Marginal Utility of Income: Price and Income Specifications
Kevin J. Furlong∗
1 Introduction
The discrete choice literature has used several specifications to describe the functional
relationship between household income and price, h(yi , pj ). However, each specification
has caveats and so it is ultimately up to the researcher to choose the best specification
in regards to their application. In what follows, I provide a detailed explanation of the
various specifications used in the literature.
For simplicity, let household i’s indirect utility function from consuming product j
be represented by the following:
V (yi , pj , xj ) = αh(yi , pj ) + βf (xj ),
where yi is household i’s income, pj is the the price for product j, and xj is a vector
of product characteristics. The function h(yi , pj ) represents the functional relationship
between income and price, and βf (xj ) represents all other elements of utility.
2 Constant Marginal Utility of Income
A common specification in the discrete choice literature is the assumption that the
marginal utility of income is constant across households. This is achieved by specifying
a linear relationship between price and income,
αh(yi , pj ) = α(yi − pj ).
∗
Kevin Furlong is a graduate student at North Carolina State University. This is part of his dissertation. All correspondence should be directed to [email protected]. This paper was last modified
November 1, 2011.
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Because only differences in utility matter in estimation, household i’s income has no effect
on the probability of product j being selected because it drops out of the expression,
h
i
Pi,j = Prob i,k − i,j < V (yi − pj , xj ) − V (yi − pk , xk )
h
i
= Prob i,k − i,j < α(yi − pj ) − α(yi − pk ) + βf (xj ) − βf (xk )
h
i
= Prob i,k − i,j < α(pk − pj ) + β f (xj ) − f (xk ) .
The marginal utility of income is therefore
M Uincome = α,
which only depends on price and thus fails to capture any income effects.
3 Nonlinear Marginal Utility of Income
To overcome this problem, a common approach is to specify a nonlinear functional
relationship between price and income. Most researchers use a log specification which
exhibits decreasing returns,
αh(yi , pj ) = α log(yi − pj ).
Again, only differences in utility matter in estimation. However, with this specification
household i’s income will not drop out of the probability expression and will, as a result,
affected the probability of product j being selected,
h
i
Pi,j = Prob i,k − i,j < V (yi − pj , xj ) − V (yi − pk , xk )
h
i
= Prob i,k − i,j < α log(yi − pj ) − α log(yi − pk ) + βf (xj ) − βf (xk )
h
i
= Prob i,k − i,j < α log(yi − pj ) − log(yi − pk ) + β f (xj ) − f (xk ) .
The marginal utility of income is,
M Uincome = α
1
,
yi − p j
which increases at a decreasing rate with respect to income, thus making high income
households less sensitive to prices relative to low income households. For some examples,
see Berry et al. (1995) and Petrin (2002).
Although this specification is appealing in theory, empirically it has many problems.
First, when analyzing markets in which products are priced in the tens of thousands
of dollars (e.g. automobiles) it is fairly common for a household to purchase a product
whose price is larger than his/her reported income, pj > yi . This is problematic because
the log of a negative number is undefined. This indicates that either 1) their reported
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income is unrepresentative of their financial constraint or 2) their reported income is
inaccurate. A second problem with this specification is the increased difficulty of estimating welfare effects since the marginal utility of income varies continuously across
households. Researchers are forced to use simulation techniques which significantly increases the computational burden.
4 Constant Marginal Utility of Income per Income Tier
A middle of the road approach is to allow the marginal utility of income to vary across
income groups, but remain constant within each group. This overcomes the criticism of
the first approach and greatly reduces the computational burden of conducting welfare
analyses in the second. For some examples, see Morey et al. (2002) and Goolsbee and
Petrin (2004).
The functional form for the price and income effect component, h(·), is the same as
the constant marginal utility of income specification,
α∗ h(yi , pj ) = α∗ (yi − pj ),
however now the marginal utility of income coefficient is a piecewise function depending
on household i’s income,

α1 if income ≤ y1




α2 if y1 < income ≤ y2
α∗ =
..


.



αm if income > ym−1 .
It is expected that α1 > α2 > . . . > αm such that households in the first income bracket
value a marginal dollar more than households in the second bracket, and so on. It is up
to the researcher to decide how many income groups, m, to specify.
5 Income as a Taste Parameter
Another common approach is to treat income as a proxy variable for tastes. For example,
this line of reasoning can be used to imply that wealthier households prefer luxury goods
not simply because they can afford it, but rather because it fits their preference type
better. With this specification two separate “income” terms must be included in the
model. The first represents “taste income,” Ii , and the second represents a “financial
constraint,” yi . For some examples, see Winston and Mannering (1984), Viton (1984),
Lareau and Rae (1989), Mannering and Winston (1995), and Train and Winston (2007).
The functional form of the price and income effect component, h(·), is the following:
αh(yi , Ii , pj ) = αp (yi − pj ) + αp/I
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(yi − pj )
.
Ii
Again, only differences in utility matter. Therefore, the difference in utility for the price
and income effect component, h(·), is the following:
αh(yi , Ii , pj ) − αi h(yi , Ii , pk )
h
(yi − pj ) i h
(yi − pk ) i
− αp (yi − pk ) + αp/I
= αp (yi − pj ) + αp/I
Ii
Ii
h
i h
(yi − pj )
(yi − pk ) i
= αp (yi − pj ) − αp (yi − pk ) + αp/I
− αp/I
Ii
Ii
h
i h
(pk − pj ) i
= αp (pk − pj ) + αp/I
.
Ii
Like the constant marginal utility of income case above, household i’s “financial constraint,” yi , drops out, however, household i’s “taste income” does not. As a result, the
marginal utility of “financial income” is,
M Uincome = αp + αp/I
1
,
Ii
which represents an average component and one that varies with “taste income.” Viton
(1985) argues that “as long as one is careful not to regard the ‘taste’ variable as measuring income, the received specifications can be found consistent with utility-maximizing
behavior.” Although this specification is commonly used in the discrete choice literature,
it can still be computationally burdensome when conducting welfare analysis since the
marginal utility of income varies continuously across households due to differences in
“taste income.”
“Taste income” is also commonly interacted with other product characteristics in
addition to price. For example, the vehicle demand literature indicates that wealthier
households prefer hybrid vehicles more than poorer households. To capture this effect,
the researcher may want to interact “taste income” with the hybrid dummy variable,
hybrid*income effect = hybrid ∗ log(“taste income”).
However, since this is “taste income” rather than “financial income,” the marginal utility
of income will be unaffected by this expression.
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References
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[2] Goolsbee, A. and A. Petrin. 2004. “The Consumer Gains from Direct Broadcast
Satellites and the Competition with Cable T.V.” Econometrica 72: 351-381.
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[4] Mannering, F. and C. Winston. 1995. “Automobile Air Bags in the 1990s: Market
Failure or Market Efficiency?” Journal of Law and Economics 38: 265-279.
[5] Morey, E.R., V.R. Sharma, and A. Karlstrom. 2003. “A Simple Method of Incorporating Income Effects into Logit and Nested-Logit Models: Theory and Application.” American Journal of Agricultural Economics 85: 248-253.
[6] Petrin, A. 2002. “Quantifying The Benefits of New Products: The Case of The
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[7] Train, K. and C. Winston. 2007. “Vehicle Choice Behavior and The Declining
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[8] Viton, P. A. 1985. “On the Interpretation of Income Variables in Discrete-Choice
Models.” Economic Letters 17: 203-206.
[9] Winston, C. and F. Mannering. 1984. “Consumer Demand for Automobile Safety.”
The American Economic Review 74: 316-319.
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