168. F. P. Ramsey to Harrod , 6 July [1929] [a]

[Replies to a letter not found]

4 Mortimer R d , Cambridge

6 July [1929] [1]

My dear Roy,

You are quite right; I'm sorry I can't have understood the point before. [2] I think I even had a muddled notion that you came to the opposite conclusion.

The only point which isn't clear is this. Your calculation supposes that firm A says to himself, if I produce another unit the price will fall by so much supposing the other firms do not change their amounts and calculates on that basis. But he might suppose that a reduction in price would diminish the production of the other firms so that he could expand himself a bit more than you allow for a given reduction of price.

The question is as to the rules of behaviour one assumes. 3 seem to be possible

(1) The usual one made in discussing competition, that each competitor takes the market price as fixed and produces what he would like to sell at that price.

This leads to price = marginal cost.

(2) Yours that each competitor takes as fixed not the market price but the production of his rivals, and allows for the fact that a change in his own production will affect the price but not that it will affect his rivals' production.

(3) That all the competitors but one act according to assumption (2) but that one who is peculiarly cunning or well established [b] allows for the reaction of a change in his production in that of his competitors, supposing that his competitors will act according to assumption (2) and modify their productions accordingly.

(3) seems to be the best one can do beyond (2), for we cannot have more than one cunning one or we get a vicious circle, since the rule of each depends on the rule of the other and no determinate rule results.

In practice I imagine a man would suppose that if he expanded his production the lowered price would drive out or reduce the production of some of his competitors, and so would never act according to (2) if he was in a strong competitive position. For instance your rule (2) can give an equilibrium between 2 producers of different efficiency, 1 in circumstances in which it would pay one to sell at a price which his rival could not compete with at all. Clearly in this case he would act according to (3) or else oust his competitor altogether.

One can put it out so:

Let there be n firms producing

at total costs

marginal costs .

Let the total product be mp

the demand price

elasticity of demand = .

Then profit of 1st firm = mp.

For a maximum differentiate with regard to

mp.

Now and according to (2) . i.e. an increment to is reckoned to give an exactly equal increment to total product z.

\

i.e. certainly greater for the smaller firms.

Curiosity corollary: no firm can have a share of total production greater than the elasticity of demand or we get nonsense.

What to some extent but not altogether vitiates the conclusion, is that if a firm acts according to assumption (3) it will take as less than 1 and we have in general

Marginal cost = .

There is a certain presumption that the biggest firm will act according to 3, and its share of production will have to be multiplied by a fractional , and so its ratio of cost price be more like that of a smaller firm.

If we have only two firms one acting according to (3) and one to (2) we can give a formula quite easily and it isn't clear that the bigger one may not have the bigger cost, though it is unlikely if it is more than twice the size of the other.

One can work everything out very easily for two firms if one supposes their supply curves and the demand curves are straight lines.

The easiest case is when p = a - bz and each firm has a constant cost for A

for B

and suppose .

Then if we assume rule (2) we get profit of firm A = .

For maximum .

i.e.

similar

\mp

mp = 

For each firm to make a positive profit this must be greater than (and so mp) \

i.e.

And then profit of firm A =

But now suppose that firm A decided to keep B out altogether by selling at a fraction under i.e. for purpose of calculation at . Then he sells an amount given by \

and makes a profit of .

This will be bigger than what he made before if

i.e. if

i.e. if

so that the equilibrium given by rule (2) would be broken in this way whenever lies between and .

But even suppose it did not pay A to drive B out altogether it might pay to squeeze him by acting according to (3); i.e. A reckons that if he fixes his production a certain amount above the equilibrium level , B will ultimately accept this and reduce his production to what will pay him best at this figure. If A fixes his production at x 1 , B fixes his at x 2 given by rule (2) which gives from (2) i.e. .

Then .

A's profit is .

For a maximum

giving a profit .

Whereas if he acted on rule (2) he only got which is certainly less.

{However for this to work mp must not become negative

\ \ i.e.

The cases are really in conclusion these:

if No 1 has an unquestioned monopoly

if it pays him to keep 2 out altogether

if it pays him to squeeze 2 according to rule (3) more than to keep him out altogether}

One merit of your idea seems to me to be that it explains how equilibrium can be maintained between several firms in an industry with considerable internal economies not yet reached; and that if there were such internal economies a tariff which led to a monopoly being established at home might lead to a fall in price. In fact it is easy to invent such a case.

I'm very sorry I didn't see the point before, I must have been very stupid.

Yours ever

Frank Ramsey

  1. 1. The year is not specified, and is inferred from context (see note 2 to this letter).

    2. Ramsey refers to his July 1928 negative judgement on Harrod's "Notes on Monopoly and Quasi-Competition" (see Ramsey's note attached to letter 154 , [jump to page] . Harrod's paper is reproduced here as essay 6 ; see in particular note 1 for full context).

    Harrod referred to this letter in a note to "future historians of thought" in 1945 (reproduced in note 1 to essay 6 ), and again in his last article on imperfect competition. After recollecting how Ramsey recommended rejection, Harrod cited the first and last paragraphs of this letter and commented: "In an eight-page letter, Ramsey had his own shot at supplementing my views, introducing the question of what Chamberlin later called `oligopoly'. It is possible that this letter from so a distinguished pen is worthy of publication" ("Increasing Returns", in R. E. Kuenne, Monopolistic Competition Theory: Studies in Impact. Essays in Honor of Edward H. Chamberlin, New York: Wiley, 1967, p. 65n).

    1. a. ALS, eight pages on eight leaves, in HP IV-963-973. The original documents often lacks punctuation, which is silently reinserted to facilitate legibility. In this transcription, in correspondence of equations lines are sometimes broken differently than in the original, when the lack of punctuation does not make this operation ambiguous. A note in Harrod's hand, dated 8 May 1945, commenting upon this letter and filed in the same folder, is reproduced in note 1 to essay 6 . Two paragraphs of this letter are cited in Harrod, "Increasing Returns": see note 2 to this letter for details.

      b. Ms: «wellestablished».


1. Ramsey marked this sentence with an undulating line in the margin, and noted: "See below".


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