The Principles of Political Economy
by John Stuart Mill

Book 3
Chapter 2

Of Demand and Supply in Their Relation to Value

    The supply of a commodity is an intelligible expression: it means the quantity offered for sale; the quantity that is to be had, at a given time and place, by those who wish to purchase it. But what is meant by the demand? Not the mere desire for the commodity. A beggar may desire a diamond; but his desire, however great, will have no influence on the price. Writers have therefore given a more limited sense to demand, and have defined it, the wish to possess, combined with the power of purchasing.  To distinguish demand in this technical sense, from the demand which is synonymous with desire, they call the former effectual demand. After this explanation, it is usually supposed that there remains no further difficulty, and that the value depends upon the ratio between the effectual demand, as thus defined, and the supply.

    These phrases, however, fail to satisfy any one who requires clear ideas, and a perfectly precise expression of them. Some confusion must always attach to a phrase so inappropriate as that of a ratio between two things not of the same denomination. What ratio can there be between a quantity and a desire, or even a desire combined with a power? A ratio between demand and supply is only intelligible if by demand we mean the quantity demanded, and if the ratio intended is that between the quantity demanded and the quantity supplied. But again, the quantity demanded is not a fixed quantity, even at the same time and place; it varies according to the value; if the thing is cheap, there is usually a demand for more of it than when it is dear. The demand, therefore, partly depends on the value. But it was before laid down that the value depends on the demand. From this contradiction how shall we extricate ourselves? How solve the paradox, of two things, each depending upon the other? . . .

 
Gardner Note: In Principles of Economics classes, we emphasize the difference between quantity demanded and demand (the relationship between quantity demanded and price, represented by the demand curve).  Why?  Because understanding that distinction prevents us from making logical errors such as this one:
An increase in demand => increase in price => decrease in demand,
So an increase in demand => a decrease in demand.  That's wrong!
Instead:
An increase in demand => an increase in equilibrium price, and that equilibrium is achieved, in part, by a decrease in quantity demanded

Book 3: Distribution
Chapter 16

Of Some Peculiar Cases of Value

    It sometimes happens that two different commodities have what may be termed a joint cost of production. They are both products of the same operation, or set of operations, and the outlay is incurred for the sake of both together, not part for one and part for the other. The same outlay would have to be incurred for either of the two, if the other were not wanted or used at all. There are not a few instances of commodities thus associated in their production. For example, coke and coal-gas are both produced from the same material, and by the same operation. In a more partial sense, mutton and wool are an example: beef, hides, and tallow: calves and dairy produce: chickens and eggs. Cost of production can have nothing to do with deciding the value of the associated commodities relatively to each other. It only decides their joint value. The gas and the coke together have to repay the expenses of their production, with the ordinary profit. . . . Cost of production does not determine their prices, but the sum of their prices. A principle is wanting to apportion the expenses of production between the two.
    Since cost of production here fails us, we must revert to a law of value anterior to cost of production, and more fundamental, the law of demand and supply. . . .

. . . Suppose ... that more coke is wanted at the present prices, than can be supplied by the operations required by the existing demand for gas. Coke, being now in deficiency, will rise in price. The whole operation will yield more than the usual rate of profit, and additional capital will be attracted to the manufacture. The unsatisfied demand for coke will be supplied; but this cannot be done without increasing the supply of gas too; and as the existing demand was fully supplied already, an increased quantity can only find a market by lowering the price. The result will be that the two together will yield the return required by their joint cost of production, but that more of this return than before will be furnished by the coke, and less by the gas. Equilibrium will be attained when the demand for each article fits so well with the demand for the other, that the quantity required of each is exactly as much as is generated in producing the quantity required of the other. If there is any surplus or deficiency on either side; if there is a demand for coke, and not a demand for all the gas produced along with it, or vice versa; the values and prices of the two things will so readjust themselves that both shall find a market. . .

Essays on Some Unsettled
Questions of Political Economy
by John Stuart Mill (1844)

It is established, that the advantage which two countries derive from trading with each other, results from the more advantageous employment which thence arises, of the labour and capital—for shortness let us say the labour—of both jointly. The circumstances are such, that if each country confines itself to the production of one commodity, there is a greater total return to the labour of both together; and this increase of produce forms the whole of what the two countries taken together gain by the trade.

    It is the purpose of the present essay to inquire, in what proportion the increase of produce, arising from the saving of labour, is divided between the two countries.

    This question was not entered into by Mr. Ricardo, whose attention was engrossed by far more important questions…

    The problem is, what are the causes which determine the proportion in which the cloth of England and the linen of Germany will exchange for each other?

    This, therefore, is a question concerning exchangeable value. There must be something which determines how much of one commodity another commodity will purchase; and there is no reason to suppose that the law of exchangeable value is more difficult of ascertainment in this case than in other cases.

    The law, however, cannot be precisely the same as in the common cases. When two articles are produced in the immediate vicinity of one another, so that, without expatriating himself, or moving to a distance, a capitalist has the choice of producing one or the other, the quantities of the two articles which will exchange for each other will be, on the average, those which are produced by equal quantities of labour. But this cannot be applied to the case where the two articles are produced in two different countries; because men do not usually leave their country, or even send their capital abroad, for the sake of those small differences of profit which are sufficient to determine their choice of a business, or of an investment, in their own country and neighbourhood.

    The principle, that value is proportional to cost of production, being consequently inapplicable, we must revert to a principle anterior to that of cost of production, and from which this last flows as a consequence,—namely, the principle of demand and supply…

    It may be considered, therefore, as established, that when two countries trade together in two commodities, the exchangeable value of these commodities relatively to each other will adjust itself to the inclinations and circumstances of the consumers on both sides, in such manner that the quantities required by each country, of the article which it imports from its neighbour, shall be exactly sufficient to pay for one another. As the inclinations and circumstances of consumers cannot be reduced to any rule, so neither can the proportions in which the two commodities will be interchanged. We know that the limits within which the variation is confined are the ratio between their costs of production in the one country, and the ratio between their costs of production in the other. Ten yards of cloth cannot exchange for more than 20 yards of linen, nor for less than 15. But they may exchange for any intermediate number. The ratios, therefore, in which the advantage of the trade may be divided between the two nations, are various. The circumstances on which the proportionate share of each country more remotely depends, admit only of a very general indication.

    It is even possible to conceive an extreme case, in which the whole of the advantage resulting from the interchange would be reaped by one party, the other country gaining nothing at all…

    But in general there will not be this extreme inequality in the degree in which the demand in the two countries varies with variations in the price. The advantage will probably be divided equally, oftener than in any one unequal ratio that can be named; though the division will be much oftener, on the whole, unequal than equal.

Gardner: Mill's theory of "reciprocal demand" was later formalized into graphs by Frances Edgeworth and Alfred Marshall in "offer curves" or "reciprocal demand curves" that are explained more fully in international economics courses: