I. Deadweight Loss of a Tax

Transcription

1 Economcs 203: Dervaton of Ramsey's Optmal Tax Formula Fall 2007 Casey B. Mullgan These notes derve Ramsey's optmal tax formula that we dscussed n lecture. Econ 203 students should understand the logc behnd the formula and how the formula mght be appled to government polces, but t s not expected that Econ 203 students would be able to reproduce ths algebra on an exam or problem set. I. Deadweght Loss of a Tax Consder a tax of t per unt of good. Assumpton Hcksan demand curves are lnear n the relevant range. For good, the demand curve s: p = a - x where x s the quantty of consumed, p s the prce pad by consumers, and a and are constants. If the taxes we study are suffcently small, Assumpton mght be justfed as a Taylor approxmaton to a nonlnear demand curve. Assumpton 2 The supply of good s nfntely elastc at prce =. You may remember from mcroeconomcs that the dead-weght loss of a tax s the area between the supply and demand curves and to the rght of the quantty demanded x ' when the tax s n place. Employng Assumptons & 2, Fgure draws the Hcksan demand curve as a downward slopng lne wth ntercept a and the supply curve as a horzontal lne wth ntercept. Wth taxes, the equlbrum prce of good s p = - the same for producers and consumers - and the quantty traded n the market s x *. Wth a tax per unt of t, the equlbrum prce of good s p = + t and the quantty of good s x '. Because we have drawn a Hcksan demand curve, the two shaded areas are what the consumer s wllng to pay to avod the tax. 2 The entre shaded area s not a loss to socety, however, because the government does get some revenue from the tax. The revenue s the tax per unt tmes the number of unts sold, T = t x ', and can be shown graphcally as the shaded box n the fgure. The net loss - or deadweght loss - from the tax on good s therefore the shaded trangle labeled For a dervaton wth mperfectly elastc supply curves, see Ramsey (927). 2 You may remember from mcroeconomcs that the area under the Hcksan demand curve s the expendture functon - the amount of money a consumer must spend to attan a gven level of utlty. Wth the hgher prce + t, the area under the demand curve ncreases by exactly the shaded area whch s the extra money the consumer needs to mantan hs utlty by Casey B. Mullgan

2 Economcs 203: Dervaton of the Ramsey Tax Formula Page 2 DWL. Fgure Geometrc Analyss of the Deadweght Loss of a Tax The formula for the good demand curve s p = a - x or, equvalently, x = (a -p )/. Snce we have a formula for the demand curve, we can compute the change n demand (x * - x ') as a result of the tax. x ( & x ) ' a & & a & & t ' t Defnton Defne the ad valorem tax rate of good to be taxes pad per unt of sales of good and denote that tax rate.

3 Economcs 203: Dervaton of the Ramsey Tax Formula Page 3 / tax pad on good pre&tax sales of good ' t x x ' t otce that ad valorem tax rates are easly compared across goods. An ad valorem tax rate of 0.0 s a 0% tax on the sales of a good. Per unt taxes are tougher to compare across goods. For example, a tax of $0.0 per unt s a very sgnfcant tax on a can of soda but a very nsgnfcant tax on an automoble. Defnton 2 Defne the magntude of the demand elastcty of good to be the percentage reducton n quantty demanded for every percentage change n the prce of good evaluated at p = and denote that elastcty η. η / & dp /0 x t ' 0 dx p ' a & The second equalty follows from our formula for the demand functon x =(a -p )/. Usng that formula for the demand functon and the geometry dsplayed n Fgure, we can compute a formula for the deadweght loss from a tax on good : DWL / 2 (base)(heght) / 2 / 2 t (t ) ( ) ' 2 c 2 τ 2 (*) otce that deadweght losses are n dollars: a tax per unt of a good (t ) tmes a quantty of goods (t / ). II. Ramsey's Problem Ramsey asked the followng two dual questons: C "For a gven amount of revenue to be extracted from a consumer, what tax system makes the consumer happest?"

4 Economcs 203: Dervaton of the Ramsey Tax Formula Page 4 C "For a gven level of happness, how can we extract the most revenue from a consumer?" These are dual questons just lke the classc consumer problems n prce theory What choces maxmze utlty subject to the budget constrant? and ts dual What choces mnmze expendture subject to a utlty constrant? Ths note sets up the frst problem, but keep n mnd that the second problem s ts dual. The frst problem can be expressed mathematcally as: mn τ, τ 2, ÿ, τ j ' DWL s.t. j T $ G ' The choce varables are the tax rates on the dfferent goods. The objectve s the total deadweght loss from all of the taxes. otce that summng deadweght losses across goods makes some economc sense because deadweght loss s measured n dollars. As long as we are only thnkng about the taxes to be leved on a sngle consumer, summng dollars also makes sense from a socal welfare pont of vew. If nstead one consumer consumes one set of goods and another consumer consumers a dfferent set of goods, then a dollar of deadweght loss on one good may not be socally equal to a dollar of deadweght loss on another good. 3 T s the tax revenue collected from taxes on good, so the constrant requres that total tax revenue be at least the requred government spendng G. To make the problem nterestng and more tractable, three assumptons are useful: Assumpton 3 Lump sum taxes are not avalable. In addton to the goods under consderaton, there s at least one other good that cannot be taxed. Assumpton 4 Hcksan cross-prce elastctes are zero. Assumpton 5 The requred government spendng G s fxed - t does not depend on the chosen mx of taxes. Assumpton 3 makes Ramsey's problem nterestng. If lump sum taxes were avalable, usng those and settng all ad valorem tax rates to zero would generate zero deadweght loss and maxmze utlty. If there were no untaxed good, makng all tax rates equal would not dstort any relatve prces and therefore not create any deadweght loss. Assumpton 4 makes the problem easer - we don't have to worry about the possblty that a tax on good mght affect the demand for good j. otce that formula for the demand functon shown n 3 For example, t may be socally desrable to tax the rch more heavly than the poor.

5 Economcs 203: Dervaton of the Ramsey Tax Formula Page 5 Assumpton satsfes Assumpton 4. Assumpton 5 also makes the problem easer, although see Mullgan (996), Becker and Mullgan (998), or our lecture notes on How the Economy Influences Polcy for an argument that the answer to Ramsey's problem wth Assumpton 5 should not gude tax polcy. Wth the Assumptons 3-5, the LaGrangan for Ramsey's problem s: / j ' 2 τ 2 b c 2 % λ G & j ' a & & The frst sum n the LaGrangan s the sum of deadweght loss for each good and uses the formula (*) for the deadweght loss derved n Secton I. The sum n the brackets s the sum of tax revenues from each good and uses the formula T = t x ' = (a -p )/. λ s the LaGrange multpler on the government's budget constrant and therefore mght be thought of as the margnal deadweght loss of government spendng. Takng the frst order condton wth respect to and solvng for, we get: 4 c 2 ' λ a & \ & 2 λ c 2 c ' λ c a & & 2 λ a & \ λ a ' & % 2 λ \ ' λ % 2 λ η (**) The frst equalty equates the margnal deadweght loss to the margnal revenue of a change n good 's tax rate. The equaton depend only on the tax, supply, and demand parameters of good and not those of other goods because we have assumed that the Hcksan cross prce elastctes are zero. The frst term on the RHS s proportonal to the quantty of demanded at p =. The second term hold. 4 You can check that, gven our assumptons, the second order condtons of the problem

6 Economcs 203: Dervaton of the Ramsey Tax Formula Page 6 reflects the tax revenue lost because a tax ncrease decreases the demand for good. Wthout the second term, we would only have the frst term - a change n the tax rate would rase revenue accordng to the quantty of x demanded. As one mght expect, the second term s mportant relatve to the frst when the elastcty of demand s of greater magntude. Ths fact can be seen n the second equalty (whch multples the frst by /(a - )) by notng that /(a - ) s the magntude of the demand elastcty. The second equalty also emphaszes that term on the LHS - the margnal deadweght loss of the tax on good - s relatvely mportant when the magntude of the demand elastcty s large. An elastcally demanded good therefore has a hgh margnal deadweght loss (the LHS term) and s a poor source of revenue (the second term on the RHS), suggestng that t s not optmal to tax an elastcally demanded good heavly. The fnal equalty derves from the defnton of η, the magntude of demand elastcty of good. I refer to (**) as Ramsey's Optmal Tax Formula.

7 Economcs 203: Dervaton of the Ramsey Tax Formula Page 7 III. Propertes of Ramsey's Soluton There are three propertes of Ramsey's soluton worth rememberng. Property Property 2 Property 3 Any good that can be taxed should be taxed to some degree. Goods wth hgher demand elastctes should be taxed less. The sze of government affects all tax rates proportonally. Property follows from Ramsey's formula, whch requres > 0 for all n the maxmzaton problem. Propertes 2 and 3 can be proved by dvdng the Ramsey tax formulas for any two goods and j: τ j ' η j η A change n the sze of government does not affect the optmal rato of the two tax rates. In terms of Ramsey's optmal tax formula, thnk of G as affectng the formula through λ. The more revenue that must be rased, the greater the margnal deadweght loss of government spendng. Accordng to (**) the margnal deadweght loss of government spendng affects all taxes proportonally. IV. References Becker, Gary S. and Casey B. Mullgan. Dead Weght Costs and the Sze of Government. BER Workng Paper, October 998. Mullgan, Casey B. Government Gets Fat on the Flat Tax. Chcago Sun Tmes. May 4, 996. Page 4. Ramsey, Frank. A Contrbuton to the Theory of Taxaton. Economc Journal. 37, March 927: 47-6.