CHAPTER I -- Section 7

In this section Hicks argues for rejecting the principle of Diminishing Marginal Utility and for replacing it with the principle of Diminishing Marginal Rate of Substitution.  Geometrically, this amounts to the rule that indifference curves must be convex to the axes.  He explains the meaning of Diminishing Marginal Rate of Substitution as follows:

Suppose we start with a given quantity of goods, and then go on increasing the amount of X and diminishing the amount of Y in such a way that the consumer is left neither better off nor worse off on balance; then the amount of Y which has to be subtracted in order to set off a second unit of X will be less than that which has to be subtracted in order to set off the first unit.  In other words, the more X is substituted for Y, the less will be the marginal rate of substitution of X for Y.

Hicks explains the need for this principle by noting that any point where it does not hold cannot be a stable equilibrium.  He notes that this is true even if the marginal rate of substitution equals the price ratio, and he illustrates it by means of a figure that looks somewhat similar to the one below:

The dashed curve doesn't appear in the book;  I've added it to help illustrate his explanation of the figure:

At the point Q on the diagram, the marginal rate of substitution equals the price-ratio, so that the price-line touches the indifference curve through Q.  But the marginal rate of substitution is increasing (the indifference curve is concave to the axes), so that a movement away from Q in either direction along LM would lead the individual on to a higher indifference curve.

The dashed curve is one such higher indifference curve.  Q obviously cannot be a point of equilibrium, because the consumer can move anywhere along the line LM and stay within his budget, therefore he would gain by moving to a point where the higher indifference curve intersects LM.

Hicks concludes this section by raising the question as to the foundation for assuming that Diminishing Marginal Rate of Substitution is a principle that is true in general.  He will deal with this question more in the next section.

CHAPTER I -- Section 6

In this section Hicks takes the first step forward in his effort to build a theory of consumer's demand from concepts not implying a quantitative measure of utility.  He begins with the concept of marginal utility.  He notes that if we fix the quantities of two commodities, then the slope of the indifference curve at that point is equal to the ratio of the marginal utilities of those goods, and the ratio is "independent of the arbitrariness in question."  He calls this quantity the marginal rate of substitution of X for Y, and defines it as "the quantity of Y that would just compensate the consumer for the loss of a marginal unit of X." For given market prices, an individual in equilibrium will have his marginal rate of substitution between any two goods be equal to the ratio of their prices.  If this were not the case, he could gain by reducing his consumption of one of the goods and spending the savings on the other good.  The condition for equilibrium on the market will therefore be written in terms of marginal rates of substitution rather than marginal utilities.  Hicks ends this section by noting that we can say that a commodity's price equals the marginal rate of substitution of that commodity for money.

CHAPTER I -- Section 5

In this short section Hicks sets out the goal of constructing a theory of consumer's demand, starting from the indifference map alone.  He points out that Pareto's work will be of no further help, as Pareto himself did not rework Marshall's conclusions in the light of his significant discovery about indifference maps.  He then notes that an important article on this topic had been published by the Russian economist and statistician E. E. Slutsky, but that it had remained relatively unknown -- probably due its having been published in Italian, as well as the fact that it was published in 1915 (i.e. during war time).  Value and Capital contains "the first systematic exploration of the territory which Slutsky opened up," according to Hicks.

It is interesting to note the role of Hicks's ability to read Italian in advancing his work.  In addition to being able to read Slutsky's article, Hicks was able to read Pareto's work in the original Italian.  In this interview Hicks relates an interesting story of how he came to know about Pareto's work.  A colleague of his at the London School of Economics, Hugh Dalton, advised him that since he read Italian he should read Pareto.  Dalton had first learned about Pareto while spending several months in a hospital in Italy after being wounded toward the end of World War I (he was serving in a British force sent to help the Italians on the Austrian front).  He wasn't seriously wounded, so he had time to learn to speak and read Italian.  Dalton was a Cambridge-trained economist, so he searched for Italian books on economics and found Pareto's Manual of Political Economy.

CHAPTER I -- Section 4

In this brief section Hicks points out that the process of working out marginal utility theory in terms of indifference curves achieves the remarkable accomplishment of arriving at the same results while leaving behind some of the original information.  Specifically, Marshall's theory assumed we know a consumer's utility surface.  The indifference maps of Pareto's theory contain less information.  They are somewhat analogous to contour lines on a map without any key that tells us how much change in elevation occurs between adjacent lines.  It turns out that this extra information was not necessary to explain market phenomena.  And Hicks argues that, "on the principle of Occam's razor, it is better to do without it."

CHAPTER I -- Section 3

The remarkable thing about indifference curves that Hicks explains in this section is the fact that they allow us to draw detailed conclusions about a consumer's optimal consumption choices without knowing details about the utility derived from that consumption.  Here he uses a simple example:

Suppose that we have a consumer with a given money income, who is spending the whole of that income on the commodities X and Y, no others entering into the picture.  Suppose that the prices of those commodities are given on the market.  Then we can read off the amounts that he will buy directly from his indifference map, without any information about the amounts of utility he derives from the goods.

He illustrates this with the following figure:

The line LM is constructed as follows:  the length OL represents the quantity of good X that the consumer could buy if he spent all his income on it.  Similarly, OM represents the quantity of Y he could afford if he spent everything on Y.  Any point along the line LM corresponds to a pair of quantities of the two commodities that would use up all his income.  The slope of the line LM corresponds to the ratio of the prices of the two goods (since it is the rate at which one commodity could be evenly exchanged for the other).  Hicks notes that the consumer's utility will be maximized at a point where an indifference curve is tangent to the price-line, "For at a point of tangency, the consumer will get on to a lower indifference curve if he moves in either direction."

Hicks concludes this section by relating the tangency between an indifference curve and price line to the principle (mentioned in an earlier post) of proportionality between marginal utilities and prices.  He does this as one simple assertion, which I had intended to discuss with mathematical expressions.  Unfortunately my attempts to paste equations into this blogging tool have been unsuccessful so far, so I'll have to rely simply on words for now.  Suffice it to say that at the point of tangency, the derivative of the indifference curve is equal to the slope of the price line.  The latter is proportional to the ratio of prices, whereas the former is proportional to the ratio of marginal utilities.  We can discuss in comments if anyone's interested.

Thanks again for reading.