A New Approach to Consumer Theory

A New Approach to Consumer Theory

Author(s): Kelvin J. Lancaster

Source: The Journal of Political Economy, Vol. 74, No. 2 (Apr., 1966), pp. 132-157 Published by: The University of Chicago Press

Accessed: 22/09/2008 02:58

In the traditional consumer model, the relationship between goods (GOODS) was two choices: same goods or different goods

The relationship between butter and margarine is the same as the relationship between shoes and boats

Are red Chevrolet and gray Chevrolet the same goods or different goods?

So we built a model in which utility is defined for features rather than goods.

Create a new model with the following three elements (p. 134)

1: Utility is a function of characteristic (previously it was a function of goods).

2: A good has one or more characteristics, and a characteristic is shared by several goods.

3: The characteristics of the combination of goods are different from those of the individual goods.

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This would allow, for example, a red Chevrolet and a gray Chevrolet to be described as "goods that share most characteristics and differ only in color characteristics.

In the Simplified model, which will be explained later, there is a one-to-one correspondence between Activity (purchase and other actions) and Good, but this is not the case in the general model, so Good is represented by x and Activity by y.

$ x_j = \sum_k a_{jk} y_k

$ x = Ay

A, which expresses this relationship between Activity and Good, is assumed to be linear and objective.

The relationship between Activity y and Characteristic z is assumed to be linear and objective as well.

$ z_i = \sum_k b_{ik} y_k

$ z = By

Utility U is a function of z. In contrast to the conventional model where it is a function of x.

(Lancaster intended to use k for later discussion as the level of activity for the purposes of later discussion.)

To sum up.

$ \text{Maximize} U(z)

$ \text{subject to} px \leq k

$ \text{with} z=By,\quad x=Ay,\quad x,y,z \geq 0

become

In the Simplified Model, where Activity and Good have a one-to-one correspondence, x and y are considered identical.

$ \text{Maximize} U(z)

$ \text{subject to} px \leq k

$ \text{with} z=Bx,\quad x,z \geq 0

become

p137

a) Since convex set on G-space is also convex on C-space, the budget constraint line is also convex on C-space.

b) Since there is not necessarily an inverse in B, any z on the C-space does not necessarily have x on the corresponding G-space

c) If there is an inverse matrix, this is also a mapping of a convex hull onto a convex hull, so the convexity of U is preserved

the structure of consumption technology

1: If the number of features is the same as the number of goods

Some conditions lead back to the conventional model.

2: If the number of features is greater than the number of goods

Bx=z, viewed as a simultaneous equation, has no solution in general because "there are more equalities than the number of variables".

So think in slices (ignoring some of too many features).

Lancaster emphasizes that convexity is maintained at this time

3: If the number of goods is greater than the number of features

At this time, there exists more than one x satisfying Bx=z

The customer then makes a choice among those multiple x's.

The EFFICIENCY of that choice is MINIMUM COST ("efficient" in the sense of the efficient market hypothesis is used).

The optimization for efficient x* for given z* is

$ \text{Minimize} px

$ \text{subject to} Bx=z^*, x\geq 0

The set of z for which px=k is satisfied when z* is moved is called the characteristic frontier

This indicates the boundary of the set of features that can be maximally obtained within a given budget constraint

This is the end of the explanation of the model, and now we will apply this model to various problems.

The space of goods is finite, which is unavoidable given the computational resources available in 1966.

In dealing with contexts such as new product development, there are a myriad of goods because the natural behavior is to "choose the one that seems to have the highest customer utility out of the myriad of products that could be created.

Or approximate in a sufficiently large dimensional space.

We are dealing with this as a linear problem.

I suppose it is inevitable because it would be analytically unsolvable, but since the quantity of goods is a continuous value, it would be possible to act "buy 1/3 3.5" HDD and 2/5 5" HDD" with a limited budget.

Realistically, it's impossible to buy half a camera, and can only be approximated by cloth, water, grain, etc. sold by weight.

This is due to the fact that we are treating the choice of goods x as a vector in the first place.

If we insert the one-hot constraint "customers buy only one good", then one point on the C-space will correspond to one point on the G-space.

Instead, it will not be able to handle the COMBINATION of goods, but it is better to be clear about what you are good at and what you are not good at.

Instead, the discussion of technology constraint lines on C-space carries more weight.

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