Do Preferences over Consumption Depend on Household Composition? A Collective Approach to Household Consumption.

Do Preferences over Consumption Depend on Household Composition? A Collective Approach to Household Consumption.

Master’s Thesis Economics&Business (Research Master) EBM897A30/ MSc Economics Thesis EBM877A20

Author : B.J. van Leeuwen Supervisor : dr. R.J.M Alessie

Date : 10-7-2015

University of Groningen

Abstract

We use a collective model of household consumption to explain the allocation of expenditure by singles and couples households. Collective models explain household

demand as the outcome of decisions by multiple persons in the household. We use a version proposed by Browning et al. (2013) which incorporates a form of household production that gives rise to economies of scale in consumption. We extend this model to allow individual preferences to depend on household composition. We test whether singles and members of couples of the same gender have the same preferences. This hypothesis is rejected,

suggesting that preferences may change when household composition changes. Based on the estimates of the collective model we produce indifference scales for singles and

members of couples consistent with changing preferences. We find that on average women need 76.37% and men need 75.69% of their former household’s expenditure to be

indifferent between living in their household and living alone.

Keywords: Collective model, Equivalence Scales, Household Demand, Indifference Scales, Preference Changes

2 1. Introduction

The focus of this paper is on explaining the expenditure patterns of households consisting of single persons and couples. In particular we seek to explain the difference between the expenditure patterns of singles and couples. We do so with references to the potentially different preferences that otherwise similar members of such household have. We take into account the possibility that preferences over the consumption of goods and services are household composition dependent. We recognize that when two people start living together there are three important changes relative to their previous situation. First, the members of the newly formed household will no longer have complete discretion over their spending. Instead both household members will typically have some influence on the allocation of household resources. We assume decisions are made in cooperation as in the model of Chiappori (1988, 1992). Second, the household members may share in the use of purchased goods. The multiple person household thereby realizes a more efficient use of goods

compared to a single person household. We adopt the approach of Browning et al. (2013): Efficiency gains are modelled as economies of scale in a household production process. Third, individual preferences over goods and services may be different under the new household composition. These arguments apply with minor (and obvious) differences when people stop living together. Preference changes remain an unexplored explanatory factor for the difference in expenditure patterns of singles and couples. Perhaps this is the case

because the assumption that preferences are an immutable aspect of an individuals

(consumers) psyche is of great practical convenience. The assumption facilitates comparison of an individual’s welfare in households of varying composition, see e.g. Browning et al. (2013), and Cherchye et al. (2012a). However, besides its practical convenience the assumption does not seem to have many virtues. Casual observation of people would suggest that little about their behavior is constant. People can be observed to change their mind regularly. For instance, observations of time inconsistency in economic behavior can be regarded as instances where individuals preferences have changed over time. In this paper we test whether the assumption that preferences are independent of household

3 Answering the aforementioned research question is important for at least three reasons. First, the assumption seems inconsistent with anecdotal evidence suggesting that an individual does not buy the same goods over their lifetime. Consider for example the demand as an individual ages for goods such as music festival tickets or insurance policies. On a more highly aggregated level, individuals can be observed to allocate different shares to broad categories of goods during their life. Changes in consumption patterns over a persons lifetime can in part be explained by changes in material conditions. Available

resources tend to increase, credit constraints tend to loosen and relative prices change over time. However, mere changes in the budget constraint seem to form an incomplete

explanation of lifetime consumption expenditure patterns. In particular, it is hard to reconcile the exclusive purchase of goods in certain lifecycle stages with price and income based explanations (e.g. corner solutions). We suggest that it is important to consider

preference changes as well. Specifically, we suggest that an important change in preferences occurs when an individual’s household changes in composition. Allowing individual

preferences to depend on household composition could help us better understand and predict changes in individual expenditure allocation. In particular, it may help explain why demand for certain goods is higher for certain household types. The paper focuses on perhaps the most common and important changes in household composition; household formation by singles and dissolution of couples. We recognize that preferences may also depend on an individual’s age. We therefore control for the effect of age on preferences when we investigate whether preferences depend on household composition. To our knowledge this is the first paper that applies the idea that preferences may depend on household composition in order to study the demand of individuals.

Second, the answer to the aforementioned question is especially important in light of the recent popular adoption of collective models of household decision making in empirical studies of household consumption (and labor supply) behavior. In empirical applications of collective models the assumption that preferences do not depend on household

composition is sometimes made to identify the model’s parameters. A non-technical explanation of this approach follows in two steps (though in practise they are often

4 of a couple have the same preferences as otherwise equivalent singles1. We then calculate estimates of the parameters reflecting economies of scale and income sharing. These estimates are calculated so that they rationalize the outcome for a couple given that its constituent members optimally satisfy their (assumed) preferences. The aforementioned assumption thus allows identification of economies of scale and a sharing rule. However, these estimates are biased if the assumption is incorrect. The research question we aim to answer thus amounts to a test of the validity of the identifying assumption of a series of empirically applied collective models. Regardless of the outcome of the tests, the paper will produce estimates of economies of scale, the response of goods expenditure shares to household characteristics, and the response to total expenditure. These estimates make the best possible use of our result regarding household composition dependence of preferences. Third, the answer to the research question has implications for the measurement of

economic inequality and for the design of policies that are aimed to adress inequality or poverty. Inequality is conventionally measured based on corrected household income. The purpose of the correction is to make economic welfare of individuals in households with different compositions comparable. Preferably we would use appropriately estimated equivalence scales for this purpose. Equivalence scales (or more accurately indifference scales in the spirit of Browning et al. (2013)) tell us how much income a person would need living alone to be as well of as he/she would be when living as part of a couple. The

prevailing practice is to use the Organization for Economic Co-operation and Development (OECD) equivalence scales to correct household income. Unfortunately, OECD equivalence scales only aim to correct for the difference in economies of scale between differently composed households. As argued above, economies of scale do not constitute the only difference between households of different compositions. Multiple person households also share resources. However, resources are not necessarily shared equally. Inequalities within the household are ignored by the prevailing practise, which implicitely assumes resources are shared equally. As Lise and Seitz (2011) have shown, the prevailing practice

underestimates the extend of inequality by as much as 50% compared to an approach that accounts for intra-household inequality. Equivalence scales based on a parameterized

1

5 collective model, as estimated in this paper, do account for intra-household inequality. The consequence is that the improved scales for women (men) would tend to be higher (lower) than conventional equivalence scales if womens’ share of total expenditure exceeds 50%. This effect reflects on the one hand the increased (decreased) benefit to women (men) of being part of a couple. On the other hand it reflects the greater (smaller) loss in welfare in the event of dissolution of the couple. Indifference scales serve to make the welfare of actual individuals comparable across differently composed households. As opposed to traditional equivalence scales which only make welfare comparable under the restrictive (and emperically rejected) assumption of equal sharing. Income corrected by the improved scales is therefore a more appropriate measure of individual welfare. On the basis of this corrected income measure of individual welfare we can measure economic inequality at the individual level. Furthermore, the indifference scales can be used to determine poverty rates or program eligibility (income) thresholds for couples when such income levels are known for singles. Also note that the earlier definition of equivalence scales implies that the scales for couples measure the loss in effective income and the loss in efficiency of consumption resulting from the loss of a partner. Equivalence scales may therefore be used to calculate appropriate levels of income to replace the pension of a deceased spouse, desired live insurance levels, and compensation for economic losses in the event of wrongful death of a spouse. This discussion should illustrate the importance of accurately estimating the inputs of equivalence scales; estimates of the efficiency gain in consumpion in larger households (economies of scale) and the relation between household characterisics and the share of consumpion each member gets (the sharing rule). We should not make the assumption that preferences of singles and members of couples are the same without testing it. Otherwise we do not know whether we can be confident in the accuracy of equivalence scales. In this paper we test the aforementioned assumption. We produce equivalence scales that are consistent when preferences do depend on household composition. The scales can be used to improve measurement of inequality and poverty rates. In summary, we estimate

equivalence scales and test the assumption that guarantees accuracy of these scales. In order to test our hypothesis we develop a collective model in the spirit of Browning et al. (2013) which does not restrict the preferences of members of couples. The model is

6 and Consumption dataset. The LISS data records expenditure of individual members of households on a number of goods and services. In addition it records the expenditure of their household on several unassignable goods. The dataset has a (relatively) large number of assignable goods. Other attractive properties of the dataset are its size and

representativeness. The individual expenditures data allows us to relax the assumption that preferences over all goods are the same for singles and members of couples. Specifically, the data allows us to estimate how preferences over assignable goods change upon moving from a single to a multiple person household (and vice versa). The assumption that individual preferences of singles and members of couples are the same for all assignable goods is testable in our model. We find that the assumption is rejected by the data for both men and women. In other words, individual preferences change upon formation/dissolution of a household. We are also able to estimate how strong economies of scale in the consumption of unassignable goods are for couples. Furthermore, we partially observe and partially infer the shares consumed by each member of a household. The estimates of economies of scale and the estimates of income shares are used as inputs to construct indifference scales. These scales are consistent with our observation that individual preferences change (but do not compensate for such changes).

7 2. Literature

Early analyses of household economic decision making pictured a household as a

representative agent satisfying its own set of preferences given a household budget. This “unitary” approach conveniently allowed economist to apply the tools of microeconomic analysis at the level of the household. However, the assumption that household have their own preference ordering might be inconsistent with one of the key ideas behind

microeconomic analysis: The units of analysis is an actor with its own set of preferences. A household consists of multiple individuals. Each of these individuals has their own set of preferences. The household preference ordering, if such a thing exists, should be derived from the preference orderings of its members.

The main problem with the unitary approach is that the prediction of income pooling is routinely rejected (see below). The unitary approach implies that the source of income is irrelevant to household behavior. Regardless of the source or recipient, all income becomes part of a single budget. The household uses this budget to satisfy its preferences. This income pooling hypothesis can be tested by varying the recipient of some form of income. If varying the recipient affects household level outcomes then we can reject the hypothesis. Amongst others, the income pooling hypothesis has been rejected in Blumberg (1988), Bourguignon et al. (1993), and Browning et al. (1994). Phipps and Burton (1998) find evidence of income pooling for some consumption goods but not for others. Factors reflecting the relative economic power of household members have also consistenly been found to affect household level outcomes. This has stimulated the development and adoption of models that explicitely consider multiple decision makers in households. In contrast to the unitary models, collective models are characterized by the fact that

household behavior is influenced by the preferences of multiple members of the household. The so-called cooperative class of collective models assumes that household members can make binding agreements with each other2. As a result household members can agree to an allocation of resources that maximizes the gains from the joint household. This can be seen as a Nash equilibrium in a cooperative game. As Chiappori (1988) has shown the decision

2

8 process can alternatively be represented as a two stage process. In the first stage household members bargain over the share of total resources each member gets to spend. In the second stage each household member allocates their share of expenditure to satisfy their own preferences. Authors have made varying assumptions about the first stage bargaining process. Specifically about the threat points which determine gains from a joint household. Manser and Brown (1980) define gains from marriage relative to the outcomes achieved if a couple divorces. Lundberg and Pollak (1993) define gains relative to a non-cooperative solution.

Chiappori (1988,1992) develops a more general model which only assumes that the outcome is Pareto efficient. The original model focused on leisure and a Hicksian composite good. The collective model of Chiappori (1992) was later extended to deal with household production in Apps and Rees (1997), multiple private goods in Chiappori (2011), public goods in

Chiappori and Ekeland (2009), and goods that are neither purely private nor purely public in Browning et al. (2013). Below we follow a large part of the collective household decision literature by assuming that the members of the household reach a Pareto efficient division of resources over members and goods.

The main challenge in explaining the differences in expenditure patterns of single and

multiple person households lies in distinguishing between the effects of preference changes, of economies of scale in consumption, and of the relative influence of the members of a multiple person household. In theory it is possible to distinguish between all three of these factors. In practice data limitations make this task infeasible. The effect of one or two of the possible explanatory factors might still be studied. To do so we need to be able to rule out the influence of at least one of the factors. Past contributions have generally restricted preferences and/or economies of scales in order to study the intra-household distribution of resources. For example Cherchye et al. (2012a) assume that preferences of members of an elderly couple are the same as elderly widow(ers). They combine couples data with

9 difference is that we simultaneously allow changing preferences and economies of scale to play a role in explaining households’ expenditure shares. However, our model does not allow the effects of preference changes and economies of scale to act on the same goods. Goods may be subject to economies of scale or household composition dependent preferences, but not both. Furthermore we test the restriction on preference changes that has been made in past contributions.

3. Theory

As discussed above there are several factors that may explain the difference in expenditure patterns of single person households and multiple person households. In general we have an identification problem: There is a continuum of explanations for the difference between singles and couples expenditure which vary in terms of the strength of economies of scale and preference changes. Preference changes may be studied in a collective model by imposing more structure on the general collective model. We thereby rule out one or more of the possible explanations for different expenditure patterns. Several possibilities can be thought of. Assumptions about the public or private nature of goods can help to identify preference changes (see Browning et al. 2013, section 6.3 for a discussion). Alternatively, restricting preferences can help to identify economies of scale in the consumption of goods. The approach of this paper is to do a little of both. In this section we consider a special case of the cooperative collective model where goods3 are either assignable and purely privately consumed or publically consumed and subject to economies of scale.

3.1. The individual model

To build a model that allows preferences to depend on household composition we assume that individual utility is additively separable in private and public goods. Such individual preferences may be represented by the direct utility function4

3

It is possible to extend the model to include the demand for leisure/supply of labor. This is outside the scope of the current paper. We consider this extension an interesting avenue for future research.

4

10 𝑈(𝒒𝒑𝒓_{, 𝒒}𝒑𝒖_{, 𝒛) = u}𝑝𝑟_(𝒒𝒑𝒓_{, 𝒛) + 𝑒}𝛾(𝒛)_u𝑝𝑢_(𝒒𝒑𝒖_{, 𝒛), (1)}

where 𝒒𝒑𝒓_{and 𝒒}𝒑𝒖 _{in equation (1) are vectors of respectively private and public good} quantities, 𝒛 is a vector of background variables assumed to affect preferences, and 𝛾(𝒛) measures the preference for public relative to private expenditure. Additive separability in private and public goods is a strong assumption and not one that is well supported by the literature. As a consequence we will have to take care in interpreting result from the additively separable model. Nonetheless, the additively separable model serves as a convenient starting point to study preference changes. The model allows us to look at

subsystems of demand for which we may have information of different quality. Consumption of public goods is per definition not observable at the individual level, although the use and benefit to individuals may sometimes be inferred from complementary data5. On the other hand, consumption of private goods can in principle be assigned to household members. In the empirical analysis we will use data about “public goods” expenditure at the household level. We will also use “private goods” expenditure available at the level of the household member. In this sense the consumption data differs in quality. Individually assignable expenditure data of the kind used in this paper is necessary to study preference changes. This important feature of the data is discussed further in the data section (4.1). The benefit of additive separability lies in the fact that we can make different assumptions for goods in the private and public subsystem regarding the role of preferences changes and economies of scale. Specifically, we will look at purely private goods in the sense that the goods are assumed to be rival and excludable. Preferences over these goods will be allowed to differ between singles and members of couples. Public goods are assumed to be subject to economies of scale. Preferences over these goods are assumed to be the same for singles and members of couples. Due to additive separability the assumption that some goods are purely private does not affect the allocation in the public goods subsystem. Likewise, the assumption that preferences for public goods do not change does not affect the allocation over private goods. These modelling choices make the most of the fact that we have

individual household member’s consumption data. The assumptions allow us to investigate whether preferences depend on household composition.

5 _{For example, while a car can be considered a household public good it may be possible to infer the benefit the}

11 In the single person household, the sole member of the household seeks to maximize an individual utility function. We may represent the problem as the optimization program (2)

max_𝒒𝒑𝒓_,𝒒𝒑𝒖𝑈(𝒒𝒑𝒓, 𝒒𝒑𝒖, 𝒛) (2)

s. t. 𝑥 = 𝑥𝑝𝑟_{+ 𝑥}𝑝𝑢 _{= 𝒑}𝒑𝒓′_𝒒𝒑𝒓_{+ 𝒑}𝒑𝒖′_𝒒𝒑𝒖_,

where 𝒑𝒑𝒓_{and 𝒑}𝒑𝒖 _{are vectors of respectively private and public good prices, and 𝑥}𝑝𝑟 _{and 𝑥}𝑝𝑢 are private and public total expenditure. Under the assumption of additive separability the single person’s allocation problem is equivalent to a two stage budgeting process. In the first stage the single person allocates expenditure between public and private goods. In the second stage the budgets for private and public goods are allocated to goods within those categories. We may therefore alternatively represent the individual’s problem in terms of maximization of the indirect utility function in program (3)

𝐹𝑖𝑟𝑠𝑡 𝑠𝑡𝑎𝑔𝑒: max_𝑥𝑝𝑟_,𝑥𝑝𝑢𝑉(𝑥𝑝𝑟, 𝑥𝑝𝑢, 𝒑𝒑𝒓, 𝒑𝒑𝒖, 𝒛) = Ψ𝑝𝑟(𝑥𝑝𝑟, 𝒑𝒑𝒓, 𝒛) + 𝑒𝛾(𝒛)Ψ𝑝𝑢(𝑥𝑝𝑢, 𝒑𝒑𝒖, 𝒛) (3) s. t. 𝑥 = 𝑥𝑝𝑟_{+ 𝑥}𝑝𝑢_,

where 𝛹𝑝𝑟 _{and 𝛹}𝑝𝑢 _{are the indirect sub-utility functions for respectively private and public} goods. We find the functions 𝛹𝑝𝑟 _{and 𝛹}𝑝𝑢 _{when we plug the solution to the following} second stage program (4) back into the direct sub-utility functions 𝑢𝑝𝑟 and 𝑢𝑝𝑢:

𝑆𝑒𝑐𝑜𝑛𝑑 𝑠𝑡𝑎𝑔𝑒: max_𝒒𝒑𝒓u𝑝𝑟(𝒒𝒑𝒓, 𝒛) (4)

s. t. 𝑥𝑝𝑟_{= 𝒑}𝒑𝒓′_𝒒𝒑𝒓

and

max_𝒒𝒑𝒖u𝑝𝑢(𝒒𝒑𝒖, 𝒛) s. t. 𝑥𝑝𝑢 _{= 𝒑}𝒑𝒖′_𝒒𝒑𝒖_.

We have explicitly recognized everywhere that the utility function depends on an individual’s background variables. Perhaps the most important of these variables is the gender of the individual. The superscript 𝑘 = 𝑐𝑓, 𝑐𝑚 is used below to distinguish between functions and variables for women and men in the (couple) household model. We

12 3.2. The household model

As the focus of the paper is on the difference between expenditure patterns of singles and couples only the two person household model is discussed. The allocation of resources within the household is assumed to be Pareto efficient. The household can therefore be seen to maximize a weighted sum of individual utilities. The household problem may be represented as

max_𝒒𝒄𝒇,𝒑𝒓_,𝒒𝒄𝒇,𝒑𝒖_,𝒒𝒄𝒎,𝒑𝒓_,𝒒𝒄𝒎,𝒑𝒖𝑈ℎ= 𝜇𝑈𝑐𝑓(𝒒𝒄𝒇,𝒑𝒓, 𝒒𝒄𝒇,𝒑𝒖) + 𝑈𝑐𝑚(𝒒𝒄𝒎,𝒑𝒓, 𝒒𝒄𝒎,𝒑𝒖), s. t. 𝑥 = 𝒑𝒑𝒓′_𝒒𝒄𝒇,𝒑𝒓_{+ 𝒑}𝒑𝒓′_𝒒𝒄𝒎,𝒑𝒓_{+ 𝒑}𝒑𝒖′_𝒒𝒄𝒇,𝒑𝒖_{+ 𝒑}𝒑𝒖′_𝒒𝒄𝒎,𝒑𝒖_,

where 𝒒𝒄𝒇,𝒑𝒓_{, 𝒒}𝒄𝒇,𝒑𝒖_{, 𝒒}𝒄𝒎,𝒑𝒓_{, 𝒒}𝒄𝒎,𝒑𝒖 _{are consumed quantities of goods (the “consumed” part is} explained below) and 𝜇 is the weight that the wife’s utility has in the household utility

function. The household problem can alternatively be represented by a two stage process. In the first stage the couple allocates expenditures over private and public systems for each member6 satisfying the program

𝐹𝑖𝑟𝑠𝑡 𝑠𝑡𝑎𝑔𝑒: max_𝑥𝑐𝑓,𝑝𝑟_,𝑥𝑐𝑓,𝑝𝑢_,𝑥𝑐𝑚,𝑝𝑟_,𝑥𝑐𝑚,𝑝𝑢𝑉ℎ= 𝜇𝑉𝑐𝑓(Ψ𝑐𝑓,𝑝𝑟(𝑥𝑐𝑓,𝑝𝑟, 𝒑𝒑𝒓), Ψ𝑐𝑓,𝑝𝑢(𝑥𝑐𝑓,𝑝𝑢, 𝒑𝒑𝒖)) (5) +𝑉𝑐𝑚_(Ψ𝑐𝑚,𝑝𝑟_(𝑥𝑐𝑚,𝑝𝑟_{, 𝒑}𝒑𝒓_{), Ψ}𝑐𝑚,𝑝𝑢_(𝑥𝑐𝑚,𝑝𝑢_{, 𝒑}𝒑𝒖_)),

s. t. 𝑥 = 𝑥𝑐𝑓,𝑝𝑟_{+ 𝑥}𝑐𝑚,𝑝𝑟_{+ 𝑥}𝑐𝑓,𝑝𝑢_{+ 𝑥}𝑐𝑚,𝑝𝑢_.

The functions Ψ𝑐𝑓,𝑝𝑟_{, Ψ}𝑐𝑓,𝑝𝑢_{, Ψ}𝑐𝑚,𝑝𝑟_{, Ψ}𝑐𝑚,𝑝𝑢 _{in program (5) are found by substituting the} solution of the following second stage program (6) back into the direct sub-utility functions u𝑐𝑓,𝑝𝑟_{, u}𝑐𝑓,𝑝𝑢_{, u}𝑐𝑚,𝑝𝑟_{, u}𝑐𝑚,𝑝𝑢_: 𝑆𝑒𝑐𝑜𝑛𝑑 𝑠𝑡𝑎𝑔𝑒: max_𝒒𝒌,𝒑𝒓u𝑘,𝑝𝑟(𝒒𝒌,𝒑𝒓, 𝒛𝒌) (6) s. t. 𝑥𝑘,𝑝𝑟_{= 𝒑}𝒑𝒓′_𝒒𝒌,𝒑𝒓 and 6

We could also have represented the household problem’s first stage in terms of two sub-stages. In the first stage the household members agree to divide total expenditure between them. In the second stage they individually choose to divide their own share between expenditure on public and private goods (they maximize

𝑉𝑘 _{with respect to 𝑥}𝑘,𝑝𝑟 _{and 𝑥}𝑘,𝑝𝑢_{). There is no loss in generality in assuming that household members}

immediately agree on a division in terms of a separate budget for private and public goods for each member.

13 max_𝒚𝒌,𝒑𝒖u𝑘,𝑝𝑢(𝒒𝒌,𝒑𝒖, 𝒛𝒌)

s. t. 𝑥𝑘,𝑝𝑢 _{= 𝒑}𝒑𝒖′_𝒚𝒌,𝒑𝒖 _{and 𝒒}𝒌,𝒑𝒖_{= 𝑭(𝒚}𝒄𝒇,𝒑𝒖_{, 𝒚}𝒄𝒎,𝒑𝒖₎

for 𝑘 = 𝑐𝑓, 𝑐𝑚

where 𝒚𝒄𝒇,𝒑𝒖 _{and 𝒚}𝒄𝒎,𝒑𝒖 _{are vectors of purchased quantities of goods}7_{. The consumption} technology function 𝑭(𝒚𝒄𝒇,𝒑𝒖_{, 𝒚}𝒄𝒎,𝒑𝒖_{) captures the relationship between purchased and} consumed quantities of goods. The distinction between purchased and consumed quantities of goods reflects the efficiency gains from sharing consumption goods in a couple. We follow Browning et al. (2013) by modelling efficiency gains in consumption as economies of scale in a household production process. Households transform purchased goods into consumable goods. The process whereby consumable goods are produced is subject to economies of scales. We assume that singles consume purchased goods (which is a normalization of sorts), but couples transform purchased goods into larger consumable quantities of the same goods8. For a given quantity of purchases by a member of a couple the corresponding consumable quantity tells us how much the member of that couple would need to purchase as a single to enjoy the same consumption. In this sense we can interpret 𝒒𝒌,𝒑𝒖 _{as the vector} of singles equivalent consumption corresponding to purchases 𝒚𝒌,𝒑𝒖 _{for 𝑘 = 𝑐𝑓, 𝑐𝑚. The} latter implies that for singles households we have 𝒒𝒌,𝒑𝒖_{= 𝒚}𝒌,𝒑𝒖 _{for 𝑘 = 𝑠𝑓, 𝑠𝑚. Also note that} per definition 𝒒𝒌,𝒑𝒓_{= 𝒚}𝒌,𝒑𝒓 _{for 𝑘 = 𝑠𝑓, 𝑠𝑚, 𝑐𝑓, 𝑐𝑚 as there are no economies of scale in}

private goods. The consumption technology captures these ideas by specifying a relationship between purchased goods and consumable goods for couples. We use a simple linear

technology given by 𝑞_𝑔𝑘,𝑝𝑢 = 𝑑𝑔−1𝑦𝑔𝑘,𝑝𝑢 for 𝑘 = 𝑐𝑓, 𝑐𝑚 and 𝑔 = 𝑚1+ 1, … , 𝑚2. This is an application of Barten (1964)’s model of demographic demand scaling, where the

demographic factor is a couple dummy. The 𝑑_𝑔 parameters can thus be interpreted as Barten scales. The parameters are assumed to lie between 0.5 (perfect public goods) and 1 (perfect

7

The choice variables in the subsystem for public goods in program 6 are purchased quantities. This reflects the fact that in general the consumption of one partner may depend on purchases by the other partner. In such a case consumable quantities are not a direct object of choice. Below we rule out this possibility and represent the problem as a choice in terms of consumable quantities.

8

14 private good). These values reflect the natural upper and lower bound to sharing in the consumption of purchased goods. A couple can jointly enjoy at most twice the consumption a single person enjoys from the same good. A value of 𝑑𝑔 < 0.5 is therefore implausible. A couple can choose not to share goods at all. A value of 𝑑𝑔 > 1 is therefore implausible. In the empirical application we assume that all 𝑑𝑔take the same value 𝑑. The parameter 𝑑 can be interpreted as an Engel scale within the public goods subsystem.

Corresponding to singles equivalent consumable quantities of goods we can find Lindahl prices. Lindahl prices are nominal prices adjusted for economies of scale in consumption. They are the effective prices paid by members of couples for a singles equivalent unit of consumption. In terms of our model and technology function we find Lindahl prices given by 𝑑_𝑔𝑝_𝑔𝑝𝑢. Due to the assumption of a linear consumption technology all couples (and members of couples) face the same set of effective prices. It turns out to be more convenient to normalize Lindahl prices to 𝑑_𝑔𝑝_𝑔𝑝𝑢_⁄𝑥𝑝𝑢_{. Rational individuals are assumed not to suffer from} money illusion. Individuals should therefore choose the same allocation over goods whether they face a budget equal to 𝑥𝑝𝑢 _{and prices equal to 𝑑}

𝑔𝑝𝑔𝑝𝑢 or their share of the total public goods budget 𝜂𝑘,𝑝𝑢 _{= 𝑥}𝑘,𝑝𝑢_⁄𝑥𝑝𝑢 _{and prices equal to 𝑑}

𝑔𝑝𝑔𝑝𝑢⁄𝑥𝑝𝑢. Applying the normalization allows us to represent the individual household member’s problem in a way that looks very similar to the second stage of the single’s allocation problem in program (4). In this

representation household members choose in terms of consumable quantities of goods. A member of a couple solves the problem given by program (7)

15 The interpretation of the general model (with non-linear consumption technology) is that each member of a couple tries to find the best response in terms of their own utility to their spouse’s allocation. The individual member’s equilibrium allocation is the best allocation they can reach if they evaluate the marginal unit cost of goods at Lindahl prices consistent with their spouse’s equilibrium allocation. With a linear consumption technology the Lindahl price is independent of the spouse’s allocation. Note that the assumption that preferences of singles and members of couples within the public goods subsystem are the same means that that u𝑐𝑓,𝑝𝑢 = u𝑠𝑓,𝑝𝑢and u𝑐𝑚,𝑝𝑢= u𝑠𝑚,𝑝𝑢. Therefore we can derive the demand functions for public goods of couples members and singles from program (4). The demand for public goods of singles and members of couples differs only in that the latter evaluate the demand function at (normalized) Lindahl prices and their share of household public goods

expenditure. We derive individual demand functions, individual budget shares, household demand functions, and household budget shares in Appendix A. Section 4 discusses the data we use to test our hypothesis. Limited price variation will necessitate us to develop a simpler model than the one discussed above. We specify a parametric model that allows

straightforward testing of the hypothesis that preferences of singles and otherwise similar members of couples are the same.

4. Estimation 4.1. Data

16 detailed information about average monthly household expenditures over the previous year. Data is available on a number of (presumed) publicly consumed goods, and a number of (presumed) privately consumed goods.

The population of interest consists of childless singles and childless heterosexual couples where each member of the household is below the age of retirement of 65 (for the interval of time considered). There are three waves of data available (2009, 2010, and 2012) with repeated observations on some households. We pool the waves of data and correct for the resulting correlation between errors of observations by using standard errors clustered at the individual level.

Household level expenditure data is recorded for the following public goods in the TUC module; mortgage interest and amortization, rent, utilities, transport and means of transport9, insurance, alimony and child support, servicing debts and loans, cleaning and maintenance of the house and garden, food consumed at home, and other public goods. The present study focuses on the allocation of total expenditure over goods and services. We therefore disregard the expenditures made to service debts and loans and expenditures on alimony and child support. These categories do not represent goods or services. Expenditures on “other” public goods are disregarded because we cannot check whether this category actually contains public expenditures. The treatment of housing expenditures requires some extra attention. Note that rent and mortgage are not really comparable. Rent is a good measure of the cost of enjoying a rented house. Mortgage interest and amortization on the other hand is a poor indicator of the cost of enjoying an owned home. The sum of amortization and interest has no relation over time with the benefits of living in a house. Furthermore, after a mortgage has been repaid the owners of a house can still use the house. We would prefer to use imputed rent. Imputed rent measures the opportunity cost of living in an owned house. However, the mortgage measure is the best data we have available. Expenditures on mortgage payments and rent are combined into a category we will simply refer to as housing. We use homeownership as a taste shifter in part to control for the effects of ignoring the difference between mortgage and rent.

9

Transport and means of transport are recorded as one category. It includes expenditures on public

17 Each respondent was asked to report expenditures for their personal consumption on a number of private goods. The TUC-module uses the following private goods categories: expenditure on food and drinks outside the house, tobacco products, clothing, personal care products and services, medical care and health care cost not covered by insurance, expenditure on leisure time activities, schooling, donations and gifts, and other private goods. Medical care costs are disregarded because they do not represent the outcome of an individual’s (own) choice. Schooling, donations and gifts, and other private goods are disregarded because the vast majority of the sample reports zero expenditures, and many of the other respondents report negligible amounts.

Respondents were also asked to provide the value of the food they personally consumed at home. We could have combined this item with food and drinks consumed outside the house (“eating out” hereafter). Such a category could have been treated as a private good. Instead we choose to treat eating at home as a separate (public) good. We suspect that eating at home is subject to economies of scale. This would be ignored by combining it with eating out. Furthermore, we believe that eating and drinking outside the home is in part enjoyable as a social activity. The social aspect makes eating out distinct from eating at home. We use the share each partner consumes of food at home as a partial observation of the sharing rule for public goods. This is discussed further below.

Not all available observations were considered, only those with complete data. There were a variety of reasons why observations were deemed incomplete. First, an observation is discarded if expenditure on any private or public good category is entirely missing. Second, we consider only observations which have total private and public goods expenditure greater than zero. Third, in couples the constituent members were given the option of deferring to their spouse when it came to reporting public expenditures. In some couples both members deferred to their spouse and these observations were disregarded10. Fourth, we considered an observation incomplete if zero expenditure was reported for the sum of mortgage and rent and the household does not consist of older homeowners. These cases

10 _{Other cases were handled as follows: When only one member of a household reports strictly positive}

18 were interpreted as mistakes. If the household is renting a home or consists of young

members with a mortgaged home, then zero expenditure is implausible11. Since housing is a major expenditure for most households, these instances of miss-measurement could not be accepted. However, a majority of the individuals involved live in cost-free housing. This is likely a sign that the individuals involved are not completely financially independent. We are interested in households with complete control over their finances. Fifth, we disregarded individuals who reported zero personal food at home expenditure. We use an individual’s share of food expenditure as a measure of their influence on allocating the public goods budget. Observations with zero food expenditure cannot be used for this purpose. We disregard these observations to keep a consistent sample throughout the paper. Sixth, preliminary analysis revealed that results were sensitive to the inclusion of certain outliers. These observations are outliers in the sense that there is a large positive difference between total goods expenditure and net household income (exceeding €10.000 per month). These households may have been substantially digging in to their savings in reality. However, given the implied level of dissaving this seems unlikely. The outliers complicate finding an answer to our hypothesis that applies to the general population. We therefore disregard them. Seventh, if the spouse of a member of a couple has incomplete private goods data, then that member’s data is not used. In Table 1 in Appendix B we report the loss of observations from each selection criterion. The final sample contains 577 observations on single women, 561 observations on single men, and 834 observations on couples. Descriptive statistics for expenditure on the public goods and private goods are reported in Table 2 in Appendix B. We present a more detailed description of expenditure shares in Table 3 in Appendix B. Table 2 also reports descriptive statistics for the taste shifters and distribution factors in our model. The taste shifters we include are a homeownership dummy, a higher education dummy and age. Homeownership may affect the allocation. For example by making

expenditure on cleaning and maintenance, furnishing, and insurance more attractive relative to other goods. Education may affect preferences by promoting healthy choices.

Alternatively it may affect preferences through an individual’s peer/social network. The higher education dummy equals 1 if the respondent has received a degree in the higher education system (higher vocational or university level). Regional differences may play a role

11 _{One possibility is that the household is delinquent in payment. Unfortunately, we have no way of}

19 in determining preferences. The difference between cities and rural areas seems the most relevant distinction within the Netherlands. We hypothesized that living in a rural or urban area may affect allocations because it affects the costs and availability of certain goods. Housing tends to be cheaper in rural areas while many other goods and services are more costly to obtain. The urban area of residence dummy measures whether the household’s residence is in an area with more than 1500 addresses per square kilometer 12. Preliminary analysis has shown that living in an urbanized area has no significant effect on expenditure shares. We therefore disregard this factor in the subsequent analysis. We also consider age as a taste shifter. Preliminary analysis has shown that age is an important explanatory factor in the public but not in the private goods subsystem. We include age in both subsystems to keep a consistent set of explanatory variables in the demand system. Preliminary analysis also clearly rejected that there is heterogeneity in the response of demand to total private expenditure. In other words we found no effect of interactions between taste shifters and total private expenditure on expenditure shares in the private goods subsystem. In the public goods system we did find strong evidence of this kind of preference heterogeneity13. Maintaining consistency between the private and public subsystem comes at too high a cost in this case. We choose to allow interactions between taste shifters and total expenditure in the public goods subsystem but not in the private goods subsystem.

4.2. Parametric specification

The dataset we use has limited price variation due to the fact that it is a short panel study. We will therefore specify a parametric model with limited interaction between prices of goods. Our starting point for utility from consumption of public or private goods is the Almost Ideal Demand System of Deaton and Muellbauer (1980). This model allows expenditure shares to depend on income in a flexible way. We restrict to zero the

parameters of products between the logarithms of prices which normally appear in such a model. We obtain an individual indirect utility function given by equation (8)

𝑉(𝑥𝑝𝑟_{, 𝑥}𝑝𝑢_{, 𝒑}𝒑𝒓_{, 𝒑}𝒑𝒖_{, 𝒛) = Ψ}𝑝𝑟_(𝑥𝑝𝑟_{, 𝒑}𝒑𝒓_{, 𝒛) + 𝑒}𝛾(𝒛)_Ψ𝑝𝑢_(𝑥𝑝𝑢_{, 𝒑}𝒑𝒖_{, 𝒛), (8)}

12

This is the definition of an urban area used by Statistics Netherlands.

13 _{This preliminary analysis is based on a model of expenditure shares with interactions between taste shifters}

20 Ψ𝑝𝑟₌ln 𝑥𝑝𝑟−ln 𝑎𝑝𝑟(𝒑𝒑𝒓,𝒛)

𝑏𝑝𝑟_(𝒑𝒑𝒓₎ , Ψ𝑝𝑢=

ln 𝑥𝑝𝑢_{−ln 𝑎}𝑝𝑢_(𝒑𝒑𝒖_,𝒛)

𝑏𝑝𝑢_(𝒑𝒑𝒖₎ ,

where we have suppressed the gender index 𝑘, we have m1 privately consumed goods and m2-m1 publically consumed goods, 𝑥𝑝𝑟 is total spending on private goods, 𝑥𝑝𝑢 is total spending on public goods, 𝒑𝒑𝒓_{= [𝑝}

1, … , 𝑝𝑚1] ′ , 𝒑𝒑𝒖= [𝑝𝑚1+1, … , 𝑝𝑚2] ′ , and ln 𝑎𝑝𝑟_{(𝒑) = 𝛼} 𝑐 𝑝𝑟_{+ ∑ 𝛼} 𝑖𝑝𝑟 𝑚1 𝑖=1 (𝒛) ln 𝑝𝑖, 𝑏𝑝𝑟(𝒑) = ∏ 𝑝_𝑖𝛽𝑖 𝑝𝑟 𝑚1 𝑖=1 , ln 𝑎𝑝𝑢_{(𝒑) = 𝛼} 𝑐𝑝𝑢+ ∑𝑚𝑖=𝑚2 1+1𝛼𝑖𝑝𝑢(𝒛) ln 𝑝𝑖, 𝑏 𝑝𝑢_{(𝒑) = ∏ 𝑝} 𝑖 𝛽_𝑖𝑝𝑢(𝒛) 𝑚2 𝑖=𝑚1+1 .

Due to additive separability of the utility function in terms of private goods and public goods we know that the cost function is the sum of cost functions for the private and public good subsystems. The cost function is given by

𝑐(𝑉, 𝒑) = 𝑐𝑝𝑟_(Ψ𝑝𝑟_{, 𝒑}𝒑𝒓_{, 𝒛) + 𝑐}𝑝𝑢_(Ψ𝑝𝑢_{, 𝒑}𝒑𝒖_{, 𝒛),}

where

ln 𝑐𝑝𝑟_(Ψ𝑝𝑟_{, 𝒑}𝒑𝒓_{, 𝒛) = ln 𝑎}𝑝𝑟_(𝒑𝒑𝒓_{, 𝒛) + Ψ}𝑝𝑟_𝑏𝑝𝑟_(𝒑𝒑𝒓_),

ln 𝑐𝑝𝑢_(Ψ𝑝𝑢_{, 𝒑}𝒑𝒖_{, 𝒛) = ln 𝑎}𝑝𝑢_(𝒑𝒑𝒖_{, 𝒛) + Ψ}𝑝𝑢_𝑏𝑝𝑢_(𝒑𝒑𝒖_{, 𝒛).}

Demand functions can be found by applying Roy’s Identity to the cost function. We multiply the demand function by 𝑝_𝑔𝑙_⁄𝑥𝑙 _{for 𝑙 = 𝑝𝑟, 𝑝𝑢 to find the budget shares}

𝑤_𝑔𝑝𝑟= 𝑎_𝑔𝑝𝑟(𝒛) + 𝛽_𝑔𝑝𝑟(ln 𝑥𝑝𝑟_{− ln 𝑎}𝑝𝑟_{(𝒑)) for 𝑔 = 1, … , 𝑚} 1

𝑤_𝑔𝑝𝑢 = 𝑎_𝑔𝑝𝑢(𝒛) + 𝛽_𝑔𝑝𝑢 (𝒛)(ln 𝑥𝑝𝑢_{− ln 𝑎}𝑝𝑢_{(𝒑)) for 𝑔 = 𝑚}

1+ 1, … , 𝑚2.

We recognize that there may be heterogeneity in preferences across individuals. To capture some of this heterogeneity we let some of the preference parameters (𝑎𝑝𝑟_{, 𝑎}𝑝𝑢 _𝛽𝑝𝑢_{, 𝛾}𝑝𝑟_and

𝛾𝑝𝑟_{) depend on the variables gender, homeownership, education and age. We differentiate} between the systems for men and women from here on by adding the superscript m for men and superscript f for women, preceded by s for singles and c for members of couples.

Dependence on homeownership, education, urban area of residence, and age is modelled as follows;

21 𝛽_𝑔𝑘,𝑝𝑟= 𝛽_𝑔,𝑐𝑘,𝑝𝑟, 𝛽𝑔𝑘,𝑝𝑢(𝒛) = 𝛽𝑔,𝑐𝑘,𝑝𝑢+ 𝜷𝒈,𝒛𝒌,𝒑𝒖′𝒛,

𝛾𝑘_{(𝒛) = 𝛾}

𝑐𝑘+ 𝜸𝒛𝒌′𝒛𝒔 for 𝑘 = 𝑠𝑓, 𝑠𝑚, 𝑐𝑓, 𝑐𝑚 and 𝑙 = 𝑝𝑟, 𝑝𝑢,

where 𝒛 contains observations on homeownership, education, and age. We need the restrictions ∑𝑚1 𝑎_𝑖,𝑐𝑘,𝑝𝑟 𝑖=1 = 1, ∑𝑚𝑖=11𝒂𝒊,𝒛𝒌,𝒑𝒓= 𝟎, ∑𝑚𝑖=𝑚2 1+1𝑎𝑖,𝑐𝑘,𝑝𝑢 = 1, ∑ 𝒂𝒊,𝒛 𝒌,𝒑𝒖 𝑚2 𝑖=𝑚1+1 = 𝟎, ∑𝑚2 𝛽_𝑖,𝑐𝑘,𝑝𝑢 𝑖=𝑚1+1 = 0, ∑ 𝜷𝒊,𝒛 𝒌,𝒑𝒖 𝑚2 𝑖=𝑚1+1 = 𝟎, ∑ 𝛽𝑖 𝑘,𝑝𝑟 𝑚1

𝑖=1 = 0 in order for the systems to satisfy adding up. When estimating the model, we drop one private and one public good so the restrictions are automatically satisfied. The parameters for the budget shares of these goods can be recovered using the restrictions.

We derive budget shares that are explicit functions of background variables and prices in Appendix A. Here we only consider the budget shares which we can estimate, those that do not depend on prices. By normalizing all prices to 1 we get the following systems of budget shares 𝑤_𝑔𝑘,𝑝𝑟= (𝑎𝑔,𝑐𝑘,𝑝𝑟− 𝑎0,𝑐𝑘,𝑝𝑟𝛽𝑔,𝑐𝑘,𝑝𝑟) + 𝛽𝑔,𝑐𝑘,𝑝𝑟𝑙𝑛 𝑥𝑝𝑟+ 𝒂𝒈,𝒛𝒌,𝒑𝒓′𝒛𝒌 for 𝑘 = 𝑠𝑓, 𝑠𝑚, 𝑐𝑓, 𝑐𝑚, 𝑤_𝑔𝑘,𝑝𝑢= (𝑎𝑔,𝑐𝑘,𝑝𝑢− 𝑎0,𝑐𝑘,𝑝𝑢𝛽𝑔,𝑐𝑘,𝑝𝑢) + 𝛽𝑔,𝑐𝑘,𝑝𝑢𝑙𝑛 𝑥𝑝𝑢+ (𝒂𝒈,𝒛𝒌,𝒑𝒖− 𝑎0,𝑐𝑘,𝑝𝑢𝜷𝒈,𝒛𝒌,𝒑𝒖) ′ 𝒛𝒌_{+ 𝜷} 𝒈,𝒛 𝒌,𝒑𝒖′_𝒛𝒌_{𝑙𝑛 𝑥}𝑝𝑢 for 𝑘 = 𝑠𝑓, 𝑠𝑚,

where we have suppressed the subscript identifying the observation to avoid clutter. If we were to fix 𝑎_𝑐𝑘,𝑙, then we can estimate the remaining parameters from a model that is linear in reduced form parameters. The econometric model would be substantially simplified. The parameter 𝑎_𝑐𝑘,𝑙 can be interpreted as the logarithm of subsistence spending on private or public goods14. We choose to set 𝑎_𝑐𝑘,𝑙 equal to the log of the 1st percentile of the subsample’s distribution of total spending on private or public goods. A justification of this approach is that the 1st percentile of expenditure can be considered an observation on subsistence spending15. Our results are robust to moderate changes16 in the choice of 𝑎_𝑐𝑘,𝑙.

14

From equation 8 we can observe that if expenditure (within a subsystem of goods) is no higher than 𝑎𝑐𝑘,𝑙 then

(sub-)utility is non-positive. Therefore 𝑎𝑐𝑘,𝑙 can be interpreted as the logarithm of subsistence spending.

15

We could have chosen to set 𝑎𝑐𝑘,𝑙 equal to minimum expenditure. However, for the private goods subsystem

this minimum is around €5 (it differs across subsamples). As a measure of subsistence expenditure this seems to be too low. Private goods include such items as food and clothing. The minimum amount of private

22 4.2.1. Individual demand

To investigate whether preferences for private goods change when moving from a single person household to a multiple person household, we compare the parameter estimates of the system for private goods for singles and members of couples (of the same gender). For the purpose of simplification we assume that assignable goods are fully private in the sense that only one person directly derives utility from them (this definition rules out paternalistic forms of altruism). The assumption implies that the same effective prices of private goods are relevant for all individuals regardless of household form. As a consequence, we do risk mixing up the effects of preference changes and economies of scales in what we assume are private goods.

We estimate the following four systems of equations 𝑤_𝑔𝑘,𝑝𝑟= 𝜃_𝑔,1𝑘,𝑝𝑟+ 𝜃_𝑔,2𝑘,𝑝𝑟𝑙𝑛 𝑝𝑟𝑖𝑣 𝑒𝑥𝑝𝑘_{+ 𝜃}

𝑔,3𝑘,𝑝𝑟ℎ𝑜𝑚𝑒𝑘+ 𝜃𝑔,4𝑘,𝑝𝑟𝑒𝑑𝑢𝑐𝑘+ 𝜃𝑔,5𝑘,𝑝𝑟𝑎𝑔𝑒𝑘+ 𝑢𝑔𝑘, (9)

for 𝑔 = 1, … , 𝑚₁− 1 and 𝑘 = 𝑠𝑓, 𝑠𝑚, 𝑐𝑓, 𝑐𝑚,

where 𝑠𝑓, 𝑐𝑓, 𝑠𝑚 and 𝑐𝑚 are used as shorthand for single women, women in couples, single men and men in couples. We choose to estimate equation (9) based on the expenditure shares of eating out, tobacco, clothing and personal care products. The model is linear in its parameters and the same explanatory variables appear in each budget share. We may therefore estimate the parameters of the system with a relatively simple procedure; by applying ordinary least squares regression to the system equation by equation, and

correcting the covariance matrix of all the system’s parameters for correlation between the error terms of different budget shares for the same observation.In addition, we cluster the standard errors at the individual level to correct for the correlation between error terms of observations of the same individual in multiple waves. The structural parameters estimates

Note that very low expenditure is not necessarily a sign of poverty. The individual may simply eat at home, not

smoke and may have bought sufficient cloths and personal care products in the past. While the 1th percentile is

a somewhat arbitrary choice, setting subsistence spending above minimum expenditure recognizes that some people in the sample may spend less than the subsistence level.

16 _{The main results are virtually unchanged by choices of 𝑎}

𝑐

𝑘,𝑙 _{that are consistent with the notion of subsistence}

expenditure. There are small quantitative (but not qualitative) differences in structural parameter estimates. There are larger quantitative differences for choices that are not consistent with the aforementioned notion.

This occurs when 𝑎𝑐𝑘,𝑙 is set higher than the logarithm of the first quartile of expenditure. However, the main

23 can be easily recovered from the reduced form estimates by using the relation

𝛉_𝒈𝐤,𝐩𝐫= 𝒉_𝒈(𝜻𝒌,𝒑𝒓_{) = [(𝑎} 𝑔,𝑐𝑘,𝑝𝑟− 𝑎0,𝑐𝑘,𝑝𝑟𝛽𝑔,𝑐𝑘,𝑝𝑟), 𝛽𝑔,𝑐𝑘,𝑝𝑟, 𝒂𝒈,𝒛𝒌,𝒑𝒓′] ′ for 𝑔 = 1, … , 𝑚1− 1 and 𝑘 = 𝑠𝑓, 𝑠𝑚, 𝑐𝑓, 𝑐𝑚, where 𝛉_𝒈𝐤,𝐩𝐫= [𝜃_𝑔,1𝑘,𝑝𝑟, 𝜃_𝑔,2𝑘,𝑝𝑟, 𝜃_𝑔,3𝑘,𝑝𝑟, 𝜃_𝑔,4𝑘,𝑝𝑟, 𝜃_𝑔,5𝑘,𝑝𝑟], and 𝜻𝒌,𝒑𝒓_{= [𝛼} 1,𝑐𝑘,𝑝𝑟, 𝜶𝟏,𝒛𝒌,𝒑𝒓′, 𝛽1,𝑐𝑘,𝑝𝑟, … , 𝛼𝑚1−1,𝑐 𝑘,𝑝𝑟 _{, 𝜶} 𝒎𝟏−𝟏,𝒛 𝒌,𝒑𝒓 ′_{, 𝛽} 𝑚1−1,𝑐 𝑘,𝑝𝑟 _].

Our hypotheses are about preferences and therefore focus on the structural parameters of the utility function. We do discuss the reduced form parameters, since they in contrast to the structural parameters are directly interpretable in terms of the behavior of individuals. A similar approach is taken to estimate the preference parameters in the singles demand system for public goods. We estimate the following systems of expenditure shares 𝑤_𝑔𝑘,𝑝𝑢= 𝜃_𝑔,1𝑘,𝑝𝑢+ 𝜃_𝑔,2𝑘,𝑝𝑢𝑙𝑛 𝑝𝑢𝑏 𝑒𝑥𝑝𝑘_{+ 𝜃}

𝑔,3𝑘,𝑝𝑢ℎ𝑜𝑚𝑒𝑘+ 𝜃𝑔,4𝑘,𝑝𝑢𝑒𝑑𝑢𝑐𝑘+ 𝜃𝑔,5𝑘,𝑝𝑢𝑎𝑔𝑒𝑘+ 𝜃𝑔,6𝑘,𝑝𝑢ℎ𝑜𝑚𝑒𝑘∗

𝑙𝑛 𝑝𝑢𝑏 𝑒𝑥𝑝𝑘_{+ 𝜃}

𝑔,7𝑘,𝑝𝑢𝑒𝑑𝑢𝑐𝑘∗ 𝑙𝑛 𝑝𝑢𝑏 𝑒𝑥𝑝𝑘+ 𝜃𝑔,8𝑘,𝑝𝑢𝑎𝑔𝑒𝑘∗ 𝑙𝑛 𝑝𝑢𝑏 𝑒𝑥𝑝𝑘+ 𝑢𝑔𝑘, (10)

for 𝑖 = 𝑚1+ 1, … , 𝑚2− 1 and 𝑘 = 𝑠𝑓, 𝑠𝑚.

We choose to estimate equation (10) based on the expenditure shares of housing, utilities, transport and insurance.

4.2.2. Testing

The main hypothesis of this study is that singles and members of couples of the same gender (and with the same background variables) have the same preferences. Having discussed the method of estimation of preference parameters, we can now formulate a test statistic for our hypothesis. The null hypotheses can be expressed as 𝜻𝒄𝒇,𝒑𝒓_{= 𝜻}𝒔𝒇,𝒑𝒓 _{for women and}

𝜻𝒄𝒎,𝒑𝒓_{= 𝜻}𝒔𝒎,𝒑𝒓 _{for men. We refer to the model in stacked form to motivate the test statistic.} We stack the observations of equation (9) for all individuals to get equation (11)

𝒘_𝒈𝒌,𝒑𝒓= 𝑿𝒌_𝜻 𝒈 𝒌,𝒑𝒓_{+ 𝒖}

24 where 𝑿𝒌 _{contains the explanatory variables of equation (9) including the constant,}

𝜻_𝒈𝒌,𝒑𝒓= [𝛼_𝑔,𝑐𝑘,𝑝𝑟, 𝜶𝒌,𝒑𝒓′_𝒈,𝒛 , 𝛽_𝑔,𝑐𝑘,𝑝𝑟], and 𝒖𝒈 is a vector of error terms. We then stack the expenditure share vectors in equation (11) for all goods, which gives equation (12)

𝒘∙𝒌,𝒑𝒓= 𝑿∙𝒌𝜻𝒌,𝒑𝒓 + 𝒖∙ for 𝑘 = 𝑠𝑓, 𝑐𝑓, 𝑠𝑚, 𝑐𝑚, (12) where 𝒘∙𝒌,𝒑𝒓= [𝒘𝟏𝒌,𝒑𝒓′, … , 𝒘𝒎𝟏−𝟏 𝒌,𝒑𝒓 ′_]′ , 𝑿∙𝒌= 𝑰𝒎𝟏−𝟏⊗ 𝑿𝒌, and 𝒖∙= [𝒖𝟏′, … , 𝒖𝒎𝟏−𝟏′] ′ .

We denote the covariance matrix of 𝜻𝒌,𝒑𝒓 _{by 𝑽}𝒌,𝒑𝒓_{. A test statistic for the hypothesis that} preferences of a single person and an otherwise similar member of a couple differ is given by 𝝉𝒇_{= (𝜻}𝒄𝒇,𝒑𝒓_{− 𝜻}𝒔𝒇,𝒑𝒓₎′_{∗ (𝑽}𝒄𝒇,𝒑𝒓_{+ 𝑽}𝒔𝒇,𝒑𝒓₎−𝟏_{∗ (𝜻}𝒄𝒇,𝒑𝒓_{− 𝜻}𝒔𝒇,𝒑𝒓_{) for women and 𝝉}𝒎_{= (𝜻}𝒄𝒎,𝒑𝒓₋

𝜻𝒔𝒎,𝒑𝒓₎′_{∗ (𝑽}𝒄𝒎,𝒑𝒓_{+ 𝑽}𝒔𝒎,𝒑𝒓₎−𝟏_{∗ (𝜻}𝒄𝒎,𝒑𝒓_{− 𝜻}𝒔𝒎,𝒑𝒓_{) for men}17

. Under the null hypothesis that preferences do not change these test statistics follow the 𝜒202 distribution.

We also consider whether the preferences of women and men are the same for; singles over private goods, for members of couples over private goods, and singles over public goods. The test statistic for the hypotheses that two sets of preference parameters are the same for private or public goods is given by 𝝉𝒌,𝒍,𝒎= (𝜻𝒌,𝒎− 𝜻𝒍,𝒎)′∗ (𝑽𝒌,𝒎_{+ 𝑽}𝒍,𝒎₎−𝟏_{∗ (𝜻}𝒌,𝒎_{− 𝜻}𝒍,𝒎_)~𝜒

202 for 𝑘, 𝑙 = 𝑠𝑓, 𝑠𝑚, 𝑐𝑓, 𝑐𝑚 and 𝑚 = 𝑝𝑟, 𝑝𝑢.

4.2.3. Couples demand

We can recover estimates of economies of scale in consumption of public goods from the expenditure shares of couples. The general form of these expenditures shares is derived in Appendix A and reproduced here

𝑤_𝑔ℎ,𝑝𝑢 =𝑝𝑔𝑝𝑢𝑦𝑔ℎ,𝑝𝑢

𝑥𝑝𝑢 = 𝜂𝑝𝑢𝑤𝑔

𝑐𝑓,𝑝𝑢_(𝝅𝒑𝒖_{, 𝜂}𝑝𝑢_{) + (1 − 𝜂}𝑝𝑢_)𝑤

𝑔𝑐𝑚,𝑝𝑢(𝝅𝒑𝒖, 𝜂𝑝𝑢) + 𝜖𝑔, (13) where 𝑦_𝑔ℎ,𝑝𝑢is household demand for 𝑔 = 𝑚1+ 1, … , 𝑚2.As mentioned earlier we use a linear consumption technology which results in effective prices given by 𝜋_𝑔𝑝𝑢 = 𝑑_𝑔𝑝_𝑔𝑝𝑢_⁄𝑥𝑝𝑢_. Estimation is complicated by the fact that there are a natural lower and upper bound to

17

25 economies of scale. A couple cannot derive more than twice the consumable quantity that a single person derives from the same purchased quantity of goods. Given the nature of the goods we consider, it is safe to assume that there are no diseconomies of scale to consuming these goods in couples. The economies of scale parameters are therefore restricted to lie between 0.5 (perfect public good) and 1 (perfect private good). These restrictions could be enforced by specifying the functional form 𝑑𝑔= (1 + 𝑒𝛿𝑔⁄(1 + 𝑒𝛿𝑔)) 2⁄ , 𝑔 = 𝑚1+ 1, … , 𝑚2for economies of scale and estimating the parameters 𝛿𝑔. However preliminary analysis has shown that the resulting non-linear model does not produce reliable estimates of economies of scale18. We simplify the analysis considerably by making the assumption that economies of scale parameters 𝑑𝑔 are all equal. In other words, we estimate an Engel scale for public goods.

The LISS panel contains observations on the amount of food consumed at home. This data is essentially a partial observation of the shares of public expenditure consumed by household members. We assume that this partial observation is a good measure of overall sharing in public goods. We set 𝜂𝑝𝑢 _{equal to the share of food consumed at home. We are concerned} that using food sharing data overestimates the share of public goods expenditure by men, who on average eat more than women. As an alternative to this approach we will use the share of private goods expenditure 𝜂𝑝𝑟 _{as a measure of sharing in public goods (and thus} total goods) expenditure. The alternative approach is meant to check the robustness of the economies of scale estimates.

The choice of sharing rule and the assumption that economies of scale are equal for all public goods allows us rewrite the household expenditure shares as

𝜔_𝑔ℎ,𝑝𝑢− 𝑤_𝑔ℎ,𝑝𝑢 = (𝜂𝑝𝑢_𝛽̂ 𝑔𝑠𝑓,𝑝𝑢+ (1 − 𝜂𝑝𝑢)𝛽̂𝑔𝑠𝑚,𝑝𝑢) ln 𝑑 + 𝜖𝑔, for 𝑔 = 𝑚1+ 1, … , 𝑚2− 1 (14) where 𝜔_𝑔ℎ,𝑝𝑢= 𝜂𝑝𝑢_(𝛼̂ 𝑔𝑠𝑓,𝑝𝑢− 𝛽̂𝑔𝑠𝑓,𝑝𝑢𝛼0𝑠𝑓,𝑝𝑢+ 𝛽̂𝑔𝑠𝑓,𝑝𝑢ln(𝜂𝑝𝑢𝑥𝑝𝑢)) + (1 − 𝜂𝑝𝑢)(𝛼̂𝑔𝑠𝑚,𝑝𝑢− 𝛽̂𝑔𝑠𝑚,𝑝𝑢𝛼0𝑠𝑚,𝑝𝑢+ 𝛽̂_𝑔𝑠𝑚,𝑝𝑢ln((1 − 𝜂𝑝𝑢_)𝑥𝑝𝑢_)).

18 _{Estimates of these models imply that goods lie at either the perfect private good or perfect public good}

26 The variables 𝛼̂𝑔𝑠𝑓,𝑝𝑢, 𝛽̂𝑔𝑠𝑓,𝑝𝑢, 𝛼̂𝑔𝑠𝑚,𝑝𝑢and 𝛽̂𝑔𝑠𝑚,𝑝𝑢 are constructed using the structural parameter estimates from equation (10). We estimate the logarithm of the Engel scale 𝑑 by applying Seemingly Unrelated Regression19 to the system in equation (14). Standard errors are clustered at the individual level.

4.2.4. Substitution between private and public expenditure

The econometric model outlined above provides us with estimates for all but one of the parameters of the model. The last parameter to be estimated captures the preference for consumption of public relative to private goods. The problem of dividing resources between private good and public good expenditure is given by program (3) or

max

𝑥𝑘,𝑝𝑟_,𝑥𝑘,𝑝𝑢ℒ

𝑘_(𝑥𝑘,𝑝𝑟_{, 𝑥}𝑘,𝑝𝑢_{, 𝒛}𝒌_{) = Ψ}𝑘,𝑝𝑟_(𝑥𝑘,𝑝𝑟_{, 𝒑}𝑘,𝒑𝒓_{, 𝒛}𝒌_{) + 𝑒}𝛾𝑘_(𝒛𝒌₎

Ψ𝑘,𝑝𝑢_(𝑥𝑘,𝑝𝑢_{, 𝒑}𝑘,𝒑𝒖_{, 𝒛}𝒌_{) +}

𝜆𝑘_(𝑥𝑘_{− 𝑥}𝑘,𝑝𝑟_{− 𝑥}𝑘,𝑝𝑢_{) for 𝑘 = 𝑠𝑓, 𝑠𝑚, 𝑐𝑓, 𝑐𝑚,}

where ℒ𝑘 represents the Lagrangian of the optimization problem. We allow for heterogeneity in the relative preference for public goods. The specification 𝛾𝑘 _{= 𝛾}

0𝑘 + 𝛾1𝑘∗

ℎ𝑜𝑚𝑒𝑘_{+ 𝛾}

2𝑘∗ 𝑒𝑑𝑢𝑐𝑘+ 𝛾3𝑘∗ 𝑎𝑔𝑒𝑘 is used. The first order conditions for constrained maximization are given by

𝜆𝑘₌ 1

𝑥𝑘,𝑝𝑟_𝑏𝑘,𝑝𝑟 (15)

𝜆𝑘_{= 𝑒}𝛾𝑘_(𝒛𝒌₎ 1

𝑥𝑘,𝑝𝑢_𝑏𝑘,𝑝𝑢_(𝒛𝒌₎. (16)

Equating the RHS of equations (15) and (16) gives 𝑥𝑘,𝑝𝑢 _{= 𝑒}𝛾𝑘_(𝒛𝒌₎ 𝑏𝑘,𝑝𝑟

𝑏𝑘,𝑝𝑢_(𝒛𝒌₎𝑥𝑘,𝑝𝑟. (17)

We plug the RHS of equation (17) into the budget constraint; this gives a relation between private expenditure and total expenditure. Noting that 𝑏𝑘,𝑝𝑟= 𝑏𝑘,𝑝𝑢(𝒛𝒌_{) = 1 when prices are} held constant, the relation between private expenditure and total expenditure can be

written as

19

27

𝑥𝑘,𝑝𝑟

𝑥𝑘 =

1

1+𝑒𝛾𝑘(𝒛𝒌)+ 𝑣, (18)

where 𝑣 is an error term. Equation (18) can be used to estimate the parameter 𝛾𝑘 by means of non-linear least squares. Note that the individual budget 𝑥𝑘 is defined differently for singles and members of couples. For singles 𝑥𝑘 _{equals total household expenditure 𝑥. For} members of couples the individual budget is 𝑥𝑘_{= 𝜂}𝑘_{𝑥 where 𝜂}𝑐𝑚_{= 1 − 𝜂}𝑐𝑓_.

4.3. Indifference scales

Indifference scales in the spirit of Browning et al. (2013) are the share of household expenditure that a specific household member needs as disposable income to make him/herself as well of in a single person household as he/she is in the multiple person household. The first step to constructing indifference scales for members of couples is therefore to find the monetary amount 𝑥∗ _{that a member needs to be indifferent between a} single person and a couple household. However, it is not immediately clear what it means for an individual to be indifferent if their preferences change. Here we interpret indifference in terms of the preferences before the household composition change. A member of a couple is indifferent between being part of a couple households with expenditure 𝑥 and a single person household with expenditure 𝑥∗ _{if he/she can attain the same indifference} curve on his/her utility function as member of a couple20. In other words 𝑥∗ _{needs to satisfy}

𝑉𝑘_(𝑥𝑘,𝑝𝑟_(𝑥∗_{), 𝑥}𝑘,𝑝𝑢_(𝑥∗_{)/1, 𝒛}𝒌_{) = 𝑉}𝑘_(𝑥𝑘,𝑝𝑟_(𝜂𝑘_{𝑥), 𝑥}𝑘,𝑝𝑢_(𝜂𝑘_{𝑥)/𝑑, 𝒛}𝒌_{) for 𝑘 = 𝑐𝑓, 𝑐𝑚, (19)}

where 𝜂𝑐𝑚 _{= 1 − 𝜂}𝑐𝑓_{. The functions 𝑥}𝑘,𝑝𝑟_{(𝑎) =} 1

1+𝑒𝛾𝑘(𝒛𝒌)𝑎 and 𝑥

𝑘,𝑝𝑢_{(𝑎) =} 𝑒𝛾𝑘(𝒛𝒌) 1+𝑒𝛾𝑘(𝒛𝒌)𝑎 in

equation (19) are the optimal solutions for private and public goods expenditure found by rewriting equation (18). We can calculate “indifference income” 𝑥∗ _{by combining observed} expenditure sharing data with preference (section 4.2.1 and 4.2.4) and economies of scale (section 4.2.3) parameter estimates. An explicit solution for 𝑥∗ _{can be derived from equation} (19). Note thatwhen prices are normalized to 1 we have

20

We could just as well have defined indifference in terms of the preferences as a single person. Note though that the relative prices of private goods do not change since they are assumed not to be subject to economies of scale. Regardless of whether the preferences over private goods are different for singles and members of couples, the indirect sub-utility function for private goods remains the same. The choice for pre or post change

28 ln 𝑎𝑝𝑟 _{= 𝛼} 𝑐 𝑝𝑟 , 𝑏𝑝𝑟 = 1, ln 𝑎𝑝𝑢 _{= 𝛼} 𝑐𝑝𝑢, 𝑏𝑝𝑢 = 1.

The indirect utility function when prices are normalized is given by 𝑉𝑘_(𝑥𝑝𝑟_{, 𝑥}𝑝𝑢_{, 𝒛}𝒌_{) = ln 𝑥}𝑝𝑟_{− 𝛼}

𝑐𝑝𝑟+ 𝑒𝛾

𝑘_(𝒛𝒌₎

(ln 𝑥𝑝𝑟_{− 𝛼}

𝑐𝑝𝑢). (20)

By substituting the optimal expenditure on private and public goods into equation (20) we get an expression for the RHS of equation (19) equal to

𝑉𝑘_(𝜂𝑘_{𝑥, 𝑑) = ln (} 1 1+𝑒𝛾𝑘(𝒛𝒌)𝜂 𝑘_{𝑥) − 𝛼} 𝑐𝑘,𝑝𝑟+ 𝑒𝛾 𝑘_(𝒛) (ln ( 𝑒𝛾𝑘(𝒛𝒌) 1+𝑒𝛾𝑘(𝒛𝒌)𝜂 𝑘_{𝑥 ) − ln 𝑑 − 𝛼} 𝑐𝑝𝑢),

whereas the LHS of equation (19) is given by 𝑉𝑘_(𝑥∗_{, 1) = ln (} 1 1+𝑒𝛾𝑘(𝒛𝒌)𝑥 ∗_{) − 𝛼} 𝑐𝑘,𝑝𝑟+ 𝑒𝛾 𝑘_(𝒛) (ln ( 𝑒𝛾𝑘(𝒛𝒌) 1+𝑒𝛾𝑘(𝒛𝒌)𝑥 ∗_{) − ln 1 − 𝛼} 𝑐 𝑝𝑢 ).

Equation (19) can be rewritten as

ln(𝑥∗_{) = ln(𝜂}𝑘_{𝑥) −} 𝑒𝛾𝑘(𝒛𝒌) 1+𝑒𝛾𝑘(𝒛𝒌)ln 𝑑.

The explicit solution for indifference income is given by equation (21)

𝑥∗_{= 𝜂}𝑘_{𝑥 𝑑}

𝑒𝛾𝑘(𝒛𝒌) 1+𝑒𝛾𝑘(𝒛𝒌)

⁄ . (21)

The income needed to be indifferent is nothing more than the expenditure available as a household member divided by the effective price of public goods consumption raised to a power equal to the optimal ratio of public to total goods expenditure. The second step to constructing indifference scales is simply to divide indifference income by total household expenditure. The indifference scales are given by 𝐼𝑆 = 𝑥∗_/𝑥.

29 5. Results

5.1. Reduced form parameters

30 Table 4a: Single women’s reduced form parameters of the private goods subsystem

Single women (N=577)

Food and drink

outside Tobacco Clothing

Personal care products Leisure activities Homeowner -0.0134 -0.0766*** 0.0801*** -0.00200 0.0119 (0.0174) (0.0206) (0.0191) (0.0126) (0.0173) Education level 0.0334* -0.0515** -0.00761 -0.0260* 0.0517*** (0.0177) (0.0206) (0.0187) (0.0146) (0.0186) Age -0.00120* 0.00112* -0.000702 0.000754 2.48e-05 (0.000654) (0.000663) (0.000769) (0.000523) (0.000624) Log(private goods expenditure) 0.0302*** 0.0493*** -0.0270* -0.0686*** 0.0161 (0.0102) (0.0113) (0.0140) (0.0117) (0.0102) Constant 0.0743 -0.170*** 0.475*** 0.524*** 0.0976 (0.0661) (0.0591) (0.0845) (0.0725) (0.0615)

Robust standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1

Table 4b: Women in couples’ reduced form parameters of the private goods subsystem

Single women (N=834)

Food and drink

outside Tobacco Clothing

Personal care products Leisure activities Homeowner -0.0168 -0.0539** 0.0405* 0.0155 0.0147 (0.0141) (0.0220) (0.0217) (0.0123) (0.0146) Education level 0.0497*** -0.0594*** -0.0194 -0.000315 0.0293** (0.0138) (0.0124) (0.0176) (0.0108) (0.0146) Age -0.00192*** 0.000215 0.000662 0.00240*** -0.00136*** (0.000484) (0.000459) (0.000613) (0.000347) (0.000444) Log(private goods expenditure) 0.0349*** 0.0203** -0.0116 -0.0650*** 0.0214*** (0.00800) (0.00833) (0.0131) (0.0107) (0.00803) Constant 0.0621 0.0106 0.397*** 0.419*** 0.110** (0.0491) (0.0406) (0.0725) (0.0533) (0.0451)