Applications I: consumer theory Lecture note 8 Outline 1. Preferences to utility 2. Utility to demand 3. Fully worked example 1
From preferences to utility The preference ordering We start by assuming that is the set of all consumable bundles. We assume completeness: for any two bundles,, exactly one of the following three must be true: ~ Next we assume reflexivity: ~ Finally, we assume transitivity:, These three assumptions allow us to define a utility function: : where 2
Positive monotonic transformations of the utility function Imagine that you ranked all the bundles in terms of preferences. A utility function is an assignment of a number to each bundle that obeys the ranking. There are an infinite way of representing each preference ordering. For example, suppose there are only three bundles,,,, in order of strict preference. Here are some admissible utility functions: 1,, 0 10000, 3, 890 Crucially, ANY positive monotonic transformation of a utility function keeps it representing the same preferences Example:,, ln This is why preferences are referred to as being ordinal. 3
Additional assumptions Standard diagrammatic treatments of indifference curves usually make the assumption of a diminishing MRS: Mathematically, this is known as convex preferences: ~ 1 0,1 4
This assumption implies quasi-concavity of the utility function (see below). It is very important to realise that while concavity of the utility function implies convexity of preferences, the reverse is not true. This is because of PMTs and the ordinality of preferences. Example:, ln ln is clearly concave, but: Usually, we assume non-satiation too Finally, we make additional assumptions that ensure that the utility function is differentiable 5
From utility to demand The Marshallian problem The consumer problem in its fullest form is: Note that 0 and 0. max.. 0 For the purposes of this course, we will make the following assumptions: is differentiable in all its elements is strictly increasing in each dimension (non-satiation) is quasi-concave. For our purposes, simply note that concavity implies quasi-concavity, and that this assumption corresponds to a diminishing MRS (or convex indifference curves) By non-satiation, we know that the budget constraint will be binding. We will also make the following Inada assumption: lim 6
This means that you always want to consume at least an infitesimal amount of each commodity (since prices are finite). Under these assumptions, we can rewrite the problem as: max.. And we know that the FOC will be necessary and sufficient. We can therefore solve the Lagrangean: The FOCs: These yield the familiar result: max,,, 7
The left-hand side is also equal to the MRS using the total differential: Diagrammatically, this is just like the tangency treatment that you did as an undergrad 8
Note that the expression, is known as the Marshallian demand function. We will rewrite it as,. We also have the indirect utility function:,, As economists, we are particularly interested in the following terms:, It turns out that to say more about them, it is useful to consider a related problem. The Hicksian problem Consider the following problem: In words: min.. 0 9
A similar argument to the above allows us to solve this using the Lagrangean method: This yields identical FOCs to before: min Diagrammatically, the reasons are obvious: 10
We refer to the solution, as Hicksian demand, which we rewrite,. Just like the Marshallian system, we have the (indirect) expenditure function:,, This problem doesn t really correspond to anything we would expect a consumer to face. Nevertheless, it turns out that there are a useful set of relationships between, and,. Comparative statics of the Hicksian problem The Hicksian problem has several useful properties. Firstly, by the envelope theorem, we have (Shepherd s lemma):., 11
Also, by the concavity of the expenditure function, we have:. 0 This can also be understood for diagrammatic reasons. 12
Duality and the consumer identities There are several ways to link the two types of demand:,,,,,,,,,,, We can differentiate through the second using the chain rule to get: 13
Using Shepherd s lemma and rearranging yields the Slutsky equation: Since 0, we have the law of demand for normal goods, as well as the standard substitution/income effect breakdown 14
Note how you get Giffen goods, both diagrammatically and mathematically 15
A fully worked example There are two commodities: For now, we treat prices and income as unknown parameters. We begin by solving the Marshallian problem, making use of a PMT 16
This yields the Marshallian demands and indirect utility function:, 3, 2 3, ln 1 3 ln3 2 3 ln 3 2 Next we solve the expenditure minimization problem: 17
We obtain the Hicksian demand and expenditure function: We can confirm Shepherd s lemma and the negative substitution effect As well as all the duality relations 18
Finally we have the Slutsky equation 19