Last time we talked about measuring preference using utility concept and then maximizing the utility subject to choice- and budget constraints. Let’s continue. (For those who just joined, The Manager has kindly put the series in order of appearances on the sidebar).
The utility that is a function of the things we consume (apple, orange and all that) is called direct utility function (DUF). That is, you infer my utility function by looking at how many apples I decide to eat. Many times, you can just infer one’s utility function by knowing how he usually responds to a given income level and the existing prices of goods considered. This is called indirect utility function (IUF). These two concepts are very closely linked. Your satisfaction from apple depends on how many apples you consume. But how many apples you consume depends on how much money you have and how much one apple costs. So, representing utility using information on prices and income level is identical to representing utility using consumption level. This is the first magic.
Both DUF and IUF are maximizing functions: you maximize your utility given your income. That is, you are answering the question of “what is the maximal level of utility you can get with your given income”. What if the question instead is “what is the minimal level of income needed to reach a given utility level”? Then you do it the other way around: you minimize your expenditure given the utility level you want to achieve. The first approach is called utility maximization problem (UMP), and the second is expenditure minimization problem (EMP). Now here is the second magic: if you want to know the optimal level of one’s consumption, you can go either way: UMP or EMP. We say, one is the dual to the other.
Note again that in UMP we try to find the optimal level of consumption by varying utility to match a given income level. The resulting demand function is called Walrasian demand function.1 In the EMP approach, we try to find the optimal level of consumption by varying income level to match a given utility level. The resulting demand function is called Hicksian demand function.2 This latter function is also called compensated demand function because in fact we imagine the individual as if we keep adjusting (‘compensating’) his income so as to let him be in the same level of satisfaction. (So you know why the former is also called uncompensated demand function).
Now we are ready for the most celebrated Law of Demand: “If the price of a good goes up, the demand for it goes down...” – many textbooks stop here; nevertheless, it should really continue with “... if the price changes are accompanied by income compensation”. This is the property of Hicksian demand. On the other hand, it is possible that when a good’s price falls, the demand for it ... decrases. This can happen with Walrasian demand, an example of which is a Giffen good.
Stay tuned for the welfare analysis of consumer demand.
1 After Leon Walras, 1874. Many textbooks inaccurately call this Marshallian demand function (after Alfred Marshall, 1920).
2 After John R. Hicks, 1939.