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Thursday, September 19, 2013:

Ramsey and Pigou: crypto-Georgists

Frank Ramsey was verballed, writes Gavin R. Putland.

In 1902, the Congregationalist Cambridge mathematician Arthur Stanley Ramsey married Mary Agnes Wilson, the socialist suffragette daughter of the Vicar of Horbling. They had two sons and two daughters. The younger son, Arthur Michael Ramsey, born on 14 November 1904, became the 100th Archbishop of Canterbury. The elder, Frank Plumpton Ramsey, born on 22 February 1903, was an atheist.

Frank Ramsey formally studied mathematics but diversified into philosophy and economics. Among economists he is famous for proving that if the tax system is to raise a given revenue with minimum deadweight, each commodity should be taxed in inverse proportion to its price-elasticity of demand (Ramsey, 1927). That is unfortunate because the cited paper does not contain any such result. It is the more unfortunate because Ramsey himself could not correct the record, having died on 19 January 1930 at the age of 26. But, now that his work is out of copyright (see References), the travesty will be harder to maintain.

Here is Ramsey's original statement (Ramsey, 1927, p.56):

For infinitesimal taxes ... the tax ad valorem on each commodity should be proportional to the sum of the reciprocals of its supply and demand elasticities.

Then, in case the implication is not sufficiently clear, Ramsey spells it out (pp.56–7):

If any one commodity is absolutely inelastic, either for supply or for demand, the whole of the revenue should be collected off it. This is independently obvious, for taxing such a commodity does not diminish utility at all. If there are several such commodities the whole revenue should be collected off them, it does not matter in what proportions.

While there is no commodity for which the demand is absolutely inelastic, there is one obvious commodity for which the supply is absolutely inelastic, namely land — provided of course that the taxable criterion of “supply” is the existence of the land and not its allocation to any particular purpose (opponents of “land tax” deliberately confuse the two).

One of the few scholars trying to set the record straight is Mason Gaffney (2009, pp.375–6). Another is Joseph Stiglitz, who notes that the “Ramsey tax rate is proportional to the sum of the reciprocals of the elasticities of supply and demand” (Stiglitz, 1986, pp.403–4; quoted by Gaffney).

As we shall see, Ramsey not only formulated a rule that leads directly to a “single tax” on land, but also anticipated the so-called Laffer curve in cases where the “single tax” is not employed. Moreover, Ramsey's rule was to be applied after any externalities had been internalized by means of appropriate taxes and bounties.

First, internalize the externalities

The first scholar who got the story right — because he posed the problem that Ramsey solved — was Ramsey's tutor, A.C. Pigou. In Chapter VIII of A Study in Public Finance, under the heading “Taxes and Bounties to correct Maladjustments”, Pigou (1947, p.94) sets the scene:

Of these maladjustments there are two principal causes. The first is that, in respect of certain goods and services, the return at the margin which resources devoted to making them yields [sic] to their makers is not equal to the full return which the community as a whole receives ... In other words, the value of the marginal private net product of resources so employed is greater or less than the value of the marginal social net product.

That is, the production of certain goods and services causes what we would now call negative or positive externalities. Pigou continues:

The second cause is that in respect of certain goods and services, the ratio, so to speak, between people's desire and the satisfaction which results from the fulfilment of desire is greater, or less, than it is in respect of other goods and services.

The “second cause” is more controversial (and, as we shall see, is not mentioned by Ramsey). As an example, Pigou cites excessive discounting of future costs and benefits. Be that as it may, Pigou concludes (p.99):

When maladjustments have come about ... it is always possible, on the assumption that no administrative costs are involved, to correct them by imposing appropriate rates of tax on resources employed in uses that tend to be pushed too far and employing the proceeds to provide bounties, at appropriate rates, on uses of the opposite class. There will necessarily exist a certain determinate scheme of taxes and bounties, which, in given conditions, distributional considerations being ignored, would lead to the optimum result.

This optimum mixture of what we would now call Pigovian taxes and bounties (subsidies) is assumed to be in place in Pigou's next chapter.

Then collect the necessary revenue

Pigou's next chapter deals with Ramsey's contribution. But before we hear it second-hand from Pigou, let's get it straight from Ramsey. Here's his opening paragraph (Ramsey, 1927, p.47):

THE problem I propose to tackle is this: a given revenue is to be raised by proportionate taxes on some or all uses of income, the taxes on different uses being possibly at different rates; how should these rates be adjusted in order that the decrement of utility may be a minimum? I propose to neglect altogether questions of distribution and considerations arising from the differences in the marginal utility of money to different people; and I shall deal only with a purely competitive system with no foreign trade. Further I shall suppose that, in Professor Pigou's terminology, private and social net products are always equal or have been made so by State interference not included in the taxation we are considering. I thus exclude the case discussed in Marshall's Principles in which a bounty on increasing-return commodities is advisable. Nevertheless we shall find that the obvious solution that there should be no differentiation is entirely erroneous.

Note the implication that the taxes to be found are in addition to any Pigovian taxes and bounties that are needed to internalize externalities. Even then, a flat consumption tax isn't optimal.

Now here's the rest of the introduction (Ramsey, 1927, pp.47–8; my emphasis):

The effect of taxation is to transfer income in the first place from individuals to the State and then, in part, back again to rentiers and pensioners. These transfers will slightly alter the demand schedules in a way depending on the incidence of the taxes and the manner of their expenditure. I neglect these alterations;¹ and I also suppose that “a given revenue” means a given money revenue, “money” being so adjusted that its marginal utility is constant.

This problem was suggested to me by Professor Pigou, to whom I am also indebted for help and encouragement in its solution.

In the first part I deal with the perfectly general utility function and establish a result which is valid for a sufficiently small revenue, and takes a peculiarly simple form if we can treat the revenue as an infinitesimal. I prove, in fact, that in raising an infinitesimal revenue by proportionate taxes on given commodities the taxes should be such as to diminish in the same proportion the production of each commodity taxed.

In the second part I assume that the utility function is quadratic, which means roughly that the supply and demand curves are straight lines, but does not exclude the most general possibilities of joint supply and joint demand. With this assumption we can show that the rule given above for an infinitesimal revenue is valid for any revenue which can be raised at all.

In the third part I give certain important special cases of these general theorems; and in part four indicate certain practical applications.


¹ The outline of a more general treatment is given in the Appendix.

Ramsey, 1927 (abridged)

(N.B.: The rest of this page requires JavaScript. If it renders too slowly, you need to upgrade your browser.)

Not wishing to overtax the reader's mathematical ability (or my own), I now offer a severe abridgment of the rest of Ramsey's paper. Omissions are indicated by ellipsis dots (...), not to be confused with continuation dots in mathematical expressions \((\ldots)\). Explanatory notes are added in square brackets [thus]. Otherwise all the words, symbols and equation numbers, and even the italics, are Ramsey's.

The first segment (Ramsey, 1927, pp.48–52) leads to the result stated in the introduction.

Part I

(1) I suppose there to be altogether \(n\) commodities on which incomes are spent and denote the quantities of them which are produced in a unit of time by \(x_1, x_2 \ldots x_n\).  ... The quantities ... can be measured in any convenient different units.

(2) We denote by \(u = F(x_1 \ldots x_n)\) the net utility of producing and consuming (or saving) these quantities of commodities. This is usually regarded as the difference of two functions, one of which represents the utility of consuming, the other the disutility of producing. But so to regard it is to make an unnecessary assumption of independence between consumption and production ... This assumption we do not require to make.

(3) If there is no taxation stable equilibrium will occur for values of the \(x\)'s which make \(u\) a maximum. Let us call these values \(\bar{x}_1, \bar{x}_2 \ldots \bar{x}_n\) or collectively the point \(P\)....

Suppose now taxes are levied on the different commodities at the rates \(\lambda_1, \lambda_2 \ldots \lambda_n\) per unit [not ad valorem] in money whose marginal utility is unity. Then .... the revenue [is]  \(R = \sum \lambda_r x_r\)   [where \(r=1,2 \ldots n\)].

We shall always suppose \(R\) to be positive, but there is no a priori reason why some of the \(\lambda\)'s should not be negative; they will then, of course, represent bounties.

(4) Our first problem is this: given \(R\), how should the \(\lambda\)'s be chosen in order that the values of the \(x\)'s ... shall make \(u\) a maximum.

. . .

(6) Suppose now \(R\) and the \(\lambda\)'s can be regarded as infinitesimals; then .... [it is shown that] \begin{equation}\tag{4} \frac{dx_1}{x_1} = \frac{dx_2}{x_2} = \cdots = \frac{dx_n}{x_n} = -\theta < 0 \end{equation} [where \(\theta\) is a positive constant]; i.e., the production of each commodity should be diminished in the same proportion.

(7) It is interesting to extend these results to the case of a given revenue to be raised by taxing certain commodities only....

Let us denote the quantities of the commodities to be taxed by \(x_1 \ldots x_n\), and those not to be taxed by \(y_1 \ldots y_m\) [not \(y_n\), which is a misprint]. If  ... \(\lambda_r\) is the tax per unit on \(x_r\), and .... if the \(\lambda\)'s are infinitesimal .... [it is shown that] \[\frac{dx_1}{x_1} = \cdots = \frac{dx_n}{x_n}\,,\] i.e., as before the taxes should be such as to reduce in the same proportion the production of each taxed commodity.

Thus equation \((4)\) — the condition that yields the given infinitesimal revenue with the minimum loss of utility — holds if all commodities may be taxed, and holds for the taxed commodities if other commodities are pre-emptively constrained to be untaxed.

In §9 and Part II of the paper (pp.52–5), Ramsey considers a non-infinitesimal revenue:

(9) Further than this it is difficult to go without making some new assumption. The assumption I propose is perhaps unnecessarily restrictive, but it still allows scope for all possible first-order relations between commodities in respect of joint supply or joint demand, and it has the great merit of rendering the problem completely soluble.

I shall assume that the utility is a non-homogeneous quadratic function of the \(x\)'s [“non-homogeneous” meaning that there is at least one linear or constant term in addition to the quadratic terms], or that the \(\lambda\)'s are linear. This assumption simplifies the problem in precisely the same way as we have previously simplified it by supposing the taxes to be infinitesimal. We shall, however, make this new assumption the occasion for exhibiting a method of interpreting our formulae geometrically in a manner which makes their meaning and mutual relations considerably clearer.

It is not, of course, necessary, nor would it be sensible to suppose the utility function quadratic for all values of the variables; we need only suppose it so for a certain range of values round the point \(P\), such that there is no question of imposing taxes large enough to move the production point (values of the \(x\)'s) outside this range. If we were concerned with independent commodities, this assumption would mean that the taxes were small enough for us to treat the supply and demand curves as straight lines.

Part II

(10) Let  \(u = \mbox{constant} + \sum \alpha_r x_r + \sum\sum \beta_{rs} x_r x_s\,\),  \((\beta_{rs} = \beta_{sr})\),  and let us regard the \(x\)'s as rectangular Cartesian co-ordinates of points in \(n\)-dimensional space. [Then] the loci \(u =\) constant [generally a different constant] are hyper-ellipsoids with the point \(P\) for centre.... and the loci \(R =\) constant [different again] are hyper-ellipsoids with the point \(Q\), whose co-ordinates are \(\frac{1}{2}\bar{x}_1, \frac{1}{2}\bar{x}_2 \ldots \frac{1}{2}\bar{x}_n\,\),  for centre.... Moreover, the hyper-ellipsoids \(u =\) constant, \(R =\) constant are all similar and similarly situated....

(11) If we are to raise a revenue \(\rho\) we must depress production to some point on the hyper-ellipsoid \(R = \rho\).

To do this so as to make \(u\) a maximum we must choose a point on this hyper-ellipsoid at which it touches an ellipsoid of the family \(u =\) constant. There will be two such points which will lie on the line \(PQ\): one between \(Q\) and \(P\) making \(u\) a maximum, the other between \(O\) [the origin] and \(Q\) making \(u\) a minimum. For the point of contact of two similar and similarly situated hyper-ellipsoids must lie on the line joining their centres. Since the maximum of \(u\) is given by a point on \(OP\) we have as before that

The taxes should be such as to diminish the production of all commodities in the same proportion.

And this result is now valid not merely for an infinitesimal revenue but for any revenue which it is possible to raise at all.

The maximum revenue will be obtained by diminishing the production of each commodity to one-half of its previous amount, i.e., to the point \(Q\).

(12) If in accordance with this rule we impose taxes reducing production from \(\bar{x}_1, \bar{x}_2 \ldots \bar{x}_n\) to \((1-k)\bar{x}_1, (1-k)\bar{x}_2 \ldots (1-k)\bar{x}_n\) .... [then the revenue is]  \(R = 4k(1-k)\times\) the maximum revenue (got by putting \(k=\frac{1}{2})\).

And it is easily seen that \(R=0\) if \(k\) is \(0\) or \(1\). So, as we crank up the tax rates, revenue reaches a maximum at \(k=\frac{1}{2}\), and falls if we increase the tax rates beyond that point.

Thus Ramsey clearly anticipates the Laffer curve — and Laffer, like me, would be appalled at the suppression of production at the point of maximum revenue. That suppression, however, depends on the underlying assumptions, one of them being that production is indeed suppressed by a non-infinitesimal margin (§12).

Ramsey notes (p.54, §13) that on those assumptions,

... some of the \(\lambda_r\) will be negative, and the most expedient way of raising a revenue will be by placing bounties on some commodities and taxes on others.

He then asks:

(14) We can now consider the more general problem: a given revenue is to be raised by means of fixed taxes ... on \(m\) commodities and by taxes to be chosen at discretion on the remainder. How should they be chosen in order that utility may be a maximum?

After another hyper-geometric argument, he answers (p.55):

And so ... the whole system of taxes must be such as to reduce in the same proportion the production of the commodities taxed at discretion.

Now we come to the part that is most quoted or, more often, misquoted (Ramsey, 1927, pp.55–8):

Part III

(15) I propose now to explain what our results reduce to in certain special cases. First suppose that all the commodities are independent and have their own supply and demand equations, i.e., we have for the \(r\)th commodity the demand price \(p_r\) ... and the supply price \(q_r\) ....

Suppose the tax ad valorem (reckoned on the price got by the producer) on the \(r\)th commodity is \(\mu_r\,\), then \(\lambda_r = \mu_r q_r\) .... Hence [it is shown that] if we denote by \(\rho_r\) and \(\epsilon_r\) the elasticities of demand and supply [positive for downward-sloping demand and upward-sloping supply] ... \begin{equation}\tag{11} \mu_r = \frac{\big(\tfrac{1}{\epsilon_r}+\tfrac{1}{\rho_r}\big)\theta} {1 - \tfrac{\theta}{\rho_r}} \end{equation} (valid provided the revenue is small enough, see §5).

For infinitesimal taxes \(\theta\) is infinitesimal and \begin{equation}\tag{12} \frac{\mu_1}{\tfrac{1}{\epsilon_1}+\tfrac{1}{\rho_1}} = \cdots = \cdots = \frac{\mu_n}{\tfrac{1}{\epsilon_n}+\tfrac{1}{\rho_n}} \end{equation} i.e. the tax ad valorem on each commodity should be proportional to the sum of the reciprocals of its supply and demand elasticities.

(16) It is easy to see

(1) that the same rule \((12)\) applies if the revenue is to be collected off certain commodities only, which have supply and demand schedules independent of each other and all other commodities, even when the other commodities are not independent of one another.

(2) The rule does not justify any bounties; for in stable equilibrium, although \(\frac{1}{\epsilon_r}\) may be negative, \(\frac{1}{\rho_r}+\frac{1}{\epsilon_r}\) must be positive.

(3) If any one commodity is absolutely inelastic, either for supply or for demand, the whole of the revenue should be collected off it. This is independently obvious, for taxing such a commodity does not diminish utility at all. If there are several such commodities the whole revenue should be collected off them, it does not matter in what proportions.

(17) Let us next take the case in which all the commodities have independent demand schedules but are complete substitutes for supply ....

We can imagine this case as that of a country in which all commodities are produced at constant returns by the application of one kind of labour only, the increase in the supply price arising solely from the increasing marginal disutility of labour, and the commodities satisfying independent needs....

Or if \(u_r\) represents the tax ad valorem and \(\rho_r\) the elasticity of demand for the \(r\)th commodity and \(\epsilon\) the elasticity of supply of things in general, we get, by a similar process to that of §15, \begin{equation}\tag{13} \mu_r = \frac{\big(\tfrac{1}{\rho_r}+\tfrac{1}{\epsilon}\big)\theta} {1 - \tfrac{\theta}{\rho_r}}\,. \end{equation} If the taxes are infinitesimal we have \begin{equation}\tag{14} \frac{\mu_r}{\tfrac{1}{\rho_r}+\tfrac{1}{\epsilon}} = \cdots = \cdots = \theta. \end{equation}

In this case we see that if the supply of labour is fixed (absolutely inelastic, \(\epsilon\rightarrow 0)\) the taxes should be at the same ad valorem rate on all commodities.

(19) If some commodities only are to be taxed it is easier to work from the result proved in §8 for an infinitesimal revenue, that the production of the commodities taxed should be diminished in the same ratio.

Suppose, then, \(x_1, \ldots x_m\) are to be taxed, \(x_{m+1} \ldots x_n\) untaxed.

Let \[dx_1=-kx_1~,~ \ldots~,~ dx_m=-kx_m\,.\] [Then it is shown that] .... \[\mu_1 = k\,\left(\frac{1}{\rho_1} + \frac{\textstyle\sum_1^m x_r} {\textstyle\epsilon\sum_1^n x_r + \sum_{m+1}^n \rho_{r\,} x_r} \right)\,,~ \mbox{etc.}\]

As before we see that of two commodities that should be taxed most which has the least elasticity of demand, but that if the supply of labour is absolutely inelastic all the commodities should be taxed equally.

Thus, under doubly unrealistic conditions of supply — perfect fungibility and perfect inelasticity — Ramsey's reasoning leads to a flat consumption tax.

Here it should be noted that the supply of labour cannot be fixed as the supply of land is, and not only because workers can work more or fewer hours. Even if workers are taxed not for working but for merely existing — by a poll tax — they can emigrate or die or reduce their rate of reproduction. Land can do none of these.

In equation \((14)\) and the last equation above, if we assume that the supply of labour is perfectly elastic \((\epsilon\rightarrow\infty)\), the tax rate on each commodity becomes inversely proportional to the elasticity of demand. But Ramsey does not consider this obviously unrealistic assumption; after the last equation he notes a qualitative inverse relationship between the tax rate and the elasticity of demand, but not a precise inverse proportionality.

In the remainder of the paper (pp.58–60), Ramsey considers applications and a generalization:

Part IV

(20) ... The sort of cases in which our theory may be useful are the following:

(21) (a) If a commodity is produced by several different methods or in several different places between which there is no mobility of resources, it is shown that it will be advantageous to discriminate between them and tax most the source of supply which is least elastic....

(b) If several commodities which are independent for demand require precisely the same resources for their production, that should be taxed most for which the elasticity of demand is least (§19). [Again there is no implication of a precise inverse proportionality.]

(c) In taxing commodities which are rivals for demand, like wine, beer and spirits, or complementary like tea and sugar, the rule to be observed is that the taxes should be such as to leave unaltered the proportions in which they are consumed (§14)....

(d) In the case of the motor taxes we must separate off so much of the taxation as is offset by damage to the roads. This part should be so far as possible equal to the damage done. The remainder is a genuine tax and should be distributed according to our theory ....

(22) (e) Another possible application of our theory is to the question of exempting savings from income-tax. We may consider two uses of income only, saving and spending, and supposing them independent we may use the result (13) in §17. We must suppose the taxes imposed only for a very short time ....

On these assumptions, since the amount of saving in the very short time cannot be sufficient to alter appreciably the marginal utility of capital, the elasticity of demand for saving will be infinite .... and we see that income-tax should be partially but not wholly remitted on savings. The case for remission would, however, be strengthened enormously by taking into account the expectation of taxation in the future [the reason is not explained].

(23) It should be emphasized in conclusion that the results about “infinitesimal” taxes can only claim to be approximately true for small taxes ....


We can also say something about the more general problem in which the State wishes to raise a revenue for two purposes; first, as before, a fixed money revenue, \(R_1\), which is transferred to rentiers or otherwise without effect on the demand schedules; and secondly, an additional revenue, \(R_2\), sufficient to purchase fixed quantities, \(a_1, a_2, \ldots a_n\) of each commodity....

Although these equations do not give such simple results as we previously obtained for an infinitesimal revenue or a quadratic utility function, in the cases considered in §15 and §17 they lead us again to the equations (11) and (13).

Nowhere is it said that the tax rate should be inversely proportional to the price-elasticity of demand.

Pigou on Ramsey

A.C. Pigou (1947, pp.105–6, 107–8) describes Ramsey's contribution as follows:

Frank Ramsey .... postulates that money income is so adjusted as to make its marginal utility ... constant. With this very important proviso we start ... from a state of things in which either there is no divergence at the margin between social and private net products, or whatever divergence there may have been is already corrected by appropriate adjustments. This ... for practical purposes implies that there are no monopolies. Ramsey further assumes that the money collected by the government in revenue is either re-transferred to holders of war loan and then allocated among different purchases in the same proportions in which its original owners would have allocated it, or, alternatively, that the government ... allocates it in these proportions. Then, provided that all the functions involved are quadratic — this implies that such independent demand and supply curves as exist are straight lines — it can be proved ... that the optimum system of proportionate taxes yielding a given revenue is one that will cut down the production of all commodities and services in equal proportions.

. . .

On the assumption — a highly unrealistic one, no doubt — that the demand and supply schedules are all completely independent, a very simple formula, built upon the elasticities of these independent demand and supply schedules in respect of the quantities that would be produced and sold in the absence of any taxation (beyond the taxes considered in the last chapter), can be found.

Remember that “the taxes considered in the last chapter” are those needed to internalize externalities. Pigou then gives a formula corresponding to Ramsey's equation \((12)\), indicating that the tax rate on each commodity is proportional to the sum of the reciprocals of the elasticities of supply and demand (except that the “sum” has become a difference because, under Pigou's sign convention, the elasticity of demand is normally negative). Pigou continues (p.108):

That is to say, the rate of tax on any commodity must be larger, the less elastic in respect of pre-tax output is the demand for it, and the less elastic, if positive, or more elastic, if negative, is the supply of it. If the elasticities of all the supplies are infinite ... the rates of tax on them must be inversely proportional to their elasticities of demand.

Thus it is Pigou, not Ramsey, who states the unrealistic special case for which Ramsey is remembered. Pigou immediately adds a more realistic case, which Ramsey also mentions, but which has been curiously forgotten:

If there is any commodity for which either the demand or the supply is absolutely inelastic, the formula implies that the rate of tax imposed on every other commodity must be nil, i.e. that the whole of the revenue wanted must be raised on that commodity.

The one commodity that meets this criterion is land; in the economic context, land is that factor of production whose supply is beyond the influence of economic agents, and there is no commodity whose demand is beyond their influence. Thus Ramsey's results imply that, in the words of Joseph Stiglitz (1986, p.568), “While a direct tax on land is nondistortionary, all the other ways of raising revenue induce distortions.”


Gaffney, M., 2009, “The hidden taxable capacity of land: enough and to spare”, International Journal of Social Economics, vol.36, no.4 (2009), pp. 328–411. Online: is.gd/gaffney2009.

Pigou, A.C., 1947, A Study in Public Finance, 3rd Ed. (London: Macmillan, 1947; reprinted 1960).

Ramsey, F.P., 1927, “A Contribution to the Theory of Taxation”, The Economic Journal, vol.37, no.145 (March 1927), pp. 47–61. Online: is.gd/ramsey1927.

Stiglitz, J.E., 1986, Economics of the Public Sector, 1st Ed. (New York: W.W. Norton, 1986). Quoted by Gaffney (q.v.).

[Last modified 23 November 2013.]

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