A New Approach to Consumer Theory

Lancaster1966 Lancaster 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 Stable URL: http://www.jstor.org/stable/1828835 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. 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.

Representation of the model in the formulas

. 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. $x_j={\sum }_k{a}_{jk}{y}_k$ $x=Ay$ When Activity and Good are in one-to-one correspondence, A is a unit matrix and $x=y$. 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.

• Utility U is not necessarily linear (law of diminishing marginal utility)

The linear budget constraint is expressed as $px\le k$ using the price vector p.

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

To sum up. $\text{Maximize}U(z)$ $\text{subject to}px\le k$ $\text{with}z=By,x=Ay,x,y,z\ge 0$ become

In the Simplified Model, where Activity and Good have a one-to-one correspondence, x and y are considered identical. $\text{Maximize}U(z)$ $\text{subject to}px\le k$ $\text{with}z=Bx,x,z\ge 0$ become

In the previous model, utility was defined in Good-space (G-space, the space of goods), so the relationship with the budget constraint could be expressed by Indifference-curve (indifference curve), but in this model, utility is defined in Characteristic -space(C-space, space of features), so a mapping is necessary.

• 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

In the Simplified model, A disappears, so B is important. This B is called consumption technology (consumer technology). The discussion is divided into three patterns depending on the shape of this B.

• 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
  • to “what does it mean in the real world that there is no exact solution for x corresponding to given z?” I feel that this is not the case. In the real world, if a consumer is not satisfied with a product on the market that satisfies his/her “desired feature z,” he/she will buy a similar product if it is satisfactory, and thus the consumer will be absorbed into the framework of utility maximization. There is no need for an exact solution.
• 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
    • $\text{Minimize}px$
    • $\text{subject to}Bx=z^{\ast },x\ge 0$
  • 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.

Impressions of

• 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.
  • (This is not an outlandish idea; recent natural language processing considers the word space to be of infinite dimension, and includes a prior distribution using the Dirichlet process, etc.)
  • 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.
  • The shattering of G-space makes it difficult to discuss budget constraint lines, but we can add the inverse of price as one of the features, as Adner 2002 does
    • Instead, the discussion of technology constraint lines on C-space carries more weight.

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