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.