Mark Prell

United States

Department of

Agriculture

Economic

Research

Service

Food Assistance

and Nutrition

Research Report

Number 39-2 

An Economic Model of WIC, 

the Infant Formula Rebate

Program, and the Retail Price

of Infant Formula

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National Agricultural Library Cataloging Record:

Prell, Mark A.

An economic model of WIC, the infant formula rebate program, and the 

retail price of infant formula.

 (Food Assistance and Nutrition Research report ; no. 39-2)

 1. Special Supplemental Nutrition Program for Women, Infants, and Children (U.S.)

 2. Infant formula industry—United States. 3. Infant formulas—Prices—United States.

 I. United States. Dept. of Agriculture. Economic Research Service. II. Title.

HV696.F6

 
United States

Department

of Agriculture

www.ers.usda.gov

Electronic Report from the Economic Research Service

December 2004

Food Assistance

and Nutrition

Research Report 

Number 39-2

An Economic Model of WIC,

the Infant Formula Rebate

Program, and the Retail

Price of Infant Formula

Mark Prell

Abstract

This report develops an economic model that provides the theoretical 

framework for the econometric analyses presented in the report’s companion

volume, WIC and the Retail Price of Infant Formula (FANRR-39). The

model examines supermarket retail prices for infant formula in a local market

area, and identifies the theoretical effects of the Special Supplemental

Nutrition Program for Women, Infants, and Children (WIC) and its infant

formula rebate program. Special attention is given to the rebate program’s

sole-source procurement system by which a single manufacturer becomes a

State’s “contract brand”—the State’s one supplier of formula to WIC

infants—in exchange for paying rebates to WIC. When a manufacturer’s

brand is designated a State’s contract brand, the model predicts that supermarkets increase that brand’s retail price. The model also predicts that an

increase in the ratio of WIC to non-WIC formula-fed infants in a local

market results in an increase in the price of the contract brand and, through

demand substitution, a relatively small price increase for noncontract brands.

Keywords: WIC program, infant formula, cost containment, rebates, food

package costs, Special Supplemental Nutrition Program for Women, Infants,

and Children, child nutrition, food assistance 

Acknowledgments

The author gratefully acknowledges the helpful comments of David Betson

of the University of Notre Dame, and Victor Oliveira, David Smallwood,

Elizabeth Frazão, William Levedahl, and James McDonald of the Economic

Research Service, USDA. Lou King provided editorial assistance.
Contents

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Structure of the WIC Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Basic Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Cost Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Demands of Out-of-Pocket Households . . . . . . . . . . . . . . . . . . . . . . . . 7

Demand of WIC Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Brand Demand for the Supermarket Sector . . . . . . . . . . . . . . . . . . . . . 12

Solution of the WIC Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Results of the WIC Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Effects of Marginal Cost, Brand Preferences, and Population . . . . . . 17

Effects of Relative Size of WIC, Without Rebate Contracts . . . . . . . . 18

Effects of Relative Size of WIC, With Open Market Contract . . . . . . 24

Effects of Relative Size of WIC, With Sole-Source Contract . . . . . . . 26

Contract Systems, Contract Brand Effects, and 

Distribution Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Effect of Market Structure Conditions . . . . . . . . . . . . . . . . . . . . . . . . 31

Summary of Econometric Specification . . . . . . . . . . . . . . . . . . . . . . . 32

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Appendix: WIC Model Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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Summary

This report develops an economic model to examine supermarket retail

prices for infant formula in local market areas, and to identify the theoretical

effects of the Special Supplemental Nutrition Program for Women, Infants,

and Children (WIC) and its infant formula rebate program. The model

adapts a multi-firm Cournot oligopoly model to a new setting that incorporates two differentiated products: heterogeneous consumers segmented by

income, and the roles of WIC and its rebate program. This report provides

the theoretical framework for the specification and interpretation of the

econometric model presented in the report’s companion volume, WIC and

the Retail Price of Infant Formula (Oliveira et al., 2004).

Using this framework, the influence of WIC in a local market area is measured by the relative size of WIC, defined as the ratio of WIC to non-WIC

formula-fed infants. While non-WIC consumers are sensitive to the price of

formula, WIC households are not price sensitive because they do not pay for

formula out of pocket. Holding other factors constant, the model predicts

that as the relative size of WIC increases, retailers increase infant formula

retail prices. The model also predicts that retail prices depend on the type of

contract used by the WIC State agency to procure infant formula. Currently,

WIC State agencies use competitive bidding to award a contract to a single

manufacturer of infant formula for the exclusive right to provide its product

to WIC participants in the State. Under sole-source or exclusive-rights

procurement, all of the WIC demand is channeled to the formula provided

by the contract-winning manufacturer and none of the WIC demand goes to

other national brands. When the retail price of the formula made by the

contract-holding manufacturer increases due to an increase in the relative

size of WIC, some non-WIC households respond by switching to other

infant formula brands. The retail prices of these other infant formula brands

increase as a result, although by a smaller amount than the initial price

increase of the contract-holding manufacturer’s formula. 

The report also examines how retail prices can be affected by different types

of distribution systems; socioeconomic factors; the degree of competition

faced by the supermarket sector (inclusive of all supermarkets) due to the

presence of discount stores; the degree of competition faced by a supermarket chain (which may own one or more stores) due to the presence of

other supermarket chains; and a formula manufacturer’s wholesale cost. The

model focuses on the retail markup, treating manufacturers’ wholesale

prices for infant formula as exogenous. 

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Introduction

This report develops an economic model (the “WIC model”) that examines

supermarket retail prices for infant formula in a local market area, and identifies the theoretical effects of the Special Supplemental Nutrition Program

for Women, Infants, and Children (WIC) and its infant formula rebate

program (hereafter, “rebate program”). The foundation of the WIC model is

a multi-firm Cournot oligopoly model, but the WIC model generalizes the

Cournot model to a new setting. The WIC model features two differentiated

products, heterogeneous consumers that are segmented by income, and the

roles of WIC and its rebate program. The WIC model provides the theoretical framework for the specification and interpretation of the econometric

model presented in this report’s companion volume WIC and the Retail

Price of Infant Formula (Oliveira et al., 2004).

This chapter describes background information on WIC and the rebate

program. Chapter 2 presents the WIC model’s assumptions and mathematical

structure, and the model’s solution is derived in Chapter 3. A central variable

of the WIC model that captures the importance of WIC in a local market area

is the relative size of WIC, measured by the ratio of WIC to non-WIC

formula-fed infants. Chapter 4 uses the model to determine the theoretical

effect on retail prices due to changes in the relative size of WIC and other

WIC-related factors.1 The WIC-related factors that are considered are:

• the presence of the WIC program (without rebates) relative to the

absence of the WIC program;

• an increase in the relative size of WIC (without rebates);

• an increase in the relative size of WIC if WIC has rebates generated by

open market contracts;

• the use of open market contracts relative to the absence of any rebate

contracts;

• an increase in the relative size of WIC if WIC has rebates generated by

sole-source contracts;

• the use of sole-source contracts relative to the use of open market

contracts;

• the use of sole-source contracts relative to the absence of the WIC

program; and

• the use of home delivery or direct distribution relative to the use of the

retail food delivery system for distribution of WIC formula.

WIC safeguards the health of low-income pregnant, breastfeeding, and postpartum women, infants, and children up to age 5 who are at nutritional risk,

by providing a package of supplemental foods, nutrition education, and

health care referrals. Although WIC encourages mothers to breastfeed, a

majority of participating infants receives infant formula through WIC. In the

mid-1980s, infant formula accounted for nearly 40 percent of total WIC

food costs, and infant formula retail prices were rising more quickly than

prices for other foods. These factors led Tennessee and other States to look

into cost containment practices to reduce infant formula costs. Tennessee

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1The WIC model resembles nearly

all other economic models in that it

does not generate predictions of the

magnitudes of price effects, but

instead yields theoretical predictions

of the direction—rise or fall—of price

effects that result from changes in 

economic, demographic, and policy

factors. The empirical analyses in the

companion volume are designed to

estimate the magnitudes of price

effects.
initiated a rebate contract system in 1987. In 1989, P.L. 100-147 required

States to use competitive bidding—or an alternate method that would yield

savings equal to or greater than those produced by competitive bidding—to

procure infant formula. Indian State agencies with 1,000 or fewer WIC

participants are exempt from this requirement. 

Currently, WIC State agencies use competitive bidding to award a contract

to a single manufacturer of infant formula for the exclusive right to provide

its product to WIC participants in the State. The contract-winning manufacturer is then billed for the amount of the rebates on the formula issued to

WIC participants. During the 1994-2000 study period covered by the empirical analyses in Oliveira et al., four manufacturers held one or more WIC

State contracts (for at least part of the study period): Mead-Johnson, Ross,

Carnation, and Wyeth. In fiscal year 2000, infant formula rebates totaled

$1.4 billion, an amount that supported 27 percent of WIC participants

(USDA, 2001).

WIC is an influential agent in the infant formula market: ERS estimates that

infants participating in WIC consume about 54 percent of all formula sold

in the United States. In most States, WIC participants use food vouchers or

food checks to purchase their infant formula, free of charge, at participating

retail grocery stores. WIC then reimburses the retail grocery stores for the

amount of infant formula purchased. Some observers have hypothesized that

WIC and its rebate program may significantly affect the retail infant formula

prices faced by non-WIC consumers, either indirectly, through their impact

on wholesale prices, or directly, through their effect on the retail markup

(the difference between the retail price and the wholesale price). The WIC

model focuses on the retail markup, with the model’s structure and language

developed to analyze supermarket pricing decisions. The manufacturers’

wholesale prices for infant formula are exogeneous in the model. 

The WIC model focuses on supermarket infant formula retail prices as

opposed to prices established by other retailers because the price data

analyzed in Oliveira et al. are obtained from supermarket surveys. The

retail price data cover many major U.S. market areas for the study period

1994-2000 and are used, in part, for price regressions for the separate

national brands of each of the major manufacturers selling formula as of

September 2000. 

In principle, WIC-related factors that could affect retail prices in a local

market area—for any given levels of manufacturers’ wholesale prices—

include:

• the relative size of WIC 

• the presence of a WIC rebate contract, and the type of contract (i.e.,

whether the contract is a sole-source contract or an open market

contract)

• the delivery system used to provide infant formula to WIC households. 

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The WIC model predicts that, holding other factors constant, profit-maximizing supermarkets establish a higher retail price if the relative size of

WIC is larger. Economic reasoning suggests that WIC lowers the price

sensitivity of demand in the infant formula market. WIC provides food

checks or food vouchers (hereafter, “vouchers”) to WIC households, thus

taking a group of (low-income) households that would otherwise be pricesensitive and enabling them to obtain infant formula without bearing the

retail price themselves. WIC makes these households insensitive to the retail

price, and retail prices are higher when a price-insensitive component of the

local market is larger. In the WIC model it turns out that retail prices are

affected not by the absolute size of the WIC program, as measured by the

number of WIC formula-fed infants, but rather the relative sizes of the WIC

and non-WIC market segments. 

The second set of WIC-related factors listed above concerns the presence

and type of WIC contract. As of September 2000, each of the 50 WIC State

agencies used a single manufacturer as a supplier of formula for WIC

households. In contrast, under an open market contract—a form of contract

that some WIC State agencies used in the early 1990s—any and all infant

formula manufacturers could voluntarily participate in a State’s rebate

program. A participating manufacturer would pay a rebate, chosen by the

manufacturer, when its particular formula was provided to a WIC household. Under the open market contract, WIC households themselves chose

which brand of formula to obtain and each manufacturer’s national brand

receives at least some of the infant formula demand of WIC households

(hereafter, “WIC demand”). In contrast, under sole-source procurement, the

formula provided by the contract-winning manufacturer receives all of the

WIC demand and all other national brands receive none of the WIC

demand.2 The WIC model predicts that retail prices depend on the type of

contract. Most notably, under a sole-source contract, the prices of the

contract and noncontract brands can be expected to respond differently to a

change in WIC demand.

The bulk of the analysis of the WIC model is devoted to the retail food

delivery system inasmuch as 48 of the 50 States utilized that system during

the study period in Oliveira et al. Under the home distribution system,

which Vermont uses, infant formula is delivered to the WIC participant’s

home. Under the direct distribution system, which Mississippi uses, WIC

participants pick up infant formula from storage facilities operated by a

State or local agency. The WIC model identifies how retail price depends on

the type of delivery system used by a WIC State agency. 

The list of WIC-related factors that may affect the retail prices (for given

wholesale prices) does not include the amount of the rebates (on either a

per-can basis or in total). Infant formula manufacturers—not retailers—bear

the legal incidence for the rebate payments. If a manufacturer increases its

wholesale price in response to its rebate payments or other WIC-related

factors, then retail prices would be affected in turn (assuming retail markups

are unchanged). This economic mechanism is discussed briefly in this

report, but it is excluded from the formal WIC model so that the model can

isolate retailer behavior. Manufacturers’ wholesale prices are included in the

2 An exception to this general statement is that the WIC State agency can

issue formula provided by a different

manufacturer when medical documentation supports the use of another

infant formula product or a noncontract brand of formula is needed for

religious reasons.

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econometric analysis in Oliveira et al. as independent variables in brandspecific regressions for retail prices. 

In addition to its examination of WIC-related factors and the wholesale

price, the WIC model identifies various socioeconomic factors and marketstructure conditions that can affect retail prices in a local market area. These

other factors include median household income; the poverty rate; the degree

of competition faced by the supermarket sector (inclusive of all supermarkets) due to the presence of discount stores; and the degree of competition

faced by a supermarket chain (which may own one or more stores) due to

the presence of other supermarket chains.

Oliveira et al. found that, within market areas, there is not a clear and

consistent relationship between a formula’s being the WIC contract brand

and its being sold at the highest average retail price. However, comparing

the retail prices of contract and noncontract brands of formula may not identify WIC-related price effects since, as just noted, other factors may affect

retail prices too. The WIC model identifies conditions under which, theoretically, the retail price of the contract brand would be higher than the prices

for noncontract brands. It also identifies alternative conditions under which

that expectation would not be met. 

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Structure of the WIC Model

The WIC model builds on existing economic theory by generalizing the

standard multi-firm single-product Cournot oligopoly model to a new

setting. The WIC model features differentiated brands of formula, heterogeneous consumers that are segmented by income, and the roles of WIC and

its rebate program. A key feature of the WIC model’s specification is the

presence of not one but two formula brands (two “products”), a feature that

is required to identify the simultaneous interactions between the prices of

contract and noncontract brands of formula under a sole-source contract. 

The theoretical device of aggregating products or brands into a “composite”

product is common in economic theory. Although there were as many as

four manufacturers in a given market area at some time during the

companion volume’s 1994-2000 study period, a two-brand model is sufficient to capture the basic theoretical interactions between a contract brand

and a single “noncontract brand” which represents an aggregation of all

other (noncontract) brands. 

Basic Assumptions

In the WIC model, the geographic extent of the market corresponds to a

local market area, i.e., retail prices are determined by factors that exist

locally.3 This treatment ignores retail prices prevailing in other market areas

and assumes, reasonably, that households do not travel to any other market

area to obtain formula. So, for example, households in San Diego would not

buy infant formula in Los Angeles. 

Households obtain formula not only from supermarkets, but also from mass

merchandisers and drugstores. Supermarkets are the major source of infant

formula, with 69 percent of infant formula products (in 2000) provided by

supermarkets, while mass merchandisers accounted for about 28 percent and

drugstores for less than 4 percent. For the empirical analysis in Oliveira et

al., data were available for estimating the number of discount stores (such as

Wal-Mart, K-Mart, and Target) that sell infant formula in a market area. The

WIC model and the empirical analysis both include the number of discount

stores (adjusted for population) as a determinant of demand for infant

formula in the supermarket sector. The opportunity for consumers to substitute between supermarkets and discount stores limits the ability of supermarkets to raise their formula prices. 

The supermarket sector is characterized by a set of chains each of which

owns a set of supermarkets (or “stores”) distinguished by geographic location. The WIC model treats a chain, rather than an individual supermarket,

as the decisionmaking “firm.”4 It is assumed that the chain establishes a

single price for the one or more supermarkets it owns: price differences

across supermarkets within a chain are ruled out. Although not all supermarkets are equally convenient to a representative consumer, it is assumed that a

representative consumer has a choice between various supermarkets

belonging to separate chains in a market area, which limits the ability of

any one chain to raise the price of formula in the supermarkets it owns. 

3 In the empirical implementation

of the WIC model, the geographic

extent of the market corresponds to a

U.S. “market area” as defined by

Information Resources, Inc. (IRI). 

4 Any measure of supermarket

concentration is based on the market

shares of “firms.” To adopt any

measure, a modeling decision must be

made for whether an individual supermarket or a chain constitute the “firm.”

The empirical analysis in Oliveira et al.

uses the shares of chains to measure

concentration, prompted by the notion

that chains “compete” with other

chains but stores within chains do not

economically “compete” with one

another due to their common ownership. (The concentration measure in

Oliveira et al. does include the share of

an individual supermarket if that store

is a so-called “independent.”)

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As in the Cournot model, market price in the WIC model is the same for all

firms (chains) and is determined as a result of a Nash game in which any

given chain treats the output (quantity of formula) of the other chains as

exogenous when choosing its own output to maximize its profits. All chains

choose their outputs simultaneously.

Previous models of oligopoly suppose either that M firms each produce a

single homogeneous product or that M firms each produce a differentiated

product of which there are M varieties, one for each firm. In either type of

model, each firm has but a single choice variable (either quantity or price)

for its single product. A distinctive feature of the WIC model is that it

contains not one but two brands of infant formula that are offered by each of

the M different supermarket chains.5 A chain must therefore choose two

levels of output—one for each brand—for its supermarkets. Thus, interdependency exists not only between firms, as in the Cournot model, but also

between brands: a chain takes into account the effect on its profits from

brand 1 when it is considering brand 2, and vice versa. The incentive a

chain has to raise or lower its price for any one brand is affected by the

extent to which its customers would substitute to the other brand on the

chain’s own shelf. The interdependency between the two brands’ prices

gives rise to a corresponding interdependency between the WIC model’s

two inter-firm Nash games. 

Cost Factors

The marginal cost to a chain for a unit (can) of formula includes its wholesale cost plus the retailing costs of inventory, shelf space, stocking, and

checkout. The wholesale cost is thought to be far larger than retailing costs.

The WIC model allows the two brands to have different marginal costs, c1

and c2 , but assumes that marginal costs do not vary across chains within

local markets. A chain’s total cost function for a brand of infant formula (as

a “stand-alone” product, separate from the chain’s many other products) is

given by:

The wholesale price schedules of infant formula manufacturers incorporate

bulk discounts. Buyers who purchase formula by the truckload (i.e., 40,000-

44,000 pounds) obtain the manufacturer’s lowest wholesale price. For

example, in 2000 (the last year of the study period in Oliveira et al.) MeadJohnson charged a truckload price of $2.94 per unit (13-ounce can) for its

milk-based liquid concentrate formula. A chain that bought half that paid a

few pennies more, totaling $2.98 per unit. Thus, in general, a chain’s

marginal cost does in fact depend on the amount purchased, in contrast to

the specification in (1).

Although a chain that bought a single 12-can case of that same formula paid

substantially more—$3.41 per unit—a key feature of the price data examined in Oliveira et al. prompts the WIC model’s assumption that marginal

cost is the same across chains. The supermarket retail price data are not

chain-specific but instead are averages across a market area’s supermarkets

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5 An appendix to the paper defines

all symbols used in the paper, in order

of introduction.

(1) C 1,2 l, , k,i k,i k k,i (q cq k i M )= == , K
based on data from a survey. To be included in the survey a chain had to

have at least $2 million in sales annually (in total for all items in all stores).

Thus, prices charged by local corner grocery stores and others with sales

below that threshold are not included. It was thought the surveyed chains

typically pay the truckload price for formula. For large-scale chains that

purchase by the truckload or more, wholesale prices can indeed be treated

as constant on the margin and identical across such chains. 

Even though the supermarket price data could include prices for mediumsized chains that pay more than the truckload price, the truckload price can

be a proxy for their wholesale costs. After all, wholesale cost differences

across chains on the order of a few pennies are small relative to the retail

price differences across market areas that the regressions seek to explain.

Furthermore, wholesale prices charged by a manufacturer tend to move

together: changes in the wholesale price paid by the medium-size firms

parallel the changes in the truckload price that was included in the regression. For these reasons, the regression specification in Oliveira et al.

included a single wholesale price for a brand—the truckload price—thereby

setting aside bulk discounts. Thus, the WIC model’s assumption that a

chain’s marginal cost is constant and equal across chains—at least for the

chains in the price survey—was thought to be reasonable for the WIC

model’s purposes of supporting and coinciding with the regression analysis.

A final issue is whether marginal cost would increase if all chains in a local

market area increase their purchases simultaneously due to an increase in

market demand. Infant formula manufacturers establish national price

schedules, so that the wholesale prices faced by any one local market area

are constant not only at the chain level but for the local market area as well.

Demands of Out-of-Pocket Households

In the market area, there are N households with one infant who is fed infant

formula. Each household is a member of one of three distinct formula-buying

groups or market segments: high-income households (H), low-income nonWIC households (L), or low-income households that receive vouchers in the

WIC program (W), where H + L + W = N. It is useful for the income

“cutoff” that divides low-income from high-income households to be set

above the income threshold for WIC income eligibility (185 percent of

poverty) instead of equal to or below that threshold. This assumption means

that, by definition of the term low-income, there is some positive number of

low-income non-WIC households (L > 0) even under full funding for WIC

and full participation by eligible households; otherwise an entire category of

households, L, awkwardly “appears” or “disappears” based on whether or not

WIC has full funding and full participation by eligible households.6

The WIC model contains two brands of infant formula. The pair of brands

will be interpreted two different ways when analyzing sole-source contracts:

• the two brands each represent two different manufacturers’ national

brands in a given market area; 

6 In this framework, the term “lowincome households” refers to the total

L + W, which includes the low-income

households that participate in WIC

(W) and the low-income non-WIC

households (L) who purchase formula

out of pocket.

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• the two brands each represent a single national brand in a given market

area, under the alternative conditions of being the contract brand or

being a noncontract brand.

The first interpretation is the most straightforward one. It will be used to

compare the retail price of a national brand (e.g., Carnation) that is the

contract brand in the market area with the retail price of a different national

brand (e.g., Mead-Johnson) that is a noncontract brand in the same market

area; alternatively, the model’s noncontract brand could represent an aggregation of all noncontract brands. The second interpretation applies to a

particular national brand (e.g., Carnation) as it transitions in a given market

area between being a noncontract brand and being a contract brand. This

second interpretation is adopted to isolate the effects of a change in contract

brand status, holding constant which national brand (and its brand-specific

factors) is examined. Under either interpretation, brand 1 will be treated as

the contract brand and brand 2 as the noncontract brand. 

Per-household demand curves for brands 1 and 2 for supermarket infant

formula purchased by the representative out-of-pocket low- and highincome households are:

where bj , j = L, H, is an own-price slope term the value of which depends

on the income level of the household, s is a cross-price slope term, u is a

parameter capturing what the model calls the tag-along effect, and a1 and a2

are brand-specific constants. Each of the parameters is considered in turn.

The own-price slope terms bL and bH in (2) reflect how readily the two

groups of out-of-pocket consumers, L and H, substitute away from (towards)

supermarket formula in response to an increase (decrease) in the supermarket price. Substitutes for supermarket formula include home-prepared

formula, the introduction of cow’s milk and/or solid foods into the infant’s

diet at an earlier age, and formula obtained from a discount store. As substitutes become closer, out-of-pocket consumers become more price-sensitive

and the two slope terms become larger (in absolute value). For example, the

greater the presence of discount stores in a local market, the greater the

price sensitivity of supermarket customers to the supermarket price, and the

larger (in absolute value) will be the own-price slope terms slope terms bL

and bH. The relative importance of the various substitutes is not identified;

however, intuition suggests that most households would use discount store

formula rather than, say, switch to home production of formula in response

to a supermarket price increase. 7 The combined strength of all substitution

possibilities is reflected in the magnitudes of bL and bH. 

8

An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

Economic Research Service/USDA

7 Another imaginable substitution

behavior in response to an increase in

the supermarket price of infant

formula is for a household to switch

from formula-feeding to breastfeeding.

However, the empirical analysis in

Oliveira et al. treats the household’s

decision to breastfeed or to formulafeed as exogenous, depending on such

factors as level of education of the

mother, supportiveness of breastfeeding by the mother’s mother and

other family and friends, social acceptance of breastfeeding activities, and

availability and ease of obtaining

breastfeeding counseling. 

( )

( )

(2a) 

(2b) 

1, 1 1 1 2

2, 1 2 2 1

q P P a u b P sP j L H

q P P a u b P sP j L H

j j

j j

, ,,

, ,,

2

2

= +− + =

= −− + =
A fundamental aspect of the model’s structure is that out-of-pocket low- and

high-income households differ in their own-price sensitivities, specifically,

that bL > bH.

8 Thus, if price rises by a given dollar amount, both types of

households purchase less supermarket formula—but a low-income non-WIC

household is relatively more responsive to the price increase, switching to

substitutes (perhaps especially discount store formula) more readily than

high-income households.9

The cross-price term s shows how readily a household’s reduction in quantity demanded for a brand (in the supermarket sector), due to an increase in

that brand’s (supermarket) price, is retained within the supermarket sector

by an increase in quantity demanded for the substitute brand off the supermarket shelf. For example, if P1 increases then q1,j falls by bj in (2a) and

q2,j rises by s in (2b). It is assumed that only part of the decrease in q1,j is

shifted to q2,j, i.e., that a (representative) household makes some use of

substitutes other than the alternative brand on the supermarket shelf

(perhaps especially formula obtained at a discount store). Mathematically,

this assumption means that bj > s for both out-of-pocket groups. Because

allowing group-specific differences in s does not yield additional insight, s

is assigned the same value for both the high-income and low-income nonWIC households (unlike the own-price terms bj

, which do differ by group).

The assumption that the own-price effect on a brand’s quantity demanded

is larger than the cross-price effect is economically reasonable. This

assumption also allows the model’s mathematical solution to yield positive

prices in equilibrium. 

The introduction of the parameter s into the WIC model makes the model

interesting and powerful. If instead s is omitted from the model by setting 

s = 0, then households exhibit no inter-brand substitution at all and respond

to a brand’s price increase in the supermarket sector by buying the same

brand in the discount store sector (or by making other substitutions). If s = 0,

demands for the two brands are independent and the two-product WIC model

separates into two one-product models, in which case supermarkets would

establish each brand’s price without any particular reference to the other

brand’s price (in the same way supermarkets establish prices for, say, apples

and hairbrushes). Although a strength of the model is that it allows for s to

be positive, it is worth noting that s is not required to be positive either for

solving the theoretical model or for estimating its empirical counterpart,

which estimates statistically from price data whether or not s is positive. 

The non-negative term u is common to the demand functions for both

brands, although it differs in sign: for brand 1 (the contract brand) a positive u adds to quantity demanded in (2a), while for brand 2 (the noncontract

brand) a positive u diminishes quantity demanded in (2b) by precisely the

same amount. The parameter u represents the combined influence of two

conceptually distinct effects that the model calls a medical promotion effect

and a shelf-space effect. Either of these effects augments demand for the

contract brand at the expense of the noncontract brand under sole-source

(but not open market) contracts. 

8 The price elasticity of demand ε,

(the percentage change in quantity

demanded that results from a 1-percent

change in price, in absolute value) is

the most common measure of price

sensitivity. In (2), out-of-pocket lowincome consumers are more price

sensitive than high-income consumers

at any given price regardless of

whether slopes or elasticities are used

to measure price sensitivity. For the

WIC model, with its Cournot-like

linear demand curves, group-specific

slopes are easier to discuss and

compare than are elasticities.

9 The own-price demand terms bL

and bH differ only between income

groups and not across brands. It was

thought that the econometric specification in Oliveira et al. would be little

changed if the WIC model were to

specify own-price terms that differ by

both income group and by brand.

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An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

Economic Research Service/USDA
When a State has a single contract brand, doctors or hospitals may tend to

promote that brand either through recommendations or the provision of

formula samples. Such promotions may lead to a brand-inducement

behavior by which the (representative) non-WIC household favors the

contract brand when making its out-of-pocket formula purchase. The model

does not require that all out-of-pocket households must behave this way, but

if some proportion of them do then u will be positive for the representative

household.10

A second, distinct effect occurs if (at least some) non-WIC households favor

the brand that has a greater presence on the supermarket shelf. Given that a

sole-source contract is in effect, and that WIC formula is estimated to

account for over half of infant formula sales, the contract brand is likely to

have more shelf space than the noncontract brand. This greater shelf space

may contribute, in itself, to greater sales to non-WIC households.11

The medical detailing and shelf-space effects are combined in the model to

form a single effect, u, which the model calls the tag-along effect. This

effect captures the extent to which the designation of brand 1 as the WIC

contract brand results in sales to non-WIC households that accompanies or

“tags along” with the sales to WIC households. The two components of the

tag-along effect were identified as theoretical possibilities for infant formula

demand by the U.S. Government Accounting Office (1998), which referred

to them as “spillover” effects. The tag-along effect is modeled as a one-forone tradeoff or substitution between the contract and noncontract brands.

The tag-along effect is zero (u = 0) if:

• consumers do not respond to additional medical promotion or shelf

space that the contract brand receives; or

• the contract brand does not receive additional medical promotion or

shelf space; or 

• there is no single contract brand, i.e., there is an open market contract or

if there is no rebate program in effect.

An interpretation of the brand-specific constants a1 and a2 in (2) is based on

considering consumption outcomes given that infant formula is free to

households (making P1 = P2 = 0) and given any of the conditions above that

make u = 0. Let z represent any household’s saturation level of formula, an

amount of formula beyond which a household would not consume (per unit

of time) even if formula were available to the household in unlimited quantities for free.12 If there were but one brand of formula, say brand 1, then

that brand’s constant in (2a) would itself equal z. However, given that there

are two brands, then on average across households the consumption level z

would be divided based somehow on households’ non-price brand preferences, for which a1 and a2 are measures. Non-price brand preferences

reflect some combination of “innate” tastes, recommendations by doctors or

hospitals, and other non-price factors. If brand preferences are “symmetric,”

then for the representative household, a1 = a2 = a = z/2, in which case each

brand would be consumed by half of the households. More generally, the

share of total formula consumption in the market that brand k would receive

would be ak/(a1 + a2).

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10 Whether a manufacturer is the

contract or a noncontract brand in a

given area, the manufacturer has an

incentive to promote its brand in the

medical community so that doctors

and hospitals may recommend the

brand to patients. Some purchases by

some patients may be attributable to

this promotion activity, and such

behavior could be incorporated in the

WIC model through the demandspecific constants a1 and a2. A relatively successful medical promotion by

the manufacturer of brand 1 would,

other factors constant, be captured by

a1 > a2. The “medical promotion

effect” that is incorporated into u

specifically represents one-for-one

demand gains and losses between

brands that are attributable strictly to

whether or not a brand is designated

the contract brand in the area.

11 Empirical work on wine sales

(Folwell and Moberg, 1993) found that

“facings” (a measure of shelf space)

had an effect, separate from the effect

of price, on the quantities purchased of

wine. A study on juices (Brown and

Lee, 1996) found that the number of

facings affected the prices of juices. 

12 In fact, saturation levels would

differ across households due to physiological differences of infants. The term

z can be thought of as the average

consumption of formula for the representative household if formula were

free and unlimited.
Demand of WIC Households 

In the absence of the WIC program, WIC households would be paying out

of pocket, in which case it will be assumed that their demand would

resemble the demand of out-of-pocket low-income households given in (2a)

and (2b). Under the WIC program, WIC households receive vouchers for a

fixed amount of formula (the WIC allocation). It will be assumed the WIC

allocation equals the saturation level z.

13 The per-household demand curve

for brands 1 and 2 of supermarket formula for the W households in WIC is

given by:

where v represents the fraction of vouchers that a representative WIC household redeems in the supermarket sector (as opposed to other retail outlets), θk

represents the share of supermarket formula demand by a representative WIC

household that is provided by brand k; and δ is a zero-one dummy variable

signifying whether or not the WIC State agency uses the retail food delivery

system to distribute WIC formula. Each parameter is considered in turn. 

The term v reflects cross-sectoral substitution behavior of WIC households.

It is likely that the opportunity to substitute between supermarket formula

and discount store formula is important mainly for non-WIC households:

WIC households have no particular reason to seek out infant formula from

discount stores (which are assumed to be more distant and less convenient,

in general) because WIC households do not pay out of pocket for formula.

If WIC households simply obtain all of their formula in supermarkets (at the

same time they do grocery shopping), then v = 1. More generally, other

values for v allow changes in the presence of discount stores in a market

area to have some impact on WIC households.

Under sole-source WIC contracts, the term θk is a dummy variable: the

contract brand (brand 1) receives all of the formula demand of WIC households, making θ1 = 1, while θ2 = 0 for the noncontract brand (brand 2). In

contrast, a WIC household uses vouchers for the brand of its choice under

either open market contracts or under the counterfactual scenario in which

WIC has no rebate program at all. The WIC model assumes for these two

cases that θk in (2c) equals ak/(a1 + a2), the same values already identified

for out-of-pocket households when formula is free based on non-price brand

preferences. Regardless of the presence and type of contract, the brandshare terms θ1 and θ2 sum to 1.

The dummy variable δ equals 1 if the WIC State agency uses the food

delivery distribution system to distribute WIC formula, and 0 otherwise. As

of September 2000, δ = 1 for all but two States.

A key feature of the specification in (2c) is that supermarket prices do not

appear in the expression. The formula demand of WIC households is

completely insensitive to price (perfectly inelastic) because the WIC

program—not the WIC household—pays for the formula. 

13 If, instead, the WIC allocation

were sufficiently close to zero, WIC

households would likely be willing to

pay the retail price (at least at sufficiently low retail prices) to supplement

their relatively “small” allocation with

out-of-pocket purchases. Because supplementation is not considered to be a

widespread phenomenon, supplementation is excluded here by the assumption the WIC allocation and the saturation level are equivalent. 

11

An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

Economic Research Service/USDA

(2c) 1,2 q vz k k,W k = = δθ ,
Even though it may be reasonable to model the demand of any one WIC

household using (2c), overall WIC demand for all W households as a group

may not be price insensitive. Suppose retail price increases are not offset by

Congressional appropriations for WIC, and—despite WIC’s priority

system—some number of eligible infants are not able to participate in WIC

as a result. W then depends negatively on retail price and overall WIC

demand is price-sensitive. Nevertheless, even if overall WIC demand is not

perfectly inelastic, it is presumed that the degree of price sensitivity is sufficiently low and/or the retail price at which these effects would be felt is

sufficiently high that this possibility can be neglected. At least to a first

approximation, it will be assumed that W is fixed and that overall WIC

demand is perfectly inelastic (at least within the relevant retail price range).

This assumption yields WIC-related price effects that are both mathematically tractable and relatively easy to interpret. Despite the inelastic specification for WIC demand, any possibility of an infinite price/infinite profits

outcome is simply ruled out a priori.

Brand Demand for the Supermarket Sector

Summing (2a), (2b), and (2c) across market segments for each brand yields

two total demand equations given by:

where the market-level terms are aggregates of per-household terms given

by A1 = (H + L)a1, A2 = (H + L)a2, U = (H + L)u, S = (H + L)s, and B =

HbH + LbL. The term QW equals Wδvz, the market-level WIC demand for

all formula; the division of QW between brands depends on the brand-share

terms θ1 and θ2.

As in any model that involves aggregation of market segments, the location

of equilibrium in relation to the “kink” points of total market demand needs

to be addressed. It will be assumed that, in Nash equilibrium, supermarket

chains serve both out-of-pocket segments (rather than just the high-income

households alone). This assumption in effect stipulates that the equilibrium

price of each brand is sufficiently low that each brand’s equilibrium output

exceeds the output level of the kink point between the demands of the H and

L households. The possibility of serving only the WIC segment (an infinite

price/infinite profits outcome) was ruled out above. All three segments—L,

H, and W—are in fact served in the actual formula market, and that observed

outcome follows from the assumption that price is sufficiently low to serve

the L households. 

The assumption that bj > s for both L and H households was already

adopted for both economic plausibility and meaningfulness of the model’s

solution. The assumption implies that B > S in (3). 

As noted previously, one factor that affects supermarket price is the ease of

switching to formula sold in the discount store sector. Thus, bL and bH in

(2) and B in (3) may depend positively on the number of discount stores,

holding other factors constant—the price sensitivity of demand for super12

An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

Economic Research Service/USDA

( ) ( ) ( )

( ) ( ) ( )

(3a) 

(3b) 

1 1 1, 1 1

2 1 2, 1 1

Q P P Hq P P Lq P P Wq A Q U BP SP

Q P P Hq P P Lq P P Wq A Q U SP BP

HL W

HL W

, ,,

, ,,

2 2 1 2 1 11 1 2 , ,W

2 2 2, 2 2,W 2 2 1 2

= + + = + +− +

= + + = + −+ −

θ

θ
market formula increases due to an increase in the number of discount

stores. However, a scaling factor is needed because market areas differ in

“size.” Oliveira et al. use the size of a market area’s total population as a

scaling factor. Thus, no effect on supermarket formula demand—and, in

turn, no effect on retailer’s profit-maximizing price—would be predicted if

one market area has double the number of discount stores of another but

also serves twice the population, making a “typical” discount store equally

convenient or accessible to the representative household in the two market

areas. Let D represent the ratio of the number of discount stores to total

population. An increase in D increases the price sensitivity of either out-ofpocket segment and increases B in (3).

Another parameter in (3) that may be affected by D is v, the fraction of

vouchers that are redeemed in supermarkets (under the assumption that WIC

households consider the availability and convenience of discount stores

when making their plans for shopping trips). An increase in D is likely to

result in WIC households making larger overall expenditures in the discount

store sector, increasing voucher redemptions at discount stores and

decreasing v.

Finally, the size of the tag-along effect U is unlikely to be a simple constant

that is independent of any other factors. In particular, the amount of additional shelf space or medical promotion gained by a brand due strictly to its

status as the contract brand may depend largely on how much formula is

provided to WIC households. For simplicity, it will be assumed that under

the sole-source contract the tag-along effect is given by:

so that U is simply proportional to the amount of WIC formula.14

Solving (3a) and (3b) simultaneously, with (4), results in market demand

curves (inverse demand functions) once prices are expressed as functions of

quantities demanded:

where the derived parameters α1, α2, β, and γ are introduced to simplify

further development of the model. The assumption that bj > s means that 

B > S, which in turn implies that β and γ are each positive. 

If consumers exhibit no substitution behavior at all between brands, so that 

s = 0 at the household level, then S = 0 in (3), γ = 0 in (5), and each brand’s

demand curve becomes a function of the brand’s quantity only and is independent of the quantity of the other brand.

14 It should be noted that the tagalong effect may be influenced by the

distribution system. After all, there can

be no shelf-space effect for the

contract brand if WIC formula is not

distributed through the retail food

delivery system. On the other hand,

the medical promotion effect may

operate independently of the distribution channel. Rather than separating

out these influences, the model simply

assumes that the tag-along effect U

requires that distribution be conducted

through the retail food delivery system

(implicitly giving full weight to the

shelf-space effect, and none to the

medical promotion effect). 

13

An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

Economic Research Service/USDA

(4) U hQ h = ≤< w,0 1

[ ] ( )( )

[ ] ( )( )

(5a) P

(5b) P

1

2

= + + ++ −

− − − − −

=− −

= + + −+ +

− − − − −

=− −

BA SA B h S h Q

B S

B

B S Q S

B S Q

Q Q

BA SA B h S h Q

B S

B

B S Q S

B S Q

Q Q

W

W

12 1 2

22 22 1 2 2 2

112

21 2 1

22 22 2 2 2 1

2 21

θ θ

αβ γ

θ θ

αβ γ
Solution of the WIC Model

The chain’s profit function for the two brands of infant formula (separate

from the other products offered by the chain) is:

For each of the M chains, the first-order conditions are15:

In contrast to the Cournot model, for which there is but one best response

function (BRF) for each firm, the WIC model’s two first-order conditions in

(7) each constitute a BRF. Each of the two BRFs exhibit two types of interdependencies. As usual in a Cournot model, (7) exhibits cross-firm interdependency for each given brand, i.e., the profit-maximizing level of a brand’s

output for a chain depends on the given levels of the brand’s output chosen

by the other chains. In addition, given cross-brand substitution behavior so

that γ > 0, each of the chain’s two BRFs exhibit cross-brand interdependency, i.e., the profit-maximizing level of output for a chain’s brand 1

depends on the firm’s own chosen level of output (and its rival’s chosen

levels of output) for brand 2, and vice versa. 

To solve in reduced form for the two optimal quantities of chain i, q1i*and

q2i*, the BRFs must be solved for both cross-firm and cross-brand interactions. As in a Cournot model in which all firms have identical constant

marginal costs, each chain’s quantity (for a given brand) is identical. Using

the result that the output of the (M – 1) rivals for chain i is simply (M –

1)qk,i , k = 1, 2, (7) becomes, in partially reduced form:

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Economic Research Service/USDA

15 The second-order conditions π11i

< 0, π22

i < 0, and π11i

π22

i - (π12i

)2 > 0

are met at the point at which the firstorder conditions are satisfied, given

typical assumptions about the magnitude of marginal cost relative to other

parameters (to exclude a corner solution of zero output). The solution to

(7) constitutes a local profit-maximizing set of output levels for the

chain. This local maximum will be the

focus of attention, since the infinite

price/infinite profits outcome was

already ruled out. 

( ) [ ] ( ) ( )

[ ] ( ) ( )

(6) 1, 2, 1 1 2 2 1

2 21 1 2

π αβ γ

αβ γ

i i i i ji j i j i

i ji j i j i

qq q q q q cq

q q q q cq

j i

j i

, , ,, , ,

, ,, , ,

= − +∑ − +∑ −

+ − +∑ − +∑ −

≠

≠

≠

≠

1 1

2 2

( )( )

( )( )

(7 ) 0

(7 ) 0

1 2 1 2, 1

2 1 2 1, 2

a q q cq q

b q qcq q

ji j ji j i i

ji j ji j i i

= −∑ −∑ −− −

= −∑ −∑ −− −

≠ ≠

≠ ≠

αβ γ γ β

αβ γ γ β

1

2

2 2

2 2

,, ,

,, ,

( ) ( )

[ ] ( )( )

( )

( ) ( )

[ ] ( )( )

( )

(8a) q

(8b) q

1, 2 2

12 1

2

2, 1 1

21 2

1

ii i

W

i

ii i

W

i

q

c

M q

BA SA B h S h Q c

M B

S

B q

q

c

M q

BA SA B h S h Q c

M B

S

B q

, , ,

, , ,

= −

+ − = + ++ − −+ −

= −

+ − = + −+ + −

+ −

α

β

γ

β

θ θ

α

β

γ

β

θ θ

1 1 1 2

2 2 2 1

1 1

1 1
Solving (8) simultaneously yields equilibrium quantities for the two brands

at the level of a chain:

Total production (in the market area) of either brand is the sum across (identical) chains of that brand’s output levels, which is simply M times greater

than the output of any one chain in (9). Equilibrium total production of each

brand can be written as:

where Y = H + L, the number of non-WIC households; w = W/Y, the ratio of

WIC to non-WIC households who buy formula; and b = B/Y, a weighted

average of the price-sensitivity parameters (slopes) of the two out-of-pocket

groups, which can be written as:

where the shares of H and L as proportions of all out-of-pocket consumers,

ωH and ωL, sum to 1.

Using (10) and (4), equilibrium market prices for brands 1 and 2 can be

expressed as functions of exogenous terms:

The assumption that b, the weighted average of the own-price terms,

exceeds s, the cross-price term, ensures that the denominators in (12) are

positive, which means prices for the contract and noncontract brands are

positive (given typical assumptions about the relative sizes of demand

parameters and marginal cost).

15

An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

Economic Research Service/USDA

( )( )

( )( )

[ ] ( )

( )

( )( )

( )( )

[ ] ( )

( )

(9 ) q 1

(9 ) q 1

1,

1

2,

2

a

c c

M

A h Q Bc Sc

M

b c c

M

A h Q Bc Sc

M

i

W

i

W

∗= −− −

− + = ++ − +

+

∗= −− −

− + = +− − +

+

βα γ α

β γ

θ

βα γ α

β γ

θ

11 22

2 2

11 2

22 11

2 2

22 1

1

1

( )

( )( )

[ ] ( )( )

( )( )

( )

( )( )

[ ] ( )( )

( )( )

(12 ) 1 1 1

(12 ) 1 1 1

1 1

2 2

a P

ba sa

M bs

M

M

c

b h s h vz

M bs

w

b P

ba sa

M bs

M

M

c

b h s h vz

M bs

w

∗

∗

= +

+ −

+

+

+

++ −

+ −

= +

+ −

+

+

+

−+ +

+ −

1 2

2 2

1 2

2 2

2 1

2 2

2 1

2 2

δ

θ θ

δ

θ θ

( )[ ] ( )

( )[ ] ( )

(10 ) Q 1

(10 ) Q

1

2

a

M

M

a h wvz bc sc Y

b M

M

a h wvz bc sc Y

∗

∗

= +++ − +

= ++− − +

11 1 2

22 21 1

θ δ

θ δ

( ) ( ) (11) L L

b H

H

b L

H

bbb = H L HH LL

+

⎛

⎝

⎜ ⎞

⎠

⎟ +

+

⎛

⎝

⎜ ⎞

⎠

⎟ = + ω ω
Results of the WIC Model

An examination of (9), (10), and (12) shows whether equilibrium levels of a

chain’s quantities, the market quantities, and prices respond positively or

negatively to a change in any one factor, holding other factors constant. This

chapter focuses on price effects, although brief attention is given to quantity

effects. It is important to distinguish which price effects are due to WIC per

se as a government-funded formula-subsidizing program, which are due to

WIC’s rebate program, and which are due to sole-source procurement. 

The chapter begins by discussing price effects associated with changes in

marginal cost, price-independent brand preferences, and population. The

chapter continues with a discussion of price effects under two scenarios:

• the presence of the WIC program (without rebates) relative to the

absence of the WIC program;

• an increase in the relative size of WIC (without rebates).

The remainder of the chapter then considers each of the following WICrelated factors in turn:

• an increase in the relative size of WIC if WIC has rebates generated by

open market contracts;

• the use of open market contracts relative to the absence of any rebate

contracts;

• an increase in the relative size of WIC if WIC has rebates generated by

sole-source contracts;

• the use of sole-source contracts relative to the use of open market contracts;

• the use of sole-source contracts relative to the absence of the WIC program; and

• the use of home delivery or direct distribution for distribution of WIC

formula relative to the use of the retail food delivery system.

A central variable of the WIC model is the relative size of WIC, w, measured by the ratio of WIC to non-WIC formula-fed infants. A challenge in

reviewing price effects occurs because there is a multiplicative interaction in

(12) between w and its coefficient, which contains the parameters θ1, θ2, δ,

and h. Thus, the price effect due to a marginal increase in w depends on

whether WIC has a rebate program, whether the rebate program uses an

open market contract or sole-source procurement, and on the distribution

system used for WIC formula.16 In the end, it turns out that price effects are

easiest to consider by reviewing them in a particular sequence.

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16 An analogous mathematical

structure is found in the familiar

Distance-Speed-Time relationship D =

ST, where the multiplictative interactions between S and T implies that the

Distance effect of a change in Speed

depends on the level of T and the

Distance effect of a change in Time

depends on the level of S. 
Effects of Marginal Cost, Brand Preferences,

and Population 

Marginal Cost. In (12), a brand’s price depends positively on the brand’s

own marginal cost. A second result of (12) is that a brand’s price does not

depend on the marginal cost of the competing brand, e.g., c2 is not a factor

that affects P1

* (hereafter, the asterisk that denotes equilibrium values will

be implicit). However, that result depends on particular features of the WIC

model: a more general model may find that P1 is affected by c2 . A third

outcome of (12) is that its price-cost relationship matches precisely the

Cournot model’s price-cost relationship. Thus, the special features of the

WIC model that distinguish it from the Cournot model do not change the

role of marginal cost: in both models, a one-unit change in ck raises Pk by

M/(M+1). 

Econometric Specification of Wholesale Cost. A brand-specific retail price

regression in Oliveira et al. includes the brand’s wholesale cost as an independent variable, since wholesale cost is a major determinant of marginal

cost. Furthermore, the price regression excludes the wholesale costs of other

brands. This econometric specification is restrictive in that, as noted above,

a more general theoretical model may find that in principle a brand’s retail

price may be affected by wholesale prices of other brands. The exclusion of

wholesale prices of other brands in the regressions is motivated less by the

results of the WIC model per se—although the WIC model identifies certain

conditions under which the specification is a correct one—and more by the

very high correlation between the wholesale prices of various manufacturers. Severe multicollinearity problems would be introduced in a retail

price regression if all wholesale prices were included in any one brand’s

regression. 

Although the WIC model, like the Cournot model, predicts that a price

regression’s coefficient on wholesale cost should be a positive fraction

equaling M/(M+1), that particular outcome depends strongly on the linear

demand specification used by the two models. Alternatively, a constant-elasticity demand formulation cannot be ruled out as a possibility (although it

cannot be readily incorporated into the formal WIC or Cournot models), in

which case a regression coefficient on wholesale cost would exceed 1. In

summary, a price regression’s coefficient on wholesale cost is certainly

expected to be positive, though not necessarily fractional. 

Price-Independent Brand Preferences. Price-independent brand preferences are measured by ak, the constant term in demand in (1). An increase

in ak results in an increase in the brand’s retail price in (12). There is no

empirical measure for ak, and the term is simply absorbed by the constant in

a brand’s price regression. It is worth identifying ak as a distinct term in the

model even though it is not empirically measured because it is useful to

note that observed cross-brand variation in retail prices may be due in part

to unobserved cross-brand variation in ak.

Population Effects. The expressions in (10) and (12) exhibit a fundamental

structural difference: the number of non-WIC households, Y, is a component of Qk but not of Pk. These results imply that the “scale” of demand—

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whether a market area’s demand is “big” or “small” on account of population size—affects only Qk but not Pk. In other words, if the numbers of H, L,

and W households were to each increase proportionately, either over time or

relative to another market area, then Qk would increase by that same proportion but Pk would remain unchanged. Given that infant formula can be sold

at a constant marginal cost (by assumption), variation in the size of a market

area does not affect equilibrium prices. 

Econometric Specification for Population. If a market area’s population

were to be included in a brand’s price regression, the coefficient is expected

to be statistically insignificantly different from zero. The sign of the variable’s coefficient may be either positive or negative due to randomness in

the data, but no systematic (statistically significant) relationship is predicted

between price and population.

Effects of Relative Size of WIC,

Without Rebate Contracts

Two Price-Increasing Effects. In this section, let θk = ak/(a1 + a2), δ = 1,

and h = 0. Two scenarios will be considered to examine price effects due to

WIC: w > 0 vs. w = 0, which compares the presence of WIC with the

absence of WIC; and a marginal increase in w. 

To keep the sequential discussion of the effects of WIC and the rebate

program organized, we temporarily assume that WIC does not have a rebate

program. This assumption is referred to as counterfactual since WIC

currently does have a rebate program; the assumption could just as well be

called an historical assumption since it describes WIC prior to the implementation of the rebate program. Examination of the counterfactual facilitates considering the effects of WIC, as a government-funded formulasubsidizing program, prior to and separately from considering the effects of

WIC’s rebate program. In the absence of the infant formula rebate program,

it is supposed that all WIC formula is distributed using the retail food

delivery system, making δ = 1 in (12). 

Under the counterfactual, WIC households themselves choose whether to

redeem vouchers for brand 1 or for brand 2. As a result, θk in (12) is positive for both brands, equaling the value of ak/(a1 + a2) as noted earlier in the

section “Demand of WIC Households”. There is no tag-along effect under

the counterfactual, so h = 0 in (12). 

It is accurate to say that WIC “adds” to the overall market demands for

infant formula. After all, WIC provides vouchers to (low-income) households which those households use to obtain more formula than they would

have paid for if they were not WIC participants. However, given the

assumption of constant marginal cost, the price effect of WIC can not be

attributed to its role in increasing demand—whether the “scale” of demand

is small or large affects only Qk—but instead to WIC’s effect on the price

sensitivity of demand. Specifically, WIC decreases the price sensitivity of

demand in the overall infant formula market, which in turn increases the Pk

established by profit-maximizing supermarket chains. 

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This effect can be separated conceptually into two aspects. One aspect was

identified in an analysis of WIC and the formula market by Salant (2003),

while another was identified by Post and Wubbenhorst (1989). 

Salant considered the behavior of a monopolistic infant formula manufacturer and examined major manufacturers’ wholesale price series. Based on a

“reservation price” monopoly model, Salant argued:

. . . by removing the portion of the population with the lowest reservation price for infant formula from the general market, the WIC

program inevitably raised the profit-maximizing monopoly price . . .

What previously restrained [the monopolist] was the recognition

that a price increase would drive away the poorer customers; but

once the WIC program absorbs these customers, the monopolist has

nothing further to lose if he raises the price . . . As more infants are

added to the WIC program, the model predicts that the [monopolist]

will continue to raise the price to non-WIC customers. 

The pricing behavior identified by Salant does not require that the firm be a

monopolist or a manufacturer: his economic reasoning also applies to the

WIC model in which multiple supermarket chains engage in (imperfect)

competition in the establishment of a retail price. 

Salant’s argument that WIC “removes” from the general market the (lowincome) households with the lowest reservation price is recast by the WIC

model as the argument that WIC “removes” from the general market the

(low-income) households that are relatively more price sensitive (bL >

bH).17 Formally, let the numerical values of W, L, and H in the absence of

the WIC program (the counterfactual) be represented by W’, L’, and H’,

where W’= 0, L’ = L + W (the WIC households all fall below the model’s

income cutoff that demarcates low- from high-income households; see the

section “Demands of Out-of-Pocket Households”), and H’ = H (the number

of high-income households is the same with or without WIC). The overall

price sensitivity of the out-of-pocket households is always a weighted

average of bL and bH, but the numerical value of that weighted average

depends critically on whether or not WIC is present. In the presence of

WIC, the numerical value of that weighted average is given by (11). Let b0

designate the value of the weighted average in the absence of WIC, where

Comparison of (13) and (11) shows that b0 > b. As WIC “removes” some

low-income households from the out-of-pocket segment of the market, the

mix of out-of-pocket households that remain is less price sensitive: the

weights on bL and bH are shifted more towards towards bH, lowering the

weighted average below b0. A decrease in the price sensitivity of out-ofpocket households raises Pk in (12)—a result that will be established

formally below. Thus, holding other factors constant, each of the M supermarket chains will find it profit-maximizing to increase the retail price of

infant formula in the presence (versus the absence) of WIC. 

17 The WIC model, like Salant,

does not consider how in the absence

of WIC some of the low-income

households may choose breastfeeding

rather than purchase formula out of

pocket. 

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( )

( )

( ) (13) 0 b H

H LW

b

L W

H LW

b = H L + +

⎛

⎝

⎜ ⎞

⎠

⎟ + +

+ +

⎛

⎝

⎜ ⎞

⎠

⎟
We call this the out-of-pocket composition effect because the effect depends

on whether out-of-pocket demand is composed of relatively few or many

low-income households. Salant himself noted that this effect is analogous to

pricing effects that seem to be found in the markets for certain pharmaceutical products. Prior to the introduction of generic pharmaceutical products

that are substitutes for brand-name products, it might have been predicted

that the entry of generic products would (as substitutes) lower the price of

the brand-name product due to the increase in “competition.” However, it

was observed that in some instances the price of the brand-name drug

increased after generic drugs entered the market. One explanation is that

those consumers who were most price sensitive switched to the generic

drug, leaving the less price sensitive (or so-called “brand-loyal”) customers

in the market for the brand-name product. In response to the decrease in

elasticity in demand facing its product, a profit-maximizing pharmaceutical

company will raise the price on its brand-name drug. 

Another mechanism by which WIC decreases the price sensitivity of

demand was identified by Post and Wubbenhorst. They argued that by

providing WIC households with vouchers, the WIC program produces a

“customer that is essentially unconcerned with the price she or he is

paying.” The WIC model calls this mechanism the voucher effect. In (12)

the voucher effect is present given that δ =1, in which case price is affected

by w through a mechanism other than the price-sensitivity term b. As w

increases from zero (in the absence of WIC) to a positive value (in the presence of WIC), the mix of demands in total market demand is changed, with

relatively fewer price-sensitive out-of-pocket households and relatively more

price-insensitive WIC households resulting in a decrease in overall price

sensitivity.18 The profit-maximizing supermarket chains respond to the

reduction in price sensitivity by raising the equilibrium retail prices of infant

formula when WIC is present.

Although the out-of-pocket composition effect and the voucher effect both

affect the mix of households in the infant formula market, the two are

different: the former changes the mix within the group of out-of-pocket

households while the latter changes the mix between the out-of-pocket

households and the WIC households.

The voucher effect reflects the WIC model’s general principle that the scale

of demand is unimportant. A change in the absolute size of WIC—whether

WIC is “big” or “small” as measured by W—does not necessarily change

price: if a change in W is accompanied by a proportionate change in Y, then

w is unchanged and price is unaffected. In contrast, a change in the relative

size of WIC, as measured by w, does affect price even if overall population

is fixed.

While WIC does “remove” a set of low-income households from the out-ofpocket segment of the retail food system—as Salant emphasized—WIC also

provides vouchers that make WIC households price insensitive—as Post and

Wubbenhorst emphasized—which “adds” those same households back into

the retail food system. A way of describing both effects at once is to state that

WIC converts out-of-pocket low income households (whose price sensitivity

is greater than for high-income households) into WIC households (whose

price sensitivity—zero—is smaller than for high-income households).19

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18 The voucher effect is also a

particular type of “composition”

effect, involving the composition of

overall market demand and the relative numbers of WIC and non-WIC

households.

19 The concept of converting lowincome households from non-WIC to

WIC participation helps clarify how

there are two answers to the seemingly

simple question: “Are WIC households

more or less sensitive to price than

high-income households?” The answer

depends on whether the price sensitivity of WIC households is considered

ex ante or ex post to their participation

in WIC.
The discussion has introduced the out-of-pocket composition effect and the

voucher effect while considering the presence vs. the absence of WIC.

Similarly, an expansion of WIC—represented by a marginal increase in w—

would also increase price as a result of the same two mechanisms.

Effects of Relative Size of WIC, Formal Derivation. The formal derivation

of the price effects of changes in w focuses on P1 in (12a) and, for generality, does not here set zero values for particular parameters such as δ or h.

The total effect on price due to a change in w is given by:

As discussed earlier, the relative size of WIC affects prices through two mechanisms. These two mechanisms correspond to the voucher effect and the outof-pocket composition effect, respectively, which are represented by the first

term on the right in (14) and by the product of the final pair of terms.

The voucher effect is given by the partial effect of w on price, which is 

non-negative:

The voucher effect is strictly positive if δ = 1, i.e., if WIC formula is distributed using the retail food delivery system.

Examination of the out-of-pocket composition effect begins by identifying

the relationship between b and w, which can be expressed as

where b0 is the weighted average price sensitivity in the absence of any

WIC program from (13). Recalling that price sensitivity is relatively large

for low-income households (making bL > bH), an increase in w reduces b,

i.e., db/dw < 0. 

The remaining term in (14) represents the partial effect of b on price:

The value of the expression is negative whether δ equals 1 or zero and h is

zero or positive. Because db/dw < 0 and the partial effect of b on P1 is also

negative, the out-of-pocket composition effect is positive. If δ = 1 the

voucher effect and the out-of-pocket composition effect both work in the

same direction, making the total effect of w on P1 unambiguously positive.

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( ) ( ) (14) 

dP w b w

dw

P

w

P

b

db

dw

1 1 1 , = ⎡

⎣

⎢ ⎤

⎦

⎥ + ⎡

⎣

⎢ ⎤

⎦

⎥

⎡

⎣

⎢ ⎤

⎦

⎥

∂

∂

∂

∂

[ ]

( )( ) (15) 1

0 2 2

∂

∂ δ P θ θw

b s vz

M bs

1 1 2 = +

+ −

≥

(16) bw b ( ) 0 ( ) H

N = − b bw L H

⎛

⎝

⎜ ⎞

⎠

⎟ −

( )( ) ( ) ( )

( )( ) (17) +1

0

2 2

1 21

2

2 22

∂

∂

P δθ δθ δ

b

b s a vzw bs a vzw b s hvzw

M bs

1 1 2 ⎡

⎣

⎢ ⎤

⎦

⎥ = − + + + + +−

−

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

<
If δ = 0, the effect of w on P1 remains positive due to the out-of-pocket

composition effect alone. 

Evaluation of the second derivative (not shown) demonstrates that P1 is

convex in w. For some analyses, such as behavior towards risk, the curvature properties of a relationship between two variables is central. However,

in the graphical analysis below, the major results of the WIC model can be

portrayed most easily by treating the relationship between P1 and w as

approximately linear.

Graphical Illustration. Figure 1 illustrates the relationship between the

equilibrium price and w on the graphical line labeled PNR for “no rebate.”

When considering PNR, the variable Pk on the vertical axis can be thought of

simply as the equilibrium price of either brand in (12); the interpretations of

other graphical curves in Figure 1 are considered below. 

Along PNR all other price-determining factors besides w are held constant.

The slope of PNR measures the price effect on Pk due to a marginal change

in w. In general, the value of this price effect does depend on which brand is

considered (inasmuch as θk in (12) can be brand specific). 

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C

B

D A

E

PB

PA

wA wB

Figure 1

Retail prices vs. relative size of WIC under various contract systems

Equilibrium supermarket price of brand k

PCB

PNR = POM*

PNCB

Ratio of WIC to non-WIC formula-fed infants

*One relationship between price and w is labeled both PNR and POM, signifying that the 

same relationship exists whether no rebates are in effect or an open market contract is in 

effect (given that marginal cost of infant formula to supermarkets is held constant).

Source: Economic Research Service/USDA.
The figure illustrates the price effects due to:

• the presence of the WIC program (under the counterfactual assumption

of no rebates) relative to the absence of the WIC program;

• an increase in the size of WIC (under the counterfactual);

When considering the presence versus the absence of WIC, the relevant

change is from a zero value of w to a positive value, say wA. When considering an expansion of WIC, the relevant change is an increase in the value

of w, say from wA to wB. In either case, the change in w results in a retail

price increase. Of course, the magnitudes can differ, with a movement from

the intercept to point A in the former case and from point A to point B in

the latter case. However, the qualitative results—the direction of price

effects—are the same for both cases. 

Econometric Specification. The price regressions in the companion volume

contain three independent variables of particular interest:

• median household income (I), as a measure of central tendency of the

income distribution

• the poverty rate (R)

• the ratio of WIC to non-WIC formula-fed infants (w).20

The regression includes I and R to capture the effects of differences across

out-of-pocket households of (income-dependent) price sensitivities and the

relative frequencies of out-of-pocket households at various income levels. 

As R increases, it is likely that the share of L out of all out-of-pocket households increases as well, increasing b in (11) and decreasing Pk. Although the

income cutoff for R is (by definition) the poverty line and the income cutoff

between the WIC model’s low- and high-income populations is set above

the income threshold at which a household qualifies for WIC (185 percent

of the poverty line), R is used in the companion volume as a rough proxy

for the overall shape of the income distribution within a market area.21 The

predicted sign for the regression coefficient on R is negative.

The inclusion of I in the price regression introduces an empirical element

that the WIC model has not fully considered. As a simplified theoretical

model, the WIC model sets but two values for price sensitivities, given by bH

and bL , and focuses on the mix of L and H households as the varying determinant of the overall price sensitivity of the out-of-pocket households (in

(11)).22 However, in actual market areas, each household’s level of income

can vary, and each level of income may be associated with a particular value

of price sensitivity: instead of just two price-sensitivity values of bH and bL

there may instead be a range of many (unobserved) income-dependent

values. Thus, the many possible levels of household incomes and price sensitivities and the frequency distribution of households together determine b, a

market area’s overall price sensitivity for out-of-pocket households. 

The variables R and I together capture, as effectively as can be done with

available data, the factors that influence b and the out-of-pocket composi20The same symbol w is used to

designate both this ratio (for the regression analysis) and the WIC model's

ratio of WIC to non-WIC formula-buying households. Strictly speaking, the

two ratios do differ slightly to the

extent that a household may have more

than one formula-fed infant. 

21Although R and I are likely to be

(negatively) correlated, the inclusion

of R in the regression does not capture

how an area's low median household

income results in a low price: R can

only capture the role of income distribution inasmuch as the regression also

contains I. If I were omitted, then the

coefficient on R would measure the

combined effects of low median

income and of income distribution. 

22These two values are set for any

given value of D, the ratio of discount

stores to population. As discussed, a

change in D does change the price

sensitivities to a new pair of values.

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tion effect. If R and I were not included in the regression, the coefficient on

w would measure a combination of the voucher effect and the out-of-pocket

composition effect. Because R and I are included in the regression, the

remaining role for w is to capture the voucher effect alone. 

Effects of Relative Size of WIC,

With Open Market Contract 

In this section, let θk = ak/(a1 + a2), δ = 1, and h = 0. A marginal increase

in w is considered given that manufacturers pay rebates under an open

market contract.

The open market contract, even when it was in use, was never as predominant as the sole-source contract. Nevertheless, an open market contract is

considered prior to the sole-source contract in order to separate as fully as

possible the price effects of a rebate system from the price effects of a solesource contract. A key feature that distinguishes a rebate system under an

open market contract from a rebate system under a sole-source contract is

that the open market contracts generate rebates without affecting the relative

demands for different brands, while sole-source contracts channel all WIC

demand to a single contract brand. 

Under the open market contract, just as under the no-rebates counterfactual

already considered, WIC households choose whether to redeem vouchers for

either brand 1 or brand 2. Under the open market contract, θk in (12) are

positive for both brands, equaling the same values of ak/(a1 + a2) introduced

for the counterfactual. Because there is no single contract brand, the tagalong effect remains zero (h = 0). Given these common specifications for

the values of θk and h, the value of the coefficient of w under the open

market contract matches the value of the coefficient under the counterfactual, given by (14). Thus, the effect of a change in the relative size of WIC

is the same regardless of whether WIC has no rebate program or WIC has a

rebate program that uses open market contracts.

Effects of Open Market Contracts. In this section, let θk = ak/(a1 + a2),

δ = 1, and h = 0. The effect of open market contracts per se is considered. 

It is important to distinguish between two different questions:

• What is the price effect of a change in the relative size of WIC if there is

an open market contract?

• What is the price effect of an open market contract (versus no open

market contract)?

The first question has already been answered, and the answer involves the

value of the slope of the relationship between Pk and w; the slope of the

relationship happens to be the same with or without an open market

contract. However, just because the slope is unaffected by the open market

contract does not mean that price is unaffected. Price can be affected by the

open market contract due to a change in:

• w, which changes the location of equilibrium price on a given curve; and

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• marginal cost c, which changes the curve’s location. 

Thus, even though the same terms appear in (12) whether the counterfactual

or open market contracts are considered, the values of two of the terms (w

and c) may depend on which of the two scenarios is considered. Initially,

marginal cost will be held constant, in which case any change in retail price

due to the open market contract occurs along the same PNR line associated

with the no-rebate counterfactual. It is for this case that Figure 1 relabels

PNR as POM for “open market” contract.

Under the counterfactual no-rebate assumption, WIC is financed by

Congressional appropriations alone. Suppose in this case that the relative

size of WIC is wA in Figure 1, with an equilibrium price of PA on the line

POM. Once rebates are received, suppose that Congress maintains the

amount of appropriations steady—so that rebates fully supplement those

appropriations—or that Congress lowers appropriations, but by less than the

amount of the rebates. Either way, total WIC funding grows and the WIC

program supports more participants.23 To the extent that more infants participate in WIC as a result of the rebates, w increases (say to wB) and retail

price increases from PA to PB on POM, holding other factors (such as

marginal cost) constant.

In this scenario, when WIC receives rebates from open market contracts, the

relative size of WIC increases and price increases. Some analysts may therefore attribute the price increase to the rebates, concluding that the rebates

“caused” a retail price increase. However, other analysts might attribute the

price increase not to the rebate program per se but instead to the increase in

the relative size of WIC. This alternative way of describing the price

increase and its “cause” is based on the view that Congress can support any

particular size of WIC it chooses—be it wA or wB—either by appropriations

alone or by some mix of appropriations and rebates. In this view, Congress

can achieve any particular w with or without rebates, making the relative

size of WIC the only critical factor in determining retail prices. This view

considers the method of financing—the mix between appropriations and

rebates—to be important for budgetary considerations, but secondary (actually, ineffectual) in determining retail price at any given value for w. 

Throughout the rest of this report the WIC model treats wholesale prices as

exogenous, but here a relaxation of that assumption is considered. The

graphical curve POM depends in part on marginal cost, which in turn

depends on manufacturers’ wholesale price. If manufacturers increase

wholesale prices in response to the payment of rebates or to the relative size

of the WIC program (which itself can depend on rebates, as just noted), then

retailers’ marginal costs rise, which would increase retail prices. Noting that

marginal cost in (12) is a component of the intercept of POM in Figure 1, an

increase in marginal cost would shift POM parallel upwards (not shown). In

this case the lines POM and PNR would differ, and retail price would be

affected not only by any increase in w associated with rebates but also by

the upward shift in POM. Having identified this possible repercussion from

the rebate program on wholesale price, the remainder of this report returns

to concentrating on retailer behavior and again treats wholesale prices as

exogenous.

23An implicit assumption is that

there had been eligible households with

infants who are not participating in

WIC that would participate if additional

funding were available, so that an

increase in total WIC funding is indeed

associated with an increase in w.

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Effects of Relative Size of WIC,

With Sole-Source Contract

In this section, θ1 = 1, θ2 = 0, δ = 1 and h = 0. Even though this section

examines sole-source contracts, which means there is a single contract

brand, the tag-along effect will be still be treated as zero; the tag-along

effect is examined separately below. 

There are four questions (each involving two prices) that will be answered

in this section and the following two sections:

• What are the effects on contract and noncontract brand prices due to an

increase in the relative size of WIC, given that a sole-source contract is

in effect?

• What are the effects on the prices of a national brand serving as either

the contract or noncontract brand as a result of changing from an open

market contract to a sole-source contract, at a given w?

• What are the effects on a national brand’s prices as it changes in contract

brand status?

• What are the effects on the prices of a national brand serving as either

the contract or noncontract brand as a result of changing from the

absence of WIC (and its rebate program) to the presence of WIC and a

sole-source contract?

Under the sole-source contract, the contract brand receives all of the WIC

demand, making θ1 = 1, while the noncontract brand receives none of the

WIC demand, making θ2 = 0. Given that δ = 1 and h = 0, (12) becomes:

Figure 2 makes a comparison of price curves for two different national

brands: the PCB and PNCB curves refer, respectively, to the curves for the

contract brand (e.g., Mead-Johnson) and the noncontract brand (e.g.,

Carnation). The figure shows that the intercept of PCB is above the intercept

of PNCB; the reverse is true for the intercept of the curve PNCB(c,a), which

will be considered below. Difference in intercepts could be due to a difference in non-price brand preferences, as measured by a1 and a2, to a difference in the marginal costs of the two brands, or both. 

In Figure 2, PCB has a positive slope, which reflects the coefficient on w in

(18a). Given that brand 1 receives all WIC demand, it is intuitive that an

increase in w results in an increase in P1. 

One might at first expect that an increase in the size of WIC will have no

effect on the price of the noncontract brand, which receives none of the

26

An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

Economic Research Service/USDA

( )

( )( ) ( ) ( )( )

( )

( )( ) ( ) ( )( )

(18 ) *

+1 1 +1

(18 ) *

+1 1 +1

1

1 2

2 2 1 2 2

2

2 1

2 2 2 2 2

a P

ba sa

M bs

M

M

c

bvz

M bs

w

b P

ba sa

M bs

M

M

c

svz

M bs

w

= +

− +

+

+ −

= +

− +

+

+ −
WIC demand (either before or after the increase in the relative size of WIC).

If there is no WIC-related price effect on the noncontract brand, its price

curve would be the (unlabelled) horizontal dashed line. What is perhaps

unexpected is that the price curve for the noncontract brand is given by

PNCB, which lies above the horizontal line. According to (18b) and its

graphical illustration as PNCB, an increase in the relative size of WIC results

in an increase in the price of the noncontract brand P2.

The mechanism by which w affects the price of the noncontract brand is

through the demand substitution behavior of the non-WIC households. An

increase in w results in profit-maximizing supermarket chains establishing a

higher P1 along PCB, and (at least some) non-WIC households respond to

this increase in the price of brand 1 by switching demand to brand 2 (strictly

speaking, the price sensitivity of demand for brand 2 must be affected, not

merely the scale or size of demand). Supermarket chains respond to the

change in the demand for brand 2—which was caused by their very own

increase P1 in the first place—by increasing P2.

24

As mentioned previously, one of the powerful features of the WIC model is

that it permits (but does not require) substitution behavior between brands.

In (2), the parameter that captures the extent to which non-WIC households

substitute between brands within the supermarket sector is s, which appears

in the numerator in (18b). If non-WIC households do not substitute at all

between the brands, then s = 0 in (18b) and the price curve for the noncon24If a nonequilibrium situation were

being examined, the increase in the

price of brand 2 would result in (at

least some) non-WIC consumers

switching to brand 1, resulting in

nudge up in the price of brand 1 and a

nudge down in the price of brand 2,

which would result in a further chain

of effects between the prices of the

two brands (with each link in the chain

growing progressively smaller).

However, an advantage of the formal

mathematical approach used by the

WIC model is that it examines the

equilibrium outcome of that entire

process (through the simultanteous

determination of the prices of the two

brands), so that the entire sequence or

chain of interactions between the two

brands results in a final outcome that

is portrayed in (14) and Figure 2. The

changes in the equilibrium prices P1

and P2 already take into account all

the interactions.

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An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

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Figure 2

Retail prices vs. relative size of WIC for two national brands

Equilibrium supermarket price of Brand K

PCB

PNCB

Ratio of WIC to non-WIC formula-fed infants

Source: Economic Research Service/USDA.

PNCB(c,a)
tract brand would be represented by the dashed horizontal line. The PNCB

curve in Figure 2 assumes that s > 0. This report adopts that assumption

hereafter. The econometric analysis in Oliveira et al. subjects this assumption to a statistical test. 

There is a difference between the positive slopes of PCB and PNCB because b

appears in (18a) and s appears in (18b). The closer s is to b—the better

brand 2 can substitute for brand 1—the closer PNCB comes to being parallel

to PCB, making the price effect of w on the contract and noncontract brands

more similar. The WIC model stipulated that b > s, making PCB steeper than

PNCB. 

Figure 2 illustrates one additional curve labeled PNCB(c,a), where the notation emphasizes how marginal cost and non-price brand preferences affect

the location of the price curve. In contrast to PNCB, PNCB(c,a) is located

above rather than below PCB. As price curves for a noncontract brand,

PNCB(c,a) and PNCB share a common slope (which is exceeded by the slope

of PCB). Figure 2 shows the price effect due to a change in w is greater for

the contract brand than for the noncontract brand regardless of whether the

contract brand’s retail price is high or low relative to the noncontract brand’s

retail price (i.e., regardless of whether the noncontract brand’s price lies on

PNCB(c,a) or PNCB). 

Oliveira et al. found that, within market areas, there is not a clear and

consistent relationship between a formula’s being the WIC contract brand

and that formula being sold at the highest average retail price. However,

comparing the retail prices of contract and noncontract brands of formula

does not necessarily identify WIC-related price effects since other factors

may affect retail prices. Retail prices of two national brands do reflect the

relative size of WIC and which of the two national brands holds the rebate

contract. But retail prices also reflect non-price brand preferences and

marginal cost. While non-price brand preferences are not measured empirically, differences across national brands in wholesale prices are measurable.

A comparison of PCB and PNCB shows how a particular national brand can

be the contract brand in a market area and have a higher retail price than a

noncontract brand due to a relatively high wholesale price for the contract

brand. In contrast, the expectation that the contract brand has the highest

retail price would not be met if PCB is compared with PNCB(c,a). Due to a

relatively low wholesale price, a national brand associated with PCB has a

retail price below the noncontract brand price on PNCB(c,a). The empirical

analysis in Oliveira et al. is designed to separate the retail price effects due

to contract brand status from the effects due to wholesale prices. 

Contract Systems, Contract Brand Effects,

and Distribution Systems

Effects of Sole-Source Contracts Compared With Open Market Contracts.

In this section, δ = 1 and h = 0 and a comparison is made between (θ1, θ2) =

(1, 0) and θk = ak/(a1 + a2).

This section begins with a comparison of retail prices under a sole-source

contract with retail prices under an open market contract. A key difference

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An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

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between the two types of contracts is whether WIC households can select

between brands 1 and 2 or if WIC households must purchase a single

contract brand.25 Then the section examines how retail prices compare

between a sole-source contract and the absence of the WIC program. 

Returning to Figure 1, the curves PCB and PNCB straddle the curve POM,

radiating from the same intercept. Now the interpretation of Figure 1 is that

it applies to any single national brand under varying conditions. Under an

open market contract, the retail price of the national brand is at point B,

given wB. Suppose that a sole-source contract is adopted instead of an open

market contract, and the national brand is awarded the rebate contract. The

price effect of a sole-source contract for the contract brand, relative to an

open market contract, is to increase retail price from point B to point C. The

reason for this price increase is not due to a “rebate system” per se: the

open market contract also generates rebates. Instead, the movement from

point B to point C is because the share of WIC demand received by the

given national brand increases from a fraction, given by ak/(a1 + a2) under

the open market contract, to 100 percent.

Suppose instead that a sole-source contract is adopted instead of an open

market contract, and the national brand is not awarded the rebate contract,

becoming the noncontract brand. The price effect of a sole-source contract

for the noncontract brand, relative to an open market contract, is to decrease

the retail price from point B to point D. The reason for this price decrease

for the noncontract brand is that its share of WIC demand drops from a fraction (under the open market contract) to 0 percent. Even though the national

brand loses its fractional share of WIC demand, its retail price does not drop

from point B to the dashed horizontal line at point E. The demand substitution from the contract brand to the noncontract brand exhibited by non-WIC

households helps sustain the retail price of the noncontract brand, making

the decrease in its retail price smaller than would be the case if there were

no substitution behavior.

Effects of Contract Brand Status. In this section, δ = 1 and h = 0 and a

comparison is made between θk = 0 and θk = 1 for a given manufacturer’s

brand.

Given that a sole-source contract is in effect, the price effect for a national

brand changing status from noncontract to contract is represented by a move

from point D to point B; alternatively, the move from point B to point D

represents the price effect of changing status from contract to noncontract.

These changes are the price effect of a change in contract brand status. The

price effect of contract brand status depends on the relative size of WIC. If

w is small, the contract brand status makes little difference in a national

brand’s retail price, whereas if w is large, the change from contract to

noncontract status—from having all of the WIC demand to having none of

the WIC demand—results in a large price effect.

25A second difference between

open market and sole-source contracts

can be important in practice. Although

both types of contracts generate

rebates, historically the amount of

rebates obtained from a sole-source

contract is larger than the amount from

an open market contract. This difference in turn could affect w and therefore retail prices. However, since the

theoretical effects of rebates were

already identified in the discussion of

the open market contracts, it will be

assumed hereafter that the rebate levels of the open market and sole-source

contracts are the same. This assumption is adopted to concentrate on the

effect that results from channeling all

WIC demand to a single contract

brand, holding other factors constant.

The size of the differences in rebates

may be of great importance in a practical comparison between the two types

of contracts. 

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An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

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Effects of Sole-Source Contract Compared With the Absence of WIC. In

this section, let (θ1, θ2) = (1, 0), δ = 1, and h = 0.

The answer to a simple question such as “What are the price effects of solesource contracts?” depends on what conditions are being compared with the

sole-source contracts. One alternative condition to a sole-source contract,

considered above, is an open market contract. This section examines a

condition in which WIC and its rebate program are absent.

Figure 1 shows that in the absence of WIC w is 0, resulting in a retail price

of the national brand at the intercept or, equivalently, at point E. If the WIC

program is adopted and a sole-source contract is used, the effect on the

retail price of the national brand depends of course on whether the national

brand is awarded the WIC contract or not. Retail price increases from E to

C for the contract brand, while retail price increases only from E to D for

the noncontract brand.

In this section, the price effect of a sole-source contract—compared with the

absence of WIC—is to increase the retail price of formula for both the

contract (E to C) and noncontract brands (E to D). As previously noted, the

price effect of a sole-source contract—compared with an open market

contract—is to increase the retail price of the contract brand (B to C) and

decrease the retail price of the noncontract brand (B to D). The seemingly

simple question “What are the price effects of using a sole-source contract?”

results in two answers that have different magnitudes and even different

signs. This is but an example of the general principle that it is important to

identify the alternative condition when investing the theoretical or empirical

effect of a change in policy, economic, or demographic variables. 

Tag-Along Effect. In this section, let (θ1, θ2) = (1, 0) and δ = 1. A comparison is made between h = 0 and h > 0.

The tag-along effect, the composite of the medical promotion effect and the

shelf-space effect, represents certain demand-shifting behaviors that favor

the contract brand at the expense of the noncontract brand. Reintroducing a

positive h shows that the tag-along effect enhances the positive effect w has

on the contract brand’s P1, entering the numerator of (18a) positively, but

(partially) offsets the positive effect w has on the noncontract brand’s price

P2, entering the numerator of (18b) negatively. 

Qualitatively, Figure 1 is not affected the tag-along effect. If the tag-along

effect is present, the price effect of a change in w is greater for the contract

brand and smaller for the noncontract brand, creating a gap in Figure 1

between the PCB and PNCB curves that widens more quickly.

The price regressions in Oliveira et al. have no separate measure for the tagalong effect, which is simply treated as a fixed parameter and absorbed into

the regression coefficients on w.

Alternative Distribution Systems. In this section, a comparison is made

between δ = 0 and δ = 1, given that θ1 = 1 and θ2 = 0.

30

An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

Economic Research Service/USDA
If a WIC State agency does not use the retail food delivery system to

distribute WIC formula, WIC consumers obtain formula either by the direct

distribution system (as in Mississippi) or the home distribution system (as in

Vermont). Alternative distribution systems involve contracts between State

WIC agencies and manufacturers but the contracts do not, strictly speaking,

involve rebates from manufacturers. Instead, State WIC agencies purchase

directly from the manufacturer. Nevertheless, direct distribution and home

distribution strongly resemble a rebate system in that they are all designed

for cost-containment. 

Under direct distribution or home distribution, a marginal change in the

relative size of WIC (or the presence of WIC vs. the absence of WIC)

increases retail prices due to the out-of-pocket composition effect alone,

which operates through changes in b. WIC “removes” low-income households from the general market, but the voucher effect is absent because WIC

does not “add” these same WIC households back into the retail food

delivery system. In this case, δ = 0 in (12), canceling the term that contains

w explicitly, and setting the partial derivative in (15) to 0. The out-of-pocket

composition effect is still positive, but because the voucher effect does not

augment it, the retail price is lower than it would be if the retail food

delivery system were used, holding other factors constant.26

The empirical analysis in Oliveira et al. focuses on price data from market

areas in which the retail food delivery system is used.

Effect of Market Structure Conditions

Discount Stores. Another factor that affects retail price is a household’s

ability to buy formula at discount stores. It was discussed earlier how an

increase in D, the ratio of the number of discount stores to total population,

can be expected to increase the price sensitivity of demand for supermarket

formula, as measured by b. An increase in b, in turn, decreases Pk. In addition, an increase in D and the convenience of discount stores to WIC households may lower v, the fraction of formula WIC households receive in

supermarkets, thus decreasing Pk further.

Concentration (M). An increase in M, the number of (equally sized) firms,

lowers concentration whether concentration is measured by the HerfindahlHirschman Index, the four-firm concentration ratio, or any other measure of

concentration. As in the Cournot model, if M = 1, the result in the WIC

model is a monopoly outcome (although here it is an outcome associated

with one monopoly supermarket chain pricing two interdependent brands).

As M increases without limit, concentration falls, market output and price

approach the outcome of a perfectly competitive market in which price

equals marginal cost (for each brand). Thus, in the WIC model an increase

in M is associated with a decrease in price (for each brand), just as in the

Cournot model.

Oliveira et al. considers how an increase in concentration may be associated

with a decrease (rather than an increase) in the retail price of infant formula,

but that discussion lies outside the scope of this report.

26While it might be thought that a

lower retail price is desirable (from a

consumers' perspective), a State that

uses direct distribution or home distribution bears the administration costs

of operating the State's formula distribution system, and such costs must be

funded somehow. An assessment of

the many pros and cons of adopting

alternative distribution systems is

beyond the scope of this report.

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An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

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Summary of Econometric Specification

An econometric specification that is consistent with the WIC model is given

by:

where

• Pk

i,t represents the retail price of brand k formula in market area i in time

period t;

• CBk

i,t represents a dummy variable that equals 1 if brand k is the

contract brand in market area i in time period t and equals zero 

otherwise;

• wi,t represents the ratio of WIC to non-WIC formula-fed infants in

market area i in time period t;

• WCk

t represents the wholesale cost for brand k in time period t;

• Di,t represents the number of discount stores relative to population in

market area i in time period t;

• HHIi represents the M-firm Herfindahl-Hirschman Index for market area

i in 2000;

• Ii,t represents median household income in market area i in time period t;

• Ri,t represents the poverty rate for market area i in time period t;

• εi,t represents an error term

It is expected that β1 > 0, measuring the price effect on the contract brand of

a change in w, and that β2 > 0, measuring the price effect on the noncontract

brand of a change in w. It is also expected that β3 > 0, β4 < 0, β5 > 0 (based

on the WIC and Cournot models), β6 > 0, and β7 < 0. 

32

An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

Economic Research Service/USDA

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

(19) 1 01 2 3

4 5 67

P CB w CB w WC

D HHI I R

i t

k

i t

k

it it

k

it t

k

it i it it it

, ,, ,,

, , ,,

=+ + − + * ( )*

+ + ++ +

ββ β β

β β ββ ε
References

Brown, M.G., and J-Y Lee. “Allocation of Shelf Space: A Case Study of

Refrigerated Juice Products in Grocery Stores.” Agribusiness. Vol. 12,

No. 2. 1996.

Folwell, R.J., and D.A. Moberg. “Factors in Retail Shelf Management

Impacting Wine Sales.” Agribusiness. Vol. 9, No. 6. 1993.

Oliveira, V., M. Prell, E. Frazao, and D. Smallwood. WIC and the Retail

Price of Infant Formula. U.S. Department of Agriculture, Economic

Research Service. Food and Nutrition Research Report No. 39. May

2004.

Oliveira, V., M. Prell, E. Frazao, and D. Smallwood. Infant Formula Prices

and Availability: ERS Final Report to Congress. U.S. Department of

Agriculture, Economic Research Service. October 2001.

Post, J.E., and W.H. Wubbenhorst III. Assessing the Infant Formula Rebate

Initiative. Boston University School of Management. April 1989. 

Salant, S. “Microeconomics in Practice.” http://www.wnorton.com/mip/. 

U.S. Department of Agriculture, Food and Nutrition Service, Office of

Analysis and Evaluation. Summary: Fiscal Year 2000 WIC Food Package

Costs. December 2001.

U.S. General Accounting Office, Resources, Community, and Economic

Development. Food Assistance: Information on WIC Sole-Source Rebates

and Infant Formula Prices. GAO/RCED-98-146. May 1998.

33

An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

Economic Research Service/USDA
Appendix: WIC Model Symbols

M the number of supermarket chains in a local market area 

ck marginal cost to the supermarket chain for formula brand k

(k = 1, 2) 

Ck,i total cost of brand k (k = 1, 2) for chain i (i = 1,...,M) 

qk,i quantity of brand k formula sold by chain i (i = 1, ..., M) 

N the number of households in a local market area that use formula 

H high-income households, used to represent both their number and

their type 

L low-income households, used to represent both their number and

their type 

qk,j quantity demanded for brand k formula (k = 1, 2) by type j households (j = L, H) 

Pk supermarket price for brand k formula (k = 1, 2) 

ak price-independent demand parameter (intercept) for brand k

formula

u demand parameter that measures the tag-along effect 

bj own-price demand parameter (slope) for type j households (j = L,

H) 

s cross-price demand parameter 

z a household’s saturation level of formula (and the WIC allocation) 

qk,W quantity demanded for brand k formula (k = 1, 2) by WIC households 

v the fraction of vouchers that a representative WIC household

redeems in supermarkets 

θk the share of supermarket formula demand by a representative WIC

household that is provided by brand k formula (k = 1, 2) 

δ dummy variable that equals 1 if WIC formula is distributed

through the food delivery distribution system and zero otherwise 

Qk Market demand for the supermarket sector for brand k formula 

(k = 1, 2) 

Ak market-level term equaling (H + L)ak

U market-level term equaling (H + L)u 

B market-level term equaling HbH + LbL

S market-level term equaling (H + L)s 

QW market-level WIC demand for all formula

D the ratio of the number of discount stores to total population 

h a constant of proportionality relating U to QW

α a derived parameter 

β a derived parameter 

γ derived parameter 

Y the number of non-WIC households 

w the ratio of WIC to non-WIC households that buy formula 

b the group-weighted average of price sensitivity terms bH and bL, if

WIC is present 

b0 the group-weighted average of price sensitivity terms bH and bL, if

WIC is absent 

34

An Economic Model of WIC, the Infant Formula Rebate Program, and the Retail Price of Infant Formula / FANRR-39-2

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