CHAPTER I - Section 2

In Section 2 Hicks summarizes the utility theory writings of Vilfredo Pareto.  After starting off with ideas similar to those developed by Marshall, Pareto turned his attention to analyzing goods that are related -- in the sense of being complementary or of being (at least partial) substitutes.  Pareto used a geometric device used earlier by the Irish-English economist Francis Ysidro Edgeworth -- the indifference curve.  For two commodities X and Y we can construct a map like that shown in the figure below (an imitation of Hicks's Figure 2).

The points in this X-Y plane correspond to quantities of the goods X and Y that are consumed -- these quantities of consumption are understood to provide utility to the consumer.  The curves shown in the figure are indifference curves:  namely, curves along which all points have the same total utility.  We can think of these curves as being similar to contour lines on a map -- they show points that are at the same "elevation" in three dimensions.  Referring to this figure, Hicks goes on to explain:

If P and P' are on the same indifference curve, that means that the total utility derived from having PM and PN [these distances correspond to quantities of the goods X and Y, respectively] is the same as that derived from having P'M' and P'N'.  If P" is on a higher indifference curve than P (the curves will have to be numbered so as to distinguish higher from lower), then P"M" and P"N" will give a higher total utility than PM and PN. 

Hicks explains that, assuming positive marginal utility (each increment increases utility, at least a little), the indifference curves must slope downwards from left to right.  One way to understand this is to realize that increasing X without decreasing Y should lead one to a higher indifference curve.  Put another way, if we increase the amount of X consumed, the only way to keep the same utility is to decrease the amount of Y.

As Hicks explains, the slope of an indifference curve at any point has a definite and important meaning, "It is the amount of Y which is needed by the individual in order to compensate him for the loss of a small unit of X." Assuming quantities are small, the decrease in utility from losing the small unit of X is given by the product (amount of X lost) x (marginal utility of X).  The gain in utility from increasing the amount of Y is given by the product (amount of Y gained) x (marginal utility of Y).  We're assuming these utilities exactly balance each other, so (amount of Y gained) x (marginal utility of Y) = (amount of X lost) x (marginal utility of X).  A small amount of algebra then gives us [apologies -- I need learn a better way to paste equations in]:

(amount of Y gained) / (amount of X lost) = (marginal utility of X) / (marginal utility of Y)

But the left-hand side of this equation is the slope, so the slope of the indifference curve equals the ratio of the marginal utilities.  (There are two things to keep in mind here -- one is that the slope of the indifference curve is changing as we consider different points along the curve;  the other is to notice which of the marginal utilities is in the numerator and the denominator on the right-hand side.)

Hicks concludes this section by examining the question of whether diminishing marginal utility implies that the indifference curves must be convex to the origin.  (One way to think about convexity here is to think of it as implying that, for any two points on the curve, the curve in between these two points must lie below a straight line connecting the two points.)  Hicks argues that, at first sight, this would seem to be true, but that it does not necessarily follow.  A counterexample is given by the case of related goods in which "the increase in X lowers the marginal utility of Y [and vice versa, and]... these cross-effects are considerable." 

Thanks for reading this far.  In the next section (a short one), Hicks notes "a really remarkable thing" about indifference curves.

CHAPTER I UTILITY AND PREFERENCE

Chapter I has nine sections, the first of which I'll discuss in this post.  Section 1 summarizes the main assumptions and conclusions of Alfred Marshall's theory of demand.  The point of that theory is to express mathematically how a consumer chooses to divide his expenditures among several consumption goods, assuming the prices of these goods are already determined.  For mathematical convenience the theory assumes the goods can be purchased in very small units.  A footnote explains:

This convenient assumption of continuity does, of course, falsify the situation a little (or sometimes more than a little) as far as the individual consumer is concerned.  But if our study of the individual consumer is only a step towards the study of a group of consumers on the market, these falsifications can be trusted to disappear when the individual demands are aggregated.

The theory makes several assumptions: that the individual spends his income so as to maximize his satisfaction (or as Marshall, Hicks and other writers have come to call it, "utility");  that utility depends on the quantities of goods purchased;  and that as one increases consumption of some good, each additional unit brings less satisfaction than the previous one -- this is called "the principle of diminishing marginal utility."

With these fairly reasonable (even obvious) assumptions, the conclusion is that "utility will be maximized when the marginal unit of expenditure in each direction brings in the same increment of utility."  In other words, the consumer will choose quantities of the various goods in a way that causes the last (tiny) unit of each good to bring the same additional amount of utility.

This conclusion seems reasonable if one considers a situation where the condition doesn't hold -- that is, in some planned set of purchases, one good has a strictly greater marginal utility than another good.  This clearly can't be the set of purchases that maximizes utility, since the consumer could increase his utility by purchasing (at least a tiny bit) less of the good that has lower marginal utility and using the extra money to buy more of  the good with the higher marginal utility.  With the ability to purchase small units, we have the additional conclusion that the marginal utilities of the various goods are proportional to their prices.  Again, thinking about this conclusion by assuming that the condition doesn't hold, we can see that shifting the planned purchases slightly in the direction of a good with higher marginal utility per unit of price would increase total utility.

In section 2, Hicks will begin to ask some probing questions about the nature of "this 'utility' which the consumer maximizes."

Interesting background on Value and Capital

Nobel Laureate Kenneth Arrow describes Value and Capital as having had a significant influence on him -- and learning about it somewhat by chance:

 I learned about general equilibrium, not from any course, but from the fact that my desk, like those of all the other graduate students, was located in the library.  I was close to where the economics books were, in the stacks, so I would simply go to the stacks, starting flipping the books around, and see if there was anything interesting.  I found a book by the English economist, John R. Hicks, titled, Value and Capital.  Nobody at the Columbia Economics Department knew anything about this book, but as you talk to people educated elsewhere, say in England, many of the economists of my generation were transformed by this book.
     That's why when many years later I was awarded the Nobel Memorial Prize in Economic Science jointly with Hicks, it was an especially great honor to be joined with one whose work I admired so much, who was so influential on me.