ(p; p 0, w 0 ). 2. Use the function µ to define an indirect utility function ˆv by ˆv(p, w) = µ(p ; p, w). 3. Invert the demand function to give (p, w) as a function of x. 4. Define the utility on the range of x by U(x) = µ(p ; p, w) where x = x (p, w). This is easier said than done, and there remain a few questions. For instance, how do we know that the differential equation has a solution? If a solution exists, how do we know that the utility U so derived generates the demand function x? We shall address these questions presently, but I find it helps to look at some examples first.
KC Border Monetary welfare measurement 13 10 Examples In order to draw pictures, I will consider two goods x and y. By homogeneity of x, I may take good y as numéraire and fix p y = 1, so the price of x will simply be denoted p. 10.1 Deriving the income compensation function from a utility For the Cobb Douglas utility function where α + β = 1, the demand functions are u(x, y) = x α y β x (p, w) = αw p, y (p, w) = βw. The indirect utility is thus The expenditure function is v(p, w) = wβ β ( α p ) α. e(p, υ) = υβ β ( p α) α. Now pick (p 0, w 0 ) and define µ(p; p 0, w 0 ) = e ( p; v(p 0, w 0 ) ) ( ) α α ) ( p = (w 0 β β p 0 β β α = w 0 ( p p 0 ) α. ) α Evaluating this at p = p 0 we have µ(p 0 ; p 0, w 0 ) = w 0. That is, the point (p 0, w 0 ) lies on the graph of µ( ; p 0, w 0 ). Figure 5 shows the graph of this function for different values of (p 0, w 0 ). For each fixed (p 0, w 0 ), the function µ(p) = µ(p; p 0, w 0 ) satisfies the (ordinary) differential equation dµ [ dp = α w 0 (p 0 ) α] p α 1 = αµ(p) = x ( p, µ(p) ). p Note that homogeneity and budget exhaustion have allowed us to reduce the dimensionality by 1. We have n 1 prices, as we have chosen a numéraire, and the demand for the n th good is gotten from x n = w n 1 i=1 p ix i.
KC Border Monetary welfare measurement 14 µ, w 3 2 1 p 1 2 3 4 Figure 5. Graph of µ(p; p 0 ; w 0 ) for Cobb Douglas α = 2/5 utility and various values of (p 0, w 0 ). 10.2 Examples of recovering utility from demand Let n = 2, and set p 2 = 1, so that there is effectively only one price p, and only one differential equation (for x 1 ) µ (p) = x ( p, µ(p) ). 1 Example In this example x(p, w) = αw p. (This x is the demand for x 1. From the budget constraint we can infer x 2 = (1 α)w.) The corresponding differential equation is µ = αµ p or µ µ = α p. (For those of you more comfortable with y-x notation, this is y = αy/x.) Integrate both sides of the second form to get ln µ = α ln p + C so exponentiating each side gives µ(p) = Kp α where K = exp(c) is a constant of integration. Given the initial condition (p 0, w 0 ), we must have w 0 = K(p 0 ) α, so K = w0 (p 0 ) α, or µ(p; p 0, w 0 ) = w0 (p 0 ) α pα.
KC Border Monetary welfare measurement 15 For convenience set p = 1, to get ˆv(p, w) = µ(p ; p, w) = w p α. To recover the utility u, we need to invert the demand function, that is, we need to know for what budget (p, w) is (x 1, x 2 ) chosen. The demand function is x 1 = αw p, x 2 = (1 α)w, so solving for w and p, we have Thus x 1 = α x 2 1 α p w = x 2 1 α x 2 = p = α. 1 α x 1 u(x 1, x 2 ) = ˆv(p, w) ( ) α x 2 x 2 = ˆv, 1 α x 1 1 α = = x 2 1 α ( ) α α x 2 1 α x 1 ( x2 1 α = cx α 1 x 1 α 2, ) 1 α ( x1 α where c = (1 α) 1 α α α, which is a Cobb Douglas utility. ) α 2 Example In this example we find a utility that generates a linear demand for x. That is, x(p, w) = β αp. (Note the lack of w.) The differential equation is µ = β αp. This differential equation is easy to solve: µ(p) = βp α 2 p2 + C For initial condition (p 0, w 0 ) we must choose C = w 0 βp 0 + α 2 p0 2, so the solution becomes µ(p; p 0, w 0 ) = βp α 2 p2 + w 0 βp 0 + α 2 p0 2. So choosing p = 0 (not really allowed, but it works in this case), we have ˆv(p, w) = µ(p ; p, w) = w βp + α 2 p2.
KC Border Monetary welfare measurement 16 Given (x, y) (let s use this rather than (x 1, x 2 )), we need to find the (p, w) at which it is chosen. We know x = β αp, y = w px = w βp + αp 2, so p = β x α, w = y + βp αp2 = y + β β x α ( ) β x 2 α. α Therefore ( β x u(x, y) = ˆv(p, w) = w α, y + β β x ( ) ) β x 2 α α α = y + β β x ( ) β x 2 α α β β x + α ( ) β x 2 α α }{{}}{{} 2 α }{{} w p p 2 = y (β x)2. 2α Note that the utility is decreasing in x for x > β. Representative indifference curves are shown in Figure 6. The demand curve specified implies that x and y will be negative for some values of p and w, so we can t expect that this is a complete specification. I ll leave it to you to figure out when this makes sense. 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 Figure 6. Indifference curves for Example 2 (linear demand) with β = 10, α = 5. 11 A general integrability theorem Hurwicz and Uzawa [13] prove the following theorem, presented here without proof. 3 Hurwicz Uzawa Integrability Theorem Let ξ : R n ++ R + R n +. Assume (B) The budget exhaustion condition p ξ(p, w) = w is satisfied for every (p, w) R n ++ R +. (D) Each component function ξ i is differentiable everywhere on R n ++ R +.
KC Border Monetary welfare measurement 17 (S) The Slutsky matrix is symmetric, that is, for every (p, w) R n ++ R +, S i,j (p, w) = S j,i (p, w) i, j = 1,..., n. (NSD) The Slutsky matrix is negative semidefinite, that is, for every (p, w) R n ++ R +, and every v R n, n n S i,j (p, w)v i v j 0. i=1 j=1 (IB) The function ξ satisfies the following boundedness condition on the partial derivative with respect to income. For every 0 a ā R n ++, there exists a (finite) real number M a,ā such that for all w 0 a p ā = Let X denote the range of ξ, ξ i (p, w) w M a,ā i = 1,..., n. X = {ξ(p, w) R n + : (p, w) R n ++ R + }. Then there exists an upper semicontinuous monotonic utility function u: X R on the range X such that for each (p, w) R n ++ R +, ξ(p, w) is the unique maximizer of u over the budget set {x X : p x w}. Moreover u has the following property (which reduces to strict quasiconcavity if X is itself convex): For each x X, there exists a p R n ++ such that if y x and u(y) u(x), then p y > p x. I have more extensive notes on this topic, including most of a sketch of the proof here. References [1] K. J. Arrow. 1950. A difficulty in the concept of social welfare. Journal of Political Economy 58(4):328 346. http://www.jstor.org/stable/1828886 [2] J. S. Chipman and J. C. Moore. 1971. The compensation principle in welfare economics. In A. M. Zarley, ed., Papers in Quantitative Economics, volume 2, pages 1 77. Lawrence, Kansas: University of Kansas Press. [3]. 1973. Aggregate demand, real national income, and the compensation principle. International Economic Review 14(1):153 181. http://www.jstor.org/stable/2526050 [4]. 1978. The new welfare economics 1939 1974. International Economic Review 19(3):547 584. http://www.jstor.org/stable/2526326
KC Border Monetary welfare measurement 18 [5]. 1980. Compensating variation, consumer s surplus, and welfare. American Economic Review 70(5):933 949. http://www.jstor.org/stable/1805773 [6]. 1980. Real national income with homothetic preferences and a fixed distribution of income. Econometrica 48(2):401 422. http://www.jstor.org/stable/1911104 [7]. 1990. Acceptable indicators of welfare change, consumer s surplus analysis, and the Gorman polar form. In J. S. Chipman, D. L. McFadden, and M. K. Richter, eds., Preferences, Uncertainty, and Optimality: Essays in Honor of Leonid Hurwicz, pages 1 77. Boulder, Colorado: Westview Press. [8] P. A. Diamond and D. L. McFadden. 1974. Some uses of the expenditure function in public finance. Journal of Public Economics 3:3 21. [9] J. E. J. Dupuit. 1844. De la l utilité and de sa mesure. Collezione di scritti inediti o rari di economisti. Turin: La Riforma Soziale. Collected and reprinted by Mario di Bernardi and Luigi Einaudi, La Riforma Soziale, Turin, 1932. [10] A. C. Harberger. 1964. The measurement of waste. American Economic Review 54:58 76. http://www.jstor.org/stable/1818490 [11] J. R. Hicks. 1936. Value and capital. Oxford, England: Oxford University Press. [12]. 1942. Consumers surplus and index-numbers. Review of Economic Studies 9(2):126 137. http://www.jstor.org/stable/2967665 [13] L. Hurwicz and H. Uzawa. 1971. On the integrability of demand functions. In J. S. Chipman, L. Hurwicz, M. K. Richter, and H. F. Sonnenschein, eds., Preferences, Utility, and Demand: A Minnesota Symposium, chapter 6, pages 114 148. New York: Harcourt, Brace, Jovanovich. [14] M. F. W. Joseph. 1939. The excess burden of indirect taxation. Review of Economic Studies 6(3):226 231. http://www.jstor.org/stable/2967649.pdf [15] N. Kaldor. 1939. Welfare propositions of economics and interpersonal comparisons of utility. Economic Journal 49(195):549 552. http://www.jstor.org/stable/2224835 [16] A. Marshall. 1890. Principles of economics. London. [17] A. Mas-Colell, M. D. Whinston, and J. R. Green. 1995. Microeconomic theory. Oxford: Oxford University Press. [18] L. W. McKenzie. 1957. Demand theory without a utility index. Review of Economic Studies 24(3):185 189. http://www.jstor.org/stable/2296067 [19] R. A. Musgrave. 1959. The theory of public finance. New York: McGraw Hill. [20] P. A. Samuelson. 1947. Foundations of economic analysis. Cambridge, Mass.: Harvard University Press.
KC Border Monetary welfare measurement 19 [21]. 1950. The problem of integrability in utility theory. Economica N.S. 17(68):355 385. http://www.jstor.org/stable/2549499 [22]. 1974. Complementarity: An essay on the 40th anniversary of the Hicks Allen revolution in demand theory. Journal of Economic Literature 12(4):1255 1289. [23] T. Scitovsky. 1941. A note on welfare propositions in economics. Review of Economic Studies 9(1):77 88. http://www.jstor.org/stable/2967640 [24] H. R. Varian. 1992. Microeconomic analysis, 3d. ed. New York: W. W. Norton & Co. [25] J. A. Weymark. 1985. Money-metric utility functions. International Economic Review 26(1):219 232. http://www.jstor.org/stable/2526537