N.H.Stern,Optimum income taxation 129 elsewhere outweighed or were outweighed by fixed costs.or necessary expendi- ture. With a net exp nditure of 12%of product and G(u) -e", the median margir.rate rises to33 The highest marginal rates for the three cases respectively are 26,21% and 39.The marginal rates rise at first but begin falling before the median is reached.Mirrlees proves that,for the log-normal distribution and where the elasticity of substitution between consumption and leisure is less than one,the marginal rate tends to zero as n tends to co.There is a higher limit in the case of the pareto distribution where with the same condition on the substitution elasticity the marginal rate tends to 1/l+asn→o when("→-+2)， Examin tion stribu of earnings(see sectio 0n3.2 suggests values of from.5 to2.5 giving limiting marginal rates Higher rates can also be produced by widening the distributi ion of skills in the log normal case is increased to 1.0(from 0.39),the median rate is 56% for the case G(u)=-e-"and a government requirement of 7%of product. Presumably with a wider distribution of skills,inequality considerations increase relative to those concerned with incentives.However.Mirrlees (1971.p.207) ts that such a a'doe not n to be at all realistic skills too vid compatible with obs earnings There are two main features of the cal tonically increasing-most of the population is in the region where they are falling-and the highest marginal rates are low.For the 'realistic'case of 0.39,applying to 5 out of 6 of Mirrlees'examples,the highest marginal rate is39 (1972)and (1973a)discusses the effect of increasing the concavity of G()a ting case of It seem clear t ha was in a influenced by the low rates in th e Mirrlees calculation tkinson(19 p.2)and (1973a,pp.390-391).The maximin criterion in the Mirrlees mod yields tax rates around 50%for the median person [see Atkinson (1972,p.28)]. We have already given the Sadka argument which explains why,for a finite 2Smanyaige7gomenat妆Romiom infinite domain,p ovided the weight in the tail is not too big.We have noted. for exar nple,that he log-norm a limitin m nal rate of ze o but the Pareto no zero limi of the marginal rat dis suggests that a declining rate at the upper end may be a feature of many del of optimum income taxation.We shall say no more (except for the spec ial case of section 5)about the shape of the tax funtion.and concentrate on the labour supply function and its relation to optimum linear taxation. otleitiobiohgm32whetherthedstnbutionotearninaseamiteadingimpresop These are annou ates rather than effective ratesN.H. Stern, Optimum income taxation 129 elsewhere outweighed or were outweighed by fixed costs, or necessary expendi￾ture. With a net government expenditure of 12% of product and G(u) = -emu, the median margirJ rate rises to 33 %. The highest marginal rates for the three cases respectively are 26 %, 21% and 39%. The marginal rates rise at first but begin falling before the median is reached. Mirrlees proves that, for the log-normal distribution and where the elasticity of substitution between consumption and leisure is less than one, the marginal rate tends to zero as n tends to co. There is a higher limit in the case of the Pareto distribution where, with the same condition on the substitution elasticity, the marginal rate tends to l/(1 +y) as n + co when (nf’/‘) + - (7 +2). Examination of distributions of earnings (see section 3.2) suggests values of y from 0.5 to 2.5 giving limiting marginal rates from 67 % to 29 %. Higher rates can also be produced by widening the distribution of skills -if ts in the log normal case is increased to 1.0 (from 0.39), the median rate is 56% for the case G(u) = -e-” and a government requirement of 7% of product. Presumably with a wider distribution of skills, inequality considerations increase relative to those concerned with incentives. However, Mirrlees (1971, p. 207) suggests that such a cr ‘does not seem to be at all realistic. . .’ since it gives a dispersion of skills too wide to be compatible with observed distributions of earnings.r4 There are two main features of the calculated tax schedules which look different from actual income tax structures.’ 5 Marginal rates are not mono￾tonically increasing - most of the population is in the region where they are falling - and the highest marginal rates are low. For the ‘realistic’ case of IJ = 0.39, applying to 5 out of 6 of Mirrlees’ examples, the highest marginal rate is 39%. Atkinson (1972) and (1973a) discusses the effect of increasing the concavity of G(a) and the limiting case of maximin. It seems clear that he was in part influenced by the low rates in the Mirrlees calculations - see Atkinson (1972, p. 2) and (1973a, pp. 390-391). The maximin criterion in the Mirrlees model yields tax rates around 50 % for the median person [see Atkinson (1972, p. 28)]. We have already given the Sadka argument which explains why, for a finite population, we should expect zero marginal tax rates at the top of the distribution. This argument may also have some intuitive force for distributions with an infinite domain, provided the weight in the tail is not too big. We have noted, for example, that the log-normal gives a limiting marginal rate of zero but the Pareto does not. The zero limit of the marginal rate for certain distributions suggests that a declining rate at the upper end may be a feature of many models of optimum income taxation. We shall say no more (except for the special case of section 5) about the shape of the tax funtion, and concentrate on the labour supply function and its relation to optimum linear taxation. I’IWe discuss in section 3.2 whether the distribution of earnings gives a misleading impression of the distribution of skills. lSThese are announced rates rather than effective rates