HOUSEHOLD PRODUCTION AND THE ELASTICITY OF
MARGINAL UTILITY OF CONSUMPTION
Disa Thureson
The Swedish National Road and Transport Research Institute (VTI),
Division of Transport Economics,
and Örebro University School of Business, Sweden
CTS Working paper 2016:10
Abstract
Chetty (2006) developed a new method of estimating the Elasticity of Marginal
Utility of Consumption (EMUC) from observed work time responses to wage
changes and derived an upper bound of 2 for this parameter. Here I show that
the omission of household production in Chetty’s model may lead to bias, and
perform a numerical sensitivity analysis of Chetty’s results in this respect. I
develop a new model that includes household production from which I derive
new, unbiased EMUC formulas. I offer empirical estimates based on current
evidence of the included parameters, suggesting a lower bound for EMUC of
about 0.9.
Keywords: Elasticity of marginal utility of consumption; Household production;
Labor supply; Coefficient of relative risk aversion; Consumer behavior
JEL codes: D10, D60, J22
Centre for Transport Studies
SE-100 44 Stockholm
Sweden
www.cts.kth.se
Household Production and the Elasticity of Marginal Utility of Consumption
1
INTRODUCTION
The link between individuals’ consumption and wellbeing is at the core of
economic analysis, from both a positive and a normative perspective. One way
to estimate this relationship is to study the tradeoffs people make between
leisure and consumption. Here I suggest an approach that adds household
production in this analysis.
The Elasticity of Marginal Utility of Consumption (EMUC) is a means for
measuring how utility changes with consumption and is of great importance for
understanding consumer behavior as well as for policy evaluation. In for
instance evaluation of climate change policies, EMUC is one of the most
influential parameters in the calculation of the social cost of carbon (Anthoff et
al., 2009). Using the Ramsey equation, discounting the costs of future disasters
should accounts for both the pure rate of time preference and the growth rate of
consumption over time; the two components are added after weighting future
consumption with EMUC (see, e.g., Baum 2009). This weight can also be
interpreted as the relative significance of consumption of current vs. future
generations, i.e., a distributional interpretation of the concept. In analyses of
issues related to risk, the EMUC coefficient corresponds to a commonly used
interpretation of the coefficient of relative risk aversion. In a review of empirical
studies of EMUC estimates, Evans (2005) reports estimates that are mostly in
the range of 1–2.
While previous analyses used the pattern of savings or information about, e.g.,
the demand for food for approximating EMUC, Ray Chetty (2006) demonstrated
that observations of how working time responds to wage changes could be an
alternative approach. He argued that if people increase their working time as
their wage increases, this implies that on average EMUC is less than unity.
Through the Slutsky decomposition of labor supply, there are two effects of
wage changes on the demand for leisure: one income effect, where a higher
wage increases income and leads to higher demand for leisure, and one
substitution effect, which increases the relative price of leisure at the margin
and hence reduces the leisure demand. If utility from leisure is also independent
of consumption level and if market work constitutes the only source of income,
an EMUC value equal to unity means that these two effects cancel each other
out. If instead EMUC is less than unity, the substitution effect dominates (leisure
demand decreases), but if EMUC is greater than unity, the income effect
dominates (leisure demand increases). Assuming that reductions (increases) in
leisure demand are realized through working time increases (reductions),
EMUC can be estimated from evidence on working time responses to wage
changes.
The starting point of Chetty’s approach is that the (uncompensated) labor
supply elasticity with respect to wage is positive on average and hence the
average EMUC is lower than unity. If, in addition to wage income, there is
unearned income and if consumption and labor are complementary, this upper
bound is relaxed. He derived EMUC formulas under such circumstances and
2
Household Production and the Elasticity of Marginal Utility of Consumption
computed corresponding numerical values based on labor supply elasticity
estimates, an concluded that the existing empirical evidence implies that EMUC
must be less than 2: “The intuition for this tight bound is simple: if the marginal
utility of wealth diminishes rapidly, why don’t people choose to work much less
when their wages rise?” (Chetty 2006, p. 1830).
Here I propose an approach to estimate EMUC that is based on an additional
basic feature of people’s day-to-day tradeoffs made as a result of wage level
changes. The idea is that individuals may respond to wage increases not only via
adjustment of their labor supply but also by reducing the time spent on
housework or other forms of unpaid work. For example, high-wage individuals
can reduce time spent in household production by buying services from lowwage individuals. I argue that effective leisure time, rather than time not spent
on paid work in the formal sector, is what is relevant when making inferences
about the utility function.
The main aim of the present study is therefore to investigate how
considerations of household production influence theoretical and empirical
conclusions regarding feasible long-term, average values of EMUC. The analysis
is restricted to the use of intensive margin responses of labor supply, and I will
not explicitly model intertemporal substitution of individuals1.
To this end, I develop a model with household production that is used to derive
some properties of the bias invoked by the omission of household production. I
further derive EMUC equations for two special cases based on the wage
elasticity of labor supply. One of these equations can also be used to estimate a
lower bound for EMUC under more general conditions.
I use the uncompensated wage elasticity of labor supply rather than – as in
Chetty’s final expression – the compensated wage elasticity in combination with
the income elasticity of labor supply. EMUC estimates from this new
specification are less sensitive to the magnitude of the assumed proportion of
unearned income to earned income than in the previous formulation. In
addition, this specification allows utilization of labor supply elasticity estimates
based on other empirical sources, notably time series and cross-country data on
mean working times and mean wages, for which it is possible to control for
household production to some extent.
The main conclusion of the analysis is that when household production and
market work can be substituted, omission of household production in an EMUC
equation will generally result in biased estimates. Examples of this are shown in
a sensitivity analysis of the differences in results with and without inclusion of
household production. In addition, the sensitivity analysis shows that the
results in Chetty (2006) are highly sensitive to the assumed proportion of
unearned income to earned income. The main results of the present study
suggest a lower bound of the average EMUC close to unity (about 0.9 at the
1
However, for the numerical estimation I will consider two cases where consumers either maximize
lifetime utility or within-period utility.
3
Household Production and the Elasticity of Marginal Utility of Consumption
lowest). In addition, I compute point estimates in the range of 0.9–1.6, but
unfortunately these latter results should be considered more as an illustration
of the proposed method than a final conclusion, since some of the underlying
assumptions are based on estimates from a single old, and possibly biased, data
source.
The next section starts with a description of the original model and continues
with my extension of the model. Section 3 reviews the literature on the relevant
parameters. In Section 4, results from the numerical analysis are presented
followed by a discussion in Section 5 and concluding remarks in Section 6.
2
THEORY
2.1 Chetty’s model
In Chetty (2006), an agent faces the following utility maximization problem
during 𝑇 time periods:
max 𝑈(𝑐1 , … , 𝑐𝑇 , 𝑀1 , … , 𝑀𝑇 )
(1)
s.t.
𝑐𝑡 ,𝑀𝑡
𝑝1 𝑐1 + ⋯ + 𝑝𝑇 𝑐𝑇 = 𝑦 + 𝑤(𝜃1 𝑀1 + ⋯ + 𝜃𝑇 𝑀𝑇 )
where 𝑐𝑡 is consumption in period 𝑡, 𝑝𝑡 is price level of consumption in each
time period, 𝑦 is unearned income (endowment), 𝑤 is mean wage, and 𝜃𝑡 is
wage share in each time period. 𝑀𝑡 is the time spent in market work (labor
supply) out of the time at one’s disposal in each time period 𝑡. This means that
market work is the opposite of leisure time and that utility is decreasing in
market work when consumption is held constant. Chetty used a two-stage
formulation to reduce expression (1) to:
where
max𝑐,𝑀 𝑢(𝑐, 𝑀) s.t. 𝑐 = 𝑦 + 𝑤𝑀,
(2)
𝑢(𝑐, 𝑀) = max 𝑈(𝑐1 , … , 𝑐𝑇 , 𝑀1 , … , 𝑀𝑇 ),
𝑐𝑡 ,𝑀𝑡
s.t.
𝑝1 𝑐1 + ⋯ + 𝑝𝑇 𝑐𝑇 = 𝑐 and 𝜃1 𝑀1 + ⋯ + 𝜃𝑇 𝑀𝑇 = 𝑀.
He further showed that when 𝑦 = 0 and consumption and labor are additively
separable in the utility function, the following result holds:
𝜕𝑀
𝜕𝑤
> 0 ⇔ 𝜂 < 1.
(3)
where 𝜂 is EMUC. The intuition is that an increase in wage implies both an
increase in income and an increase in the price of leisure relative to the
consumption numeraire. When EMUC is 1 (and market work constitutes the
only source of income and the utility of leisure is independent of consumption),
the two effects cancel out and leisure demand remains constant. When EMUC is
less than 1, the price effect dominates and leisure demand decreases.
4
Household Production and the Elasticity of Marginal Utility of Consumption
Chetty then broadened the model to allow for complementarity between market
work and consumption and for unearned income to make it possible to
empirically derive an upper bound of EMUC. He defined the level of
complementarity between market work and consumption as:
lim
∆𝑐
∆𝑀→0 𝑐
/
∆𝑀
𝑀
.
(4)
This can be interpreted as the compensated elasticity of consumption with
respect to labor supply. This would correspond to the consumption response
due to a part-time unemployment shock that is fully compensated, for instance,
by unemployment insurance.2 For simplicity, I will denote the compensated
elasticity of consumption with respect to working time as 𝜀̃𝑐𝑀 .
The resulting expression of EMUC was equation (9) in Chetty (2006):
𝜂̂ =
𝑤𝑀 −𝜀𝑀𝑦
)
𝑦 𝜀̃𝑀𝑤
𝑦
(1−(1+
)𝜀̃ )
𝑤𝑀 𝑐𝑀
(1+
,
(5)
where 𝜀̃𝑀𝑤 is the compensated elasticity of market work with respect to wage,
defined by:
𝜀̃𝑀𝑤 =
and:
𝜕𝑀𝑐
𝜕𝑤
=
𝜕𝑀𝑐
𝜕𝑤
𝜕𝑀
𝑤
∙ 𝑀,
(6)
𝜕𝑀
− 𝑀 𝜕𝑦 ,
𝜕𝑤
(7)
according to the Slutsky decomposition of compensated labor supply.
Equation (5) consist of the following three factors: an unearned income factor,
−𝜀𝑀𝑦
𝑤𝑀
(1 + 𝑦 ), an income response factor, 𝜀̃ , and a complementarity factor,
1
𝑀𝑤
. Intuitively, the core part of equation (5) is the income response
𝑦
)𝜀̃ )
𝑤𝑀 𝑐𝑀
(1−(1+
factor, which is balanced by the proportions between earned and unearned
income through the unearned income factor. Lastly, the possibility of
complementarity (or substitutability) between market work and consumption
is controlled for through the complementarity factor (when 𝜀̃𝑐𝑀 = 0 the
complementarity factor collapses to 1). A few notes can be made regarding the
income response factor. First, the elasticity of market work w.r.t. unearned
income, i.e. 𝜀𝑀𝑦 , is generally assumed to be negative, so the numerator is
positive. Also, compensated elasticity of labor supply w.r.t. wage, i.e. 𝜀̃𝑀𝑤 , is
assumed to be positive, and hence the income response factor is positive. One
can also see that EMUC estimates (through the income response factor) are
proportional in size to 𝜀𝑀𝑦 and inversely proportional to 𝜀̃𝑀𝑤 .
2
If a person wants to consume more in periods when she works more relative to periods when she
has more leisure, there exists a positive complementarity between labor and consumption according
to this definition.
5
Household Production and the Elasticity of Marginal Utility of Consumption
I will now review Chetty’s results from both a theoretical and empirical point of
𝑤𝑀
view. The unearned income factor (1 + 𝑦 ) is obviously sensitive to small
values of 𝑦 in relation to 𝑤𝑀 (as 𝑦 approaches zero, the unearned income factor
goes to infinity). As I will show in Section 3 using disaggregate data,
(proportionally) small values of 𝑦 seem more plausible than the values
suggested in Chetty’s study.
I will further show that since potential substitution between market work and
household production is not controlled for in Chetty’s model, both the income
response factor and the complementarity factor may be biased. The bias that
arises from the omission of household production will typically lead to an
underestimation of the income response factor, whereas for the
complementarity factor both underestimation and overestimation can occur.
Related to the issue of omission of household production, I will also consider the
empirical possibility that the uncompensated wage elasticity of labor supply is
𝜕𝑀
negative instead of positive, i.e. that 𝜕𝑤 < 0. Through the inequality in equation
(3), this will have implications for the possibility to estimate an upper or a lower
bound.
2.2 Introducing household production
I now define a model that is similar to expression (2) but includes household
production. The agent faces the following maximization problem:
max 𝑢(𝑐, 𝐿),
(8)
𝐿,𝑀,𝐻
s.t.
= 𝑦 + 𝑤𝑀 + 𝑔(𝐻),
𝜏 = 𝐿 + 𝑀 + 𝐻,
where
𝑔′(𝐻) > 0, 𝑔′′ (𝐻) < 0,
where 𝑐 is consumption, 𝐿 is leisure time, 𝑦 is unearned income (endowment),
𝑤 is wage, 𝜏 is total time at disposal, 𝑀 is time spent on market work (labor
supply), and 𝐻 is time spent on household production. 𝑔(𝐻) is the household
production function and represents the consumption value of total household
production for the agent. It captures substitution between market work and
household production through the following three channels: substitution
between market goods and home produced goods, labor division in household
production (hiring of staff or work sharing with other household members), and
investments in household production capital. I restrict the model by defining 𝜂
for the normalization3 𝑔(𝐻0 ) = 0.
The reason for the normalization is twofold. Most importantly, in the standard definitions of 𝜂, only
market goods and services are included in the consumption numeraire. From this perspective, the
total value of the household production needs to be normalized to zero if EMUC values are to be
3
6
Household Production and the Elasticity of Marginal Utility of Consumption
Leisure time includes all activities other than market work and household
production. The focus is on intensive margin responses of labor supply and
therefore I restrict the model to agents that to some extent participate in market
work. It is also assumed that some housework tasks are not feasible to
substitute for market work, that is, lim (𝑔′ (𝐻)) = ∞.4 It follows that an optimal
𝐻→0
solution must satisfy:
𝑐 > 0,
𝐿 > 0,
𝑀 > 0,
𝐻 > 0.
(9a)
(9b)
(9c)
(9d)
Figure 1 gives a visual interpretation of the optimization problem defined in
equation (8).
𝑐
𝑤𝑀
Indifference
curve
𝑐 = 𝑤𝑀
2
1
𝑐 = 𝑔(𝐻0 ) = 0
𝐻0
𝐿0
𝑔(𝐻)
𝐻
𝐿
𝑀
𝜏
Figure 1: Optimization problem of the agent.
The agent has 𝜏 time to allocate between leisure time, 𝐿, market work, 𝑀, and
household production, 𝐻. She first optimizes the proportion of household
production in relation to market work, which will be the point where the slopes
of 𝑔(𝐻) and 𝑤𝑀 equate (equilibrium 1). She then chooses the optimal level of
comparable in size to the standard definition. The normalization also implies simplification of the
model.
4
Another possible formulation of a model that includes household production is max 𝑢(𝑐, 𝐿, 𝑔),
𝐿,𝑀,𝐻
where 𝑐 = 𝑦 + 𝑤𝑀 and 𝑔 = 𝑔(𝐻), which is only partly tradable in the market. However, as long as g
and c are additive, this does not make a considerable difference, since 𝑔(𝐻) is general enough to
allow for extremely low substitutability between g and c for some tasks, which in practice will mean
that these tasks will not be possible to substitute for market work.
7
Household Production and the Elasticity of Marginal Utility of Consumption
consumption in relation to leisure, which is at the point where the slope of the
indifference curve equates the slope of 𝑤𝑀 (equilibrium 2).
I now put forward some propositions regarding solutions for the utilitymaximizing behavior of the agent described by expression (8) and expression
(9) and regarding the estimator implications of that behavior. Proposition 1
describes how the chosen level of household production is influenced by wage
changes. This is important for the understanding of the mechanisms in the
model and for the interpretation of the results later on. For a derivation of
Proposition 1, see Appendix 1.
Proposition 1:
The derivative of time spent in household production with respect to wage
is inversely proportional to the second derivative of the household
production function, and hence negative:
𝜕𝐻
𝜕𝑤
1
= 𝑔′′ (𝐻) < 0.
(11)
There are two limiting cases of equation (11) w.r.t. 𝑔′′ (𝐻):
lim
𝑔′′ (𝐻)→0
−∞,
𝜕𝐻
(𝜕𝑤) =
(12a)
lim
𝑔′′ (𝐻)→−∞
𝜕𝐻
(𝜕𝑤) = 0.
(12b)
Equation (12a) represents a case of perfect substitution between household
production and market production. In this case there is no internal solution, but
instead either of the two corner solutions of 100% household production or
100% market work will arise, which violates assumption (9c) and assumption
(9d), respectively. The implication for an internal solution is that if the level of
concavity of the household production function is low, a small increase in wage
can cause a great shift between unpaid and paid work.
Equation (12b) implies that when the level of concavity of the household
production function is high, the shift between household production and market
production will be small and can be approximated to zero.
Proposition 2 provides an estimator of EMUC in a model that includes
household production. For the derivation of Proposition 2, see Appendix 2.
Proposition 2:
In a model with household production, EMUC can be expressed as:
𝑤𝑀 −𝜀𝑀𝑦
)
𝑦 𝑃 𝜀̃
𝑀 𝑃𝑤
𝑦 𝑀
(1+(1+
) 𝜀̃ )
𝑤𝑀 𝐿 𝑐𝐿
(1+
𝜂=
(13)
where:
8
Household Production and the Elasticity of Marginal Utility of Consumption
𝑃 = 𝑀 + 𝐻 is total production time,
𝜀̃𝑃𝑤 =
where
𝜕𝑃 𝑐
𝜕𝑤
𝜕𝑃 𝑐
𝜕𝑤
and 𝜀̃𝑐𝐿 =
𝑤
∙ 𝑃 is the compensated production time elasticity w.r.t. wage,
=
𝜕𝑀𝑐
𝜕𝑤
𝜕𝑐 𝑐 𝐿
𝜕𝐿
𝜕𝐻
+ 𝜕𝑤 ,
∙ 𝑐 is the compensated consumption elasticity w.r.t. leisure.
The difference between equation (13) and equation (5) is that, loosely speaking,
household work is now also included in the elasticities that involve market
work in Chetty’s model. More precisely, when making inference about the utility
function (with respect to tradeoffs between consumption and leisure), the
relevant variable is how supply in total production time is changed as wage
𝜕𝑃
𝜕𝑀
𝜕𝐻
𝜕𝑀
changes, i.e. the relevant variable is 𝜕𝑤 = 𝜕𝑤 + 𝜕𝑤. When 𝜕𝑤 is used as a proxy
for
𝜕𝐻
𝜕𝑤
𝜕𝑃
𝜕𝑤
, total production time responses will be positively biased by the size of
.
Proposition 3 describes properties of the bias caused by omission of household
production in Chetty’s model. For the derivation of Proposition 3a and
Proposition 3c, see Appendix 3.
Proposition 3a:
The error factor of an EMUC estimator given by Chetty’s model, without
household production, and defined as the ratio between the estimator given
by Chetty’s model and the true value, is:
̂
𝜂
𝜂
𝑦
𝜕𝑀
𝜕𝐿
)𝜀̃ ∙ /(− )
𝑤𝑀 𝑐𝑀 𝜕𝑤
𝜕𝑤
𝑦
1−(1+
)𝜀̃
𝑤𝑀 𝑐𝑀
1−(1+
.
= (1 +
𝜕𝐻
𝜕𝑤
𝜕𝑀𝑐
𝜕𝑤
)∙
(14)
Proposition 3b follows directly from equation (14):
Proposition 3b:
If there is approximately no change in time spent on household production
in response to a wage change, Chetty’s model without household
production will yield an unbiased estimator of EMUC:
𝜂
̂≈1
≈
0
⇒
𝜕𝑤
𝜂
𝜕𝐻
(15)
In order to facilitate sensitivity analysis, it is convenient to split equation (14)
into two factors:
𝜂
̂
𝜂
𝜂 𝐴
𝜂 𝐵
= (𝜂̂) ∙ (𝜂̂) ,
(16a)
where
𝜂 𝐴
(𝜂̂) = 1 +
𝜕𝐻
𝜕𝑤
𝜕𝑀𝑐
,
(16b)
𝜕𝑤
9
Household Production and the Elasticity of Marginal Utility of Consumption
is the income response error factor, and
𝜂 𝐵
(𝜂̂) =
𝑦
𝜕𝑀
𝜕𝐿
)𝜀̃ ∙ /(− )
𝑤𝑀 𝑐𝑀 𝜕𝑤
𝜕𝑤
𝑦
1−(1+
)𝜀̃
𝑤𝑀 𝑐𝑀
1−(1+
,
(16c)
is the complementarity error factor.
Proposition 3c:
In general, the income response error factor will be within the range zero
to unity:
𝜂 𝐴
(𝜂̂) = 1 +
with
∈ (0,1),
(17a)
𝜂 𝐴
lim
|
𝜕𝐻
𝜕𝑤
𝜕𝑀𝑐
𝜕𝑤
𝜕𝑀𝑐
𝜕𝐻
|→
𝜕𝑤
𝜕𝑤
(𝜂̂) = 0.
(17b)
Proposition 3c implies that the size of household production responses in
relation to the compensated labor supply responses w.r.t. wage determines the
𝜂 𝐴
size of the bias given by (̂) . The interpretation is that the bias is driven by the
𝜂
share of the compensated labor supply response that is caused by a
counteracting change in household production, as a change in labor supply
cannot be identified as a substitution between leisure and consumption.
Proposition 3d follows directly from equation 16c:
Proposition 3d:
The size of the percental error originating from the complementarity error
factor, i.e.:
𝜂 𝐵
|(̂) − 1| =
𝜂
|
|
𝜕𝑀
𝑦
(1+
)𝜀̃𝑐𝑀 (1− 𝜕𝑤
𝜕𝐿 )
𝑤𝑀
(−
)
𝜕𝑤
𝑦
1−(1+
)
𝑤𝑀
|,
|
(18)
𝜕𝐻
𝜕𝑀
is an increasing function5 of |𝜕𝑤|, given a fixed value of 𝜕𝑤 .
Note that in the special case when utilities from leisure and consumption are
𝜕2 𝑈
𝜂 𝐵
additive, i.e.,
= 0, it follows that 𝜀̃ = 0 and (̂) = 1, and the relative bias
𝑐𝑀
𝜕𝑐𝜕𝐿
5
Remember that
𝜕𝐿
𝜕𝑤
= −(
𝜕𝐻
𝜕𝑤
+
𝜕𝑀
𝜕𝑤
).
10
𝜂
Household Production and the Elasticity of Marginal Utility of Consumption
𝜂
̂
𝜂
̂ 𝐴
𝜂
𝜂
of 𝜂̂ will reduce to the more convenient expression = ( ) = 1 +
𝜕𝐻
𝜕𝑤
𝜕𝑀𝑐
𝜕𝑤
≤ 1, that
is EMUC will be underestimated when household production is omitted.
𝜂 𝐵
Since the complementarity error factor, i.e. (̂) generally is a rather complex
𝜂
expression, its implications will be investigated in more detail numerically6 in
Section 4.1.
Propositions 3b, 3c, and 3d show that there will be an estimator bias if
household production changes as wage changes. According to Proposition 1,
changes in household production w.r.t. wage are in turn determined by the
concavity of the household production function. As can be seen from combining
Propositions 1, 3b, 3c, and 3d, if substitutability between household production
and market work is low then the bias will be small and vice versa.
Above, I have now put forward some general propositions on the bias that
arises in a model that does not include household production, if, in fact,
household production is important. In the subsequent Section 2.4, I will derive
formulas of more practical use, which I will later (in Section 4) estimate using
empirical evidence (from Section 3).
2.3 New EMUC formulas
I will now propose new point estimators for two special cases. One of these
formulas can also be used to estimate a lower bound of EMUC under some
circumstances.
Proposition 4 presents a formula of EMUC for the special case
coincides with the limit of a lower bound when either
restrictive assumption
Appendix 4.
𝜕𝑀
𝜕𝑤
𝜕𝑃
𝜕𝑤
𝜕𝑃
𝜕𝑤
= 0, which
< 0 or the more
< 0 holds. For the derivation of Proposition 4, see
Proposition 4a:
If there is no change in total work supply in response to a wage change, i.e.
𝜕𝑃
= 0, then EMUC can be estimated as:
𝜕𝑤
𝜂=
𝑦
)
𝑤𝑀
𝑦 𝑀
(1+(1+
) 𝜀̃ )
𝑤𝑀 𝐿 𝑐𝐿
(1+
.
(19)
Proposition 4b:
6
𝜂 𝐴
𝜕𝐻
𝜂
𝜕𝑤
Because the formula of (̂) is linear in
/
𝜕𝑀𝑐
𝜕𝑤
𝜂 𝐴
, an analogous analysis for (̂) would be trivial and
𝜂
will therefore not be performed.
11
Household Production and the Elasticity of Marginal Utility of Consumption
If instead the response is a decrease in total work supply, i.e.
𝜕𝑀
𝜕𝑃
𝜕𝑤
< 0, or
more restrictively, in market work, i.e. 𝜕𝑤 < 0, then equation (19) can be
used to estimate the limit of a lower bound of EMUC.
Proposition 5 presents a formula of EMUC for the special case of additive
𝜕2 𝑈
utilities from leisure and consumption, i.e. when 𝜕𝑐𝜕𝐿 = 0. For the derivation of
Proposition 5, see Appendix 5.
Proposition 5:
If leisure and consumption are additively separable in the utility function,
𝜕2 𝑈
i.e. 𝜕𝑐𝜕𝐿 = 0, then EMUC can be estimated as:
𝜂≈
3
1+
𝑦
𝑤𝑀
.
(20)
𝜕𝐻
𝜕𝑤 )
1+𝜀𝑀𝑤 (1+ 𝜕𝑀
𝜕𝑤
EMPIRICAL EVIDENCE FOR PARAMETER VALUES
This section reviews the empirical evidence for the uncompensated labor
supply elasticity; for changes in shares of market work, household production,
and leisure; for the compensated consumption elasticity w.r.t. leisure; for the
proportion of market work to leisure; and for the proportion unearned income,
as these are the parameters included in the formulas in Section 2.4.
3.1 Uncompensated labor supply elasticity
Evers et al. (2008) summarized 209 estimates of the uncompensated labor
supply elasticity w.r.t. wage that were obtained from 30 different studies based
on individual data from various OECD countries. A vast majority of the estimates
were positive, but the ranges were rather wide, from −0.08 to 2.87 for women
and from −0.24 to 0.45 for men, although the authors only selected studies with
high credibility8.
Moreover, as Blundell and MaCurdy (1999) and Evers et al. (2008)
demonstrated, the estimation of such elasticities is a demanding task involving
numerous potential pitfalls. For example, Evers et al (2008) explained, “The
standard model of labour supply does not distinguish between the effect of
wages and taxes on the decision to participate (the extensive margin) and the
7
Values above 2 come from studies performed during the period 1966–1979. As women’s
participation rate has changed significantly since then, these values might be outdated. After 1980,
the maximum value is 1.2.
8
The authors have selected studies from reputable journals and studies that do not use the OLS
method because this method is known to produce unreliable results.
12
Household Production and the Elasticity of Marginal Utility of Consumption
decision regarding hours worked (intensive margin)” (p. 29). They further cited
Mroz (1987) who found that “when this effect is neglected, the estimated wage
elasticity is biased upwards, because hours of work conditional on participation
are relatively inelastic with respect to the net wage, while the participation
decision is quite elastic with respect to the net wage.” (p. 29). Evers et al. (2008)
showed that the estimation results from various studies are sensitive to the
specification of participation for men and are sensitive to the specification of
family size and children for women.
On the macro-economic level, there are empirical indications that when wages
in society increase, working time decreases. High-income countries tend to have
shorter working times (excluding the unemployed) than low-income countries
(see for example, Figure 1 in Morley et al. (2010)). Burgoon and Baxandall
(2004) and Huberman and Minns (2007) presented regression results of
working time using mean income9 as an independent variable for 14–18 OECD
countries using time series data for 22 and 31 years, respectively10 (with about
a century in differences between the two studies).11 Both studies showed a
significant negative relationship. Huberman and Minns (2007) provided
estimates of uncompensated labor supply elasticity in the range −0.11 to −0.16
for the full sample.
However, none of these studies controlled for the diminishing productivity of
workers with long working hours. Pencavel (2014)12 suggested that marginal
productivity is rather constant up to about 50 hours/week for factory workers.
According to Huberman and Minns (2007), that threshold was passed (from
above) around 1920 for all the OECD countries considered, and this means that
later studies of the uncompensated wage elasticity are more reliable in this
respect.
Vandenbroucke (2009) estimated a simple general equilibrium model
calibrated on base-level data describing the US economy in the year 1900 to
make an ex-post prediction of the development of labor supply during the first
half of the 20th century. The model predicted 82% of the observed decline in
working time between 1900 and 1950, and it was concluded that most of the
decline was caused by the increases in wages.
9
Burgoon & Baxandall (2004) used GDP/capita and Huberman & Minns (2007) used wage.
The results of Huberman & Minns (2007) were estimated using data from 1870–1900, and
Burgoon & Baxandall (2004) used data from 1979–2000.
11
Note that in Burgoon & Baxandall (2004), estimation of the relationship between working times
and income was not the primary aim of the study, but GDP/capita was used as a control variable.
Also note that Huberman and Minns (2007) studied firm-level data with multiple controls associated
with the country in question, and hence it was not a strict macroeconomic study, but rather a semimacroeconomic study. In addition, of the around 20,000 observations included, the vast majority
were from the US. They controlled for sex of workers in each firm-level observation. Examples of
country/time-specific controls included, for example, the average age of the population, the capital
to labor ratio, and the state of the business cycle. Burgoon and Baxandall (2004) controlled for
unemployment, tax rate, inequality, and political regime.
12
Note that the results from Pencavel (2014) were not peer-reviewed and the sample was rather
small (<200 observations).
10
13
Household Production and the Elasticity of Marginal Utility of Consumption
3.2 Changes in household production and leisure
The theoretical analysis in Section 2.2 indicates that it may be important to
consider household production if EMUC is to be estimated from labor supply
response data. Although the use of macro data may reduce bias linked to the
omission of household production, it does not eliminate it. Even though work
sharing may be controlled for in the macro case, the problem of substituting
household production time for household production capital remains.13
The studies concerning the development of household production and leisure
are based on data from various survey studies that is linked together. A
complicating factor is that sometimes the division between household
production and leisure is not straightforward, for example, when it comes to
child care and education. Also, there have been structural changes such as
changes in demographics (an older population with fewer children),
dramatically increased gender equality (both in household production and in
market work, according to Ramey and Francis (2009)), and longer average
educations.
Two recent studies of the development of ∆𝐻 over time (as wages have
increased) in the United States came to partly different conclusions, although
using similar methods14. Aguiar and Hurst (2007), whose analysis was
restricted to respondents aged 21–64 years who were neither students nor
retirees, found substantial decreases in both market work and non-market work
and increases in leisure between 1965 and 2003.15
Ramey and Francis (2009), who estimated time spending for all demographic
ages for the years 1900–2005, found the same trends as Aguiar and Hurst for
the total population. But contrary to common expectations16, Ramey and Francis
(2009) found that the allocation of time spent on market work, household
production, and leisure averaged over both sexes in the age group 25–54 years
was basically the same in 2005 as a century ago17.
Table 1 shows the discrepancy18 in leisure development between the two
studies during the period 1965–2003.
13
An additional issue is that GDP/capita is negatively correlated with the size of the informal sector
(Ihrig & Moe, 2000), which implicates an additional source of negative bias for the income response
factor of the EMUC estimator. Informal sector biases might also be present in micro-economic data.
14
Both studies were mainly based on linking together various previous time-use surveys (from which
some sources are utilized both in Aguiar and Hurst and in Ramey and Francis).
15
If child care was excluded from the non-market work measure (so that it counts as leisure), the
decreases of market work were of the same magnitude as the decreases in household production.
However, if child care was included in the non-market work measure, the size of changes in nonmarket work was around half of the size of changes in market work because child care has increased.
16
Ramey and Francis (2009) claimed that the consensus previous to their study was that there had
been a dramatic increase in leisure time.
17
Although fluctuating somewhat in-between the examined time span.
18
Ramey and Francis (2009) briefly commented on the discrepancy in results between the two
studies as: “Our results for this population imply somewhat smaller increases in leisure from 1965 to
2003, mainly because we follow the literature and classify child care as home production rather than
leisure, we do not count meals at work as “work” in the early sample, and we use CPS data for our
14
Household Production and the Elasticity of Marginal Utility of Consumption
Aguiar and Hurst
(2007)
Age group (years)
21–65
Increase in leisure (hours/week)
4.6–8.1
Ramey and Francis
(2009)
Age group (years)
Increase in leisure (hours/week)
18–24
25–54
55–64
3.66
−2.5
1.12
Table 1: Comparison of results between Aguiar and Hurst (2007) and Ramey and Francis
(2009).
3.3 The compensated consumption elasticity w.r.t. leisure
No studies that estimate the compensated consumption elasticity w.r.t. leisure,
i.e., 𝜀̃𝑐𝐿 , have been found. When it comes to the compensated consumption
elasticity w.r.t. market work, i.e., 𝜀̃𝑐𝑀 , the studies cited in Chetty (2006) showed
very small if any signs of any relationship (i.e., they indicated that 𝜀̃𝑐𝑀 ≈ 0).
𝑀
𝜕𝐻
However, 𝜀̃𝑐𝑀 can be used as a proxy of − 𝐿 𝜀̃𝑐𝐿 only if 𝜕𝑤 ≈ 0. Otherwise, the
compensated consumption elasticity w.r.t market work needs to be adjusted by
the ratio between the response in market work to the response in leisure 19, i.e.
𝜕𝑀
𝜕𝐿
/
(−
), which in turn is hard to find reliable data on.
𝜕𝑤
𝜕𝑤
One crude approach to approximate 𝜀̃𝑐𝐿 is to assume that consumption
responses to income-compensated leisure shocks are proportional to the share
of leisure-related expenditures. When using this approach, it is also necessary to
estimate what proportion of leisure time gives rise to extra leisure expenditures
𝑀
and to insert only that part of the leisure into the ratio 𝐿 in expression (13) or
(19).
Vandenbroucke (2009) reported that the leisure share of expenditures
increased from 3.0% in 1900 to 5.8% in 1950 in the United States. Weagley and
Huh (2005) reported leisure expenditure shares of 8.1%–8.6 % for near retired
and retired people in the United states based on data from 1995.20
In line with these statistics, Pawlowskia & Breuera (2012)21, Thompson and
Tinsley (1979), among others concluded that leisure expenditure is mostly a
hours of work measures.” (p. 211). However, it is somewhat unclear what they mean since Aguiar
and Hurst used two measures of market work, one that included meals at work and one that did not,
and four different measures of leisure, from which two did not include child care. Nevertheless,
comparing the estimate results of the studies, it is clear that there is a considerable difference of
estimates of core market work around 1965, where Ramey and Francis estimates are lower than
Aguiar and Hurst estimates.
19
Remember that
𝑀
𝐿
𝜀̃𝑐𝐿 =
𝜕𝑀
𝜕𝑤
/(
𝜕𝐿
𝜕𝑤
) ∙ 𝜀̃𝑐𝑀 (see Appendix 3).
20
Note that only the consumption bought in the market is included in these studies, but it is
preferable that household-produced goods are also included.
21
For Germany, Pawlowskia & Breuera (2012) found that the leisure expenditure share was about
3% in 2006.
15
Household Production and the Elasticity of Marginal Utility of Consumption
luxury good, and this might also indicate that complementarity between leisure
and consumption might increase with increased income.
16
Household Production and the Elasticity of Marginal Utility of Consumption
3.4 Proportion between market work and leisure
𝑀
The proportion between market work and leisure, i.e. 𝐿 , during the 20th century
among the age group 25–54 years in the United States, is calculated using Tables
𝑀
2, 4, and 5 in Ramey & Francis (2009). The mean ratio over the century is 𝐿 =
0.25 if leisure is calculated by all non-production time (including educational
𝑀
time, commuting time, and personal care time, e.g. sleep) and 𝐿 = 0.85 if leisure
is calculated as core (recreational) leisure only.
3.5 Proportion of unearned income
𝑦
When it comes to the proportion of unearned income to earned income, 𝑤𝑀,
Chetty used the ratio between total capital income to total labor income in the
United States (equal to 0.5) 22 because he argued that they should be equal on an
aggregate level23. He further argued that the skewed distribution of wealth
𝑦
implies that 𝑤𝑀 < 0.5 for most households.
𝑦
However, in the present study, 𝑤𝑀 is calculated with more precision (and only
for individuals participating in market work) for two example countries, the
United States and Sweden.24 For the United States, the median 𝑦 is divided by
median 𝑤𝑀 (data from U.S. Census Bureau) for each sex and year between 1960
𝑦
and 1968 and restricted to full-time workers. The result is that 𝑤𝑀 is in the
range 0.00–0.08, with a tendency for higher values in the latter period25.
For Sweden, mean 𝑦 is divided by mean 𝑤𝑀 (data from Statistics Sweden) for
five different worker categories and for each year from 2003 to 201326 . For
𝑦
white and blue-collar workers, 𝑤𝑀 ranges from 0.08 to 0.18, and there is a
tendency that
𝑦
𝑤𝑀
is decreasing in salary for a given year. For farmers/self𝑦
employed and other unspecified workers, 𝑤𝑀 reaches as high as 0.36.
22
A similar method to estimate
𝑦
𝑤𝑀
is to base it on labor shares, that is, the ratio between aggregate
𝑦
costs of labor to total GDP. Using the formula
≈ (1 − 𝑙𝑎𝑏𝑜𝑢𝑟_𝑠ℎ𝑎𝑟𝑒 )/𝑙𝑎𝑏𝑜𝑢𝑟_𝑠ℎ𝑎𝑟𝑒, where
𝑤𝑀
𝑙𝑎𝑏𝑜𝑢𝑟_𝑠ℎ𝑎𝑟𝑒 ranges from about 60% to 80% in developed countries and 50% to 70% in developing
𝑦
countries based on Serres et al (2002) and Figures 31 and 32 in Global Wage Report 2012/13,
𝑤𝑀
seems to be in the range 0.25 to 0.67 in developed countries and in the range 0.43 to 1.0 in
developing countries during the years 1970–2010. The United States is of special interest because
𝑦
many other estimates used in the present study are based on U.S. data. For the United States,
𝑤𝑀
based on labor share is about 0.56–0.67.
23
Unearned income for individuals includes transfers, but on an aggregate level these are assumed
to cancel so that the only source left is capital income.
24
These countries were chosen due to good availability of official data. The US is important because
many other estimates used in the present study are based on US data.
𝑦
𝑦
25
Between 1960 and 1968,
is in the range 0.00–0.03, and between 1969 and 2014
is in the
𝑤𝑀
𝑤𝑀
range 0.01–0.08.
26
Except for two outlier years, 2007 and 2008, after a new government took power.
17
Household Production and the Elasticity of Marginal Utility of Consumption
3.6 Summary of the empirical evidence
The empirical evidence from the previous section is summarized in Table 2.
Parameters
Uncompensated labor
supply elasticity
Changes in household
production and leisure
Compensated
consumption elasticity
w.r.t. leisure
Proportion between
market work and
leisure
Source
Description
Results
Evers et al (2008)
Review, 30 studies,
OECD
−0.24 to 2.8, mostly
>0
Macroeconomic
studies
2 regressions,
Descriptive studies of
the development of
division of time
2 studies based on
overlapping survey
studies from the U.S.
Mostly <0
Aguiar and Hurst
(2007)
Development in the
U.S. 1965–2003,
among prime-age
individuals, based on
multiple survey
studies.
Household
production has
decreased, leisure
has increased
Ramey and Francis
(2009)
Development in the
U.S. 1900–2005,
based on various
data sources, mainly
survey studies.
Household
production has
decreased, leisure
has increased for
total population, no
changes for primeage population
Chetty (2006)
Summary of previous
studies
Vandenbroucke
(2009),
Leisure expenditure
shares in the US
Leisure expenditures
increased from 3% in
1900 to around 8%
in 1995
1900–2000 for age
group 25–54 years
𝑀
1 GE model
Weagley and Huh
(2005)
Ramey and Francis
(2009)
<0
Ramey and Francis
(2009) showed
unchanged time
spent on labor supply
from prime-age
individuals
𝑐
𝜀𝑐,𝑀
≈0
𝐿
= 0.85 based on
core leisure only
𝑀
𝐿
= 0.25 if leisure =
non-production time
Proportion unearned
𝑦
income,
U.S. Census Bureau
Full-time workers, US
0.00–0.08
Statistics Sweden
Working population,
Sweden
0.08–0.18 for white
and blue-collar
workers
𝑤𝑀
Table 2: Summary of empirical evidence for relevant parameter values.
18
Household Production and the Elasticity of Marginal Utility of Consumption
3.7 Discussion of parameter values
When reviewing the results from the labor supply studies, it seems that the
uncompensated labor supply elasticity w.r.t. wage is positive. However, these
studies are often contextual, since they do not control for what the alternative to
market work is. For example, for an individual who has housekeeping as her
main task, and who has a market wage that is close to the price of such services,
i.e. 𝑔′′ (𝐻) ≈ 0, it is rational to have a high labor supply elasticity w.r.t. wage
irrespective of her EMUC value (as indirectly shown in equation (12a)). 27 Since
the underlying aim of many labor supply studies is to uphold the tax base when
reforms are made, it is not a problem that the labor supply elasticity of an
individual is high because she has the option to hire staff. But if these contextual
values are used to make inference about the utility function, the results will be
biased, as shown in Section 2.3.28
It is worth noting that Ramey and Francis’ (2007) results suggest that, for the
age group 25–54 years old, it is possible that decreases in working times are
largely driven by an increase in women’s participation rate. If the average wage
is strongly correlated with women’s participation, then it is possible that
estimates of the uncompensated labor supply elasticity from the
macroeconomic studies could be negatively biased. However, the long-run
historical trend is not an increase of mean working time, even for the prime-age
𝜕𝑀
population, which suggests the relationship 𝜕𝑤 ≤ 0.
The analysis in Section 2.4 shows that in order to calculate a point estimate of
EMUC based on labor supply responses to wage changes, household production
responses need to be estimated, which is not necessary for a lower bound
according to Section 2.3. However, it is difficult to draw conclusions on utility
parameters from changes over long time periods as there may have been
structural changes, for instance, due to changes in household production
technology, changes in social norms towards division of labor within the
household, etc. In other words, the function 𝑔(𝐻) is likely to have changed
somewhat. This means that the point estimates are much less reliable than the
lower bound estimates.29
Ramey and Francis found that leisure time has increased, whereas both
household production and market work have decreased for the total population,
while time allocation has remained stable in the long run for the prime-age
population. The implied difference in elasticities between the total population
and the prime-age population relates to partly different questions. The
This scenario; i.e. 𝑔′′ (𝐻) ≈ 0, could explain why the extensive margin responses in labor supply
are larger than the intensive margin responses (see equation (12a)).
28
Also note that since both the models in the present study and in Chetty’s (2006) use a linear
budget restriction, it might be that these models are somewhat unsuitable to estimate with
individual data in cases where the individuals face non-linear budget restriction due to kinks in the
income tax scheme.
29
In principle, changes in productivity of household production is also relevant for the lower bound,
but only if household production productivity increases are of at least the same magnitude as the
general productivity increase in the market-based economy, which I find unrealistic.
27
19
Household Production and the Elasticity of Marginal Utility of Consumption
estimates for the age group 25–54 years old relate to within-time-period
optimization of leisure and consumption, whereas the estimates from the total
population roughly relate to lifetime optimization. Because it is not
straightforward which one of them to choose, both approaches will be
considered in Section 4 (estimation results).
𝑀
For the proportion between market work and leisure, i.e. 𝐿 , one might think that
it makes a huge difference if
𝑀
𝐿
= 0.25 or
𝑀
𝐿
= 0.85. Fortunately, however, the
proposed EMUC formulas are not very sensitive to
𝑀
𝐿
when the complementarity
𝑀
between leisure and consumption is modest. For this reason, I will use 𝐿 = 0.85,
based on core leisure only, for the main analysis. The rationale for this is that
the share of leisure-related expenditures is used as a proxy for the
complementarity between leisure and consumption30, and it is reasonable that
only core leisure time influences the demand for leisure-related expenditures.
𝑦
When it comes to the proportion of unearned income, 𝑤𝑀, there is a large
difference between proxy estimates based on aggregate data and the more
detailed estimates based on more disaggregate data for the working population,
especially for the US case.31 The Swedish numbers for white and blue-collar
workers range from 0.08 to 0.18, which is still notably higher than the US range
of 0.00–0.08. One reason can be that Swedish data are not restricted to full-time
workers. Another probable explanation is that Sweden has a general welfare
system that includes a general monthly child support of about 100 euros per
child32.
As a conclusion of this section, evidence from historical macroeconomic data
suggests that uncompensated labor supply elasticity w.r.t wage is negative, i.e.,
𝜕𝑀
≤ 0. Evidence for the development of leisure time and household production
𝜕𝑤
is inconclusive, but none of the studies cited indicate that time spent on
household production has increased. The leisure expenditure share of
consumption has increased from about 0.03 at the beginning of the 20th century
to about 0.08 at the end of the century in the United States. The mean ratio of
market work to leisure has been about 0.85 when only core leisure is included,
and the ratio of unearned income to earned income has ranged from about 0.00
to 0.18 for the working population in the United States and Sweden during the
30
In turn, the rationale for using leisure-related expenditures as a proxy for complementarity
between leisure and consumption is the assumption that sleep does not generate extra expenses
and that the cost of commuting, personal care, and education are fairly inelastic w.r.t. nonproduction time, or at least marginal in size on average.
31
For the Swedish case, some of the highest estimates based on disaggregate data are in line with
the lowest estimates based on aggregate data for developing countries. On the other hand, the
highest estimates from the detailed data are from self-employed/farmers and other unspecified
workers. When it comes to being self-employed, the division between labor income and capital
income is not straightforward, and because it is more favorable from a tax avoidance perspective to
declare income as capital income, these figures may not be reliable. However, numbers for white
and blue-collar workers can be considered more reliable because they are based on third-party
reporting (employers, banks, etc.).
32
Possibly in combination with a less skewed capital distribution.
20
Household Production and the Elasticity of Marginal Utility of Consumption
last decades. Based on these stylized facts, it is possible to derive estimates of
EMUC, as will be done in the following section.
4
RESULTS
In this section I present numerical estimation results of EMUC – first in the form
of a sensitivity analysis of Chetty’s results, and second in the form of point
estimates and lower bounds based on the new model that includes household
production.
4.1 Sensitivity analysis of Chetty’s results
In this section I perform a sensitivity analysis of Chetty’s (2006) result, starting
with the bias given by the complementarity error factor (equation (16c)). I will
also look at the sensitivity to unearned income.
𝜂 𝐵
In Table 3, the values of the complementarity error factor, ( 𝜂̂) =
𝑦
𝜕𝑀
𝜕𝐿
)𝜀̃ ∙ /(− )
𝑤𝑀 𝑐𝑀 𝜕𝑤
𝜕𝑤
𝑦
1−(1+
)𝜀̃
𝑤𝑀 𝑐𝑀
1−(1+
𝑦
𝑤𝑀
, for different values of the input parameters
𝜕𝑀
𝜕𝐿
/ (− 𝜕𝑤),
𝜕𝑤
, and 𝜀̃𝑐𝑀 are presented.
Response proportions
(x units)
𝝏𝑴
𝝏𝒘
1
1
1
1
0
−1
1
1
1
1
1
1
𝝏𝑯
𝝏𝒘
−1
−1.015
−1.1
−2
−1
−1
0
−0.5
~−0.67
−0.9
−0.985
−1
𝝏𝑳
𝝏𝒘
+0
0.015
0.1
1
1
2
−1
−0.5
~−0.33
−0.1
−0.015
−0
𝜼 𝑩
Complementarity error factor,( 𝜼̂)
𝒚
𝒘𝑴
𝝏𝑴
𝝏𝑳
/ (−
)
𝝏𝒘
𝝏𝒘
0
𝜺𝒄𝒄,𝑴
−∞
~−66.67
−10
−1
0
0.5
1
2
3
10
~66.67
+∞
0.5
−0.1
0.1
−0.01
0.01
−∞
−5.15
0.00
0.82
0.91
0.95
1.00
1.09
1.18
1.82
6.97
+∞
+∞
8.52
2.22
1.22
1.11
1.06
1.00
0.89
0.78
0.00
−6.30
−∞
−∞
0.00
0.84
0.97
0.99
0.99
1.00
1.01
1.03
1.13
1.97
+∞
+∞
2.03
1.17
1.03
1.02
1.01
1.00
0.98
0.97
0.86
0.00
−∞
𝜂 𝐵
Table 3: The value of the complementarity error factor,( ̂) , for different parameter values.
𝜂
The first three columns show some (plausible) proportions in time allocation
𝜕𝑀
𝜕𝐿
responses to a wage change that result in a particular value of 𝜕𝑤 / (− 𝜕𝑤),
21
Household Production and the Elasticity of Marginal Utility of Consumption
which in turn will determine the value of the complementarity error factor33,
𝜂 𝐵
( ̂) . This error factor is calculated for two type cases. One is a reference case
𝜂
where 𝜀̃𝑐𝑀 = ±0.1 and there is no unearned income. The other type case is
constructed to match the findings in Chetty (2006) that 𝜀̃𝑐𝑀 is small in absolute
terms and unearned income is half that of earned income.
We now turn to sensitivity of Chetty’s results with respect to the proportion of
unearned income. When EMUC is estimated from the ratio between the
elasticity of production time w.r.t. unearned income and the compensated
production time elasticity w.r.t. wage, the formula includes the unearned
𝑤𝑀
𝑤𝑀
income factor (1 + 𝑦 ). The occurrence of (1 + 𝑦 ) is irrespective of whether
household production is included in the model or not, that is, (1 +
𝑤𝑀
𝑦
) is
included both in Chetty’s formula (5) and in equation (13), but equation (13) is
not used for empirical estimation in the present study34. In Figure 2, the
𝑤𝑀
unearned income factor (1 + 𝑦 ) is plotted as a function of the ratio between
𝑦
unearned and earned income, 𝑤𝑀.
(1+𝑤𝑀/𝑦)
50
40
0,03; 34,3
30
20
0,08; 13,5
10
0,18; 6,6
0,5; 3
1; 2
0
0
0,1
0,2
0,3
0,4
0,5
0,6
Figure 2: The unearned income factor (1 +
(vertical axis), as a function of
33
𝑦
𝑤𝑀
0,7
𝑤𝑀
𝑦
0,8
0,9
1
𝑦/𝑤𝑀
) of EMUC, included in equations (5) and (13)
(horizontal axis).
𝜂 𝐵
Please keep in mind that ( ̂) = 1 means no bias.
𝜂
34
The formulas used for empirical estimations in the present study are based on the uncompensated
𝑤𝑀
elasticity of production time, and in these cases the ratio of
is reversed so that the unearned
income factor instead is (1 +
𝑦
𝑤𝑀
𝑦
) and is thus not as sensitive to small values of
22
𝑦
𝑤𝑀
.
Household Production and the Elasticity of Marginal Utility of Consumption
4.2 Empirical estimates of EMUC, including household production
Based on the broad picture that emerges from the empirical evidence, it is
possible to calculate lower bound and point estimates of EMUC using the
equations derived in Sections 2.3 and 2.4.
Point estimates for the special case that total production time is insensitive to
𝜕𝑃
wage changes, i.e., 𝜕𝑤 = 0, coincide with lower bound estimates of EMUC when
𝜕𝑃
total production time decreases with wage, i.e., 𝜕𝑤 ≤ 0, or when the more
restrictive assumption that also market work decreases with wage holds, i.e.,
𝜕𝑀
≤ 0. It is assumed that the proportion between market work and leisure is
𝜕𝑤
𝑀
= 0.85 based on estimates of market work and core leisure from Ramey and
Francis (2009).35 These estimates are calculated using Proposition 4 with
different levels of compensated consumption elasticity w.r.t. leisure, 𝜀̃𝑐𝐿 , and
different proportions of unearned income, 𝑦/(𝑤𝑀). The results are shown in
Table 4:
𝐿
𝜺𝒄𝒄,𝑳
0.00
0.05
0.10
0.15
𝒚/(𝒘𝑴)
0.00
1.00
0.96
0.92
0.89
0.03
1.03
0.99
0.95
0.91
Table 4: EMUC estimates for the special case
𝜕𝑃
𝜕𝑤
0.08
1.08
1.03
0.99
0.95
0.18
1.18
1.12
1.07
1.03
= 0 based on Proposition 4.
𝜕𝑃
That 𝜕𝑤 = 0 is suggested by the leisure development for the age group 25–54
years in the United States during the 20th century, according to Ramey and
Francis (2009). However, this is not supported by the parallel trend for the total
population. The columns denote the proportion of unearned income, where
0.00, 0.03, and 0.08 are based on the median income of full-time workers in the
United States over the years 1960–2014 (0.00 is the minimum value, 0.03 is the
average, and 0.08 is the maximum) and 0.08 and 0.18 denote the lowest and
highest mean values, respectively, for white and blue-collar workers in Sweden
between 2003 and 2013. Values of 𝑀/𝐿 are based on the age group 25–54 years
in the United States between 1900 and 2005 in Ramey and Francis (2009). The
values of 𝜀̃𝑐𝐿 are loosely related to observed shares of leisure-related
expenditures in the United States during the 20th century36.
35
The effect of
marginal. For
𝑀
𝐿
𝑀
𝐿
= 0.25 and
𝑀
𝐿
= 1 has also been examined, and the differences in results are
= 0.25, EMUC estimates are about 0.03 to 0.10 units higher, and thus do not
influence the lower bound. For
𝑀
𝐿
= 1 (which represents the extreme value in the data set), EMUC
𝑀
estimates are 0.01 to 0.03 units lower. Also note that when 𝜀̃𝑐𝐿 = 0, the value of does not
𝐿
influence the value of the EMUC estimates.
36
The maximum value was 0.08 in 1996. But by including somewhat higher values, I allow for the
likely event that the share has continued to grow, as well as adding some safety margins since the
link to complementary is uncertain. Also, 𝜀̃𝑐𝑀 = 0.15 was the maximum value used in Chetty (2006),
so including 𝜀̃𝑐𝐿 = 0.15 facilitates comparison between studies.
23
Household Production and the Elasticity of Marginal Utility of Consumption
Table 5 shows point estimates of 𝜂 for the special case of additive utilities
𝜕2 𝑢
between leisure and consumption, i.e. 𝜕𝑐𝜕𝐿 = 0, based on Proposition 5.
𝝏𝑯 𝝏𝑴
/
𝝏𝒘 𝝏𝒘
𝜺𝑴,𝒘
−0.106
−0.124
−0.160
−0.106
−0.124
−0.160
−0.106
−0.124
−0.160
0
0.5
1
𝒚/(𝒘𝑴)
0.00
0.03
0.08
1.12
1.14
1.19
1.19
1.23
1.32
1.27
1.33
1.47
1.15
1.18
1.23
1.22
1.27
1.36
1.31
1.37
1.51
1.21
1.23
1.29
1.28
1.33
1.42
1.37
1.44
1.59
𝜕2 𝑢
Table 5: Estimates of 𝜂 for the special case
= 0, restricted to observed values of
𝜕𝑐𝜕𝐿
uncompensated labor supply w.r.t. wage, i.e. 𝜀𝑀𝑤 , and the ratio between unearned and earned
𝑦
income, , based on Proposition 5.
𝑤𝑀
Estimates of 𝜀𝑀,𝑤 are from Huberman and Minns (2007) and are based on firmlevel data for 14 OECD countries, predominantly the United States, in the period
1870–1900 (restricted to full sample estimates with different model
specifications37). The proportion of unearned income is based on median values
of earned and unearned income for each sex and year between 1960 and 2014
𝑦
in the United States38. 𝑤𝑀 = 0 denotes the sample minimum (women in 1965),
0.03 is the sample mean over the years (same for both sexes), and 0.08 is the
𝜕𝐻 𝜕𝑀
sample maximum (women in 2006). 𝜕𝑤 / 𝜕𝑤 = 0 is not an empirical value, but it
𝜕𝐻
𝜕𝑀
is included for comparison. The other two figures of 𝜕𝑤 / 𝜕𝑤 correspond roughly
to the parallel trends in household production and market work during 1965–
2003 in the United States, controlled for demographic changes, given by Aguiar
𝜕𝐻 𝜕𝑀
and Hurst (2007). 𝜕𝑤 / 𝜕𝑤 = 0.5 corresponds to the case that child care is
defined as household production, and
where child care is defined as leisure.
𝜕𝐻
𝜕𝑤
𝜕𝑀
/ 𝜕𝑤 = 1 corresponds to the case
The values of 𝜀𝑀,𝑤 are based on the parameter estimates for log wage in Tables 4 and 5 in
Huberman and Minns (2007) where −0.106 is the minimum value among 8 different model
specifications, −0.124 is the mean value, and −0.160 is the maximum value. Note that the minimum
of −0.106 denotes the specification that includes the most control variables, all of which are
significant.
38
Since 𝜀𝑀,𝑤 are old and based on data mostly from the United States, the more recent Swedish
𝑦
estimates of
do not seem relevant.
37
𝑤𝑀
24
Household Production and the Elasticity of Marginal Utility of Consumption
5
DISCUSSION
𝜂 𝐵
In Table 3, the value of the complementarity error factor, ( 𝜂̂) =
𝑦
𝜕𝑀
𝜕𝐿
)𝜀̃ ∙ /(− )
𝑤𝑀 𝑐𝑀 𝜕𝑤
𝜕𝑤
𝑦
1−(1+
)𝜀̃
𝑤𝑀 𝑐𝑀
1−(1+
, is estimated. Note that when the uncompensated labor
𝜕𝑀
supply elasticity w.r.t wage is negative, i.e.,
relationship holds:
𝜂
̂ 𝐵
𝜕𝑀
𝜕𝑤
𝜕𝑤
𝜕𝐿
< 0, then the following
/ (− 𝜕𝑤) ∈ [0,1]. This means that the size of the error
from ( 𝜂) will be less than 11% when 𝜀̃𝑐𝑀 < 0.1 and there is no unearned
income, and it will be less than 2% when 𝜀̃𝑐𝑀 < 0.01 and unearned income is
half of earned income.
On the other hand, among the studies cited by Chetty,
and the values of
𝜕𝐿
𝜕𝑤
well behaved close to
𝜂 𝐵
𝜕𝑀
𝜕𝑤
was mostly positive
𝜂 𝐵
were not known. Table 3 and Figure 2 show that ( 𝜂̂) is
𝜕𝑀
𝜕𝑤
/ (−
𝜕𝐿
𝜕𝑤
) = 1, but when |
𝜕𝐿
𝜕𝑤
𝜕𝑀
| is small compared to |
𝜕𝑤
|,
the function ( 𝜂̂) becomes unstable, ranging from 0 to ±∞. From this
𝜕𝐿
perspective it is worrisome that there is some indication that 𝜕𝑤 might be close
to 0 (Ramey and Francis (2009)).
In addition, theoretical analysis (equation (17b)) shows that the income
𝜕𝐻
𝜕𝑀𝑐
response error factor tends to zero as |𝜕𝑤| approaches 𝜕𝑤 , which means that
𝜂 𝐴
(𝜂̂) is also sensitive to large values of |𝜕𝑤|. Lastly, I showed in Figure 3 that
𝜕𝐻
Chetty’s results are sensitive to small values of the share of unearned income,
which are supported by empirical data. This sensitivity is not exclusive to
Chetty’s model, but occurs because EMUC is estimated from the ratio between
the uncompensated labor supply elasticity w.r.t. wage and the labor supply
elasticity w.r.t. unearned income (instead of only the uncompensated labor
elasticity w.r.t. wage). From this perspective, it is recommended that formula
(13) be directly estimated only with great caution and only if reliable estimates
of the proportion of unearned income are available with high precision.
Unfortunately, using a simplified39 version of Proposition 4 (as in the present
study) Chetty (2006, p. 1829) gives the impression that the EMUC estimates in
formula (5) are also decreasing along with the proportion of unearned income,
when in fact the opposite relationship holds. In principle, this undermines the
upper bounds of EMUC estimated by equation (9) in that study, since Chetty
𝑦
does not consider any proportions of unearned income other than 𝑤𝑀 = 0.5.
Let us now turn to the new estimates of EMUC from the present study. The
results presented in Table 4 suggest that EMUC is higher than 0.9. In the two
𝜕𝑀
𝑦
Chetty’s (2006) version of the inequality is
≥ 0 ⇔ 𝜂̂ < 1 + , that is, household production
𝜕𝑤
𝑤𝑀
and complementarity, are left out compared to Proposition 4.
39
25
Household Production and the Elasticity of Marginal Utility of Consumption
𝜕𝑃
𝜕2 𝑈
special cases of 𝜕𝑤 = 0 and 𝜕𝑐𝜕𝐿 = 0, EMUC is estimated to be in the range 0.9–
1.6. None of these results contradict Chetty’s main empirical findings that EMUC
is bound below 2. However, the average estimate of that study for the case when
there is no complementarity between labor supply and consumption was 0.71 40,
which is below the lowest bound estimated here. This estimate was driven
down by a predominance of micro-data studies. The only two macro studies41
cited in Chetty (2006) yielded significantly higher estimates (1.74 and 1.78),
which are higher than all the estimates in the present study. Also, had Chetty
𝑦
𝑦
used, for example, 𝑤𝑀 = 0.03 instead of 𝑤𝑀 = 0.5, all results of that study would
have been about 11 times higher, as can be seen in Figure 2.
Let us now turn to the new estimate results from the present study. There are
two obvious problems with the estimates in Table 5. The first is that that the
wage elasticity of labor supply used is derived from old data42, which
unfortunately means that they could be biased due to diminishing productivity
of work hours. The second is that the observed estimates of the other included
parameters are more recent. Interestingly, EMUC estimates in Table 5
correspond well to EMUC estimates in previous studies (Evans, 2005).
It is worth noting that Table 4 suggests that the lower bound of EMUC is close to
unity, which is the central value used in several important recent CBA
applications, for example, The Stern Review (Stern, 2006) and the UK Treasury
(Evans, 2006).
6
CONCLUSION
In the present study, I have shown theoretically that the results in the original
study by Chetty (2006) are sensitive to the proportion of unearned income and
that the omission of household production may lead to bias. In particular, if the
observed compensated labor supply responses are partly counteracted by
unobserved changes in household work, and if the compensated consumption
elasticity w.r.t. leisure is negligible (as was suggested in Chetty’s study), this
model will produce a negative bias, which in turn will compromise the upper
40
The mean estimate of 0.71 is low compared to the EMUC estimates in former studies (Evans,
2005), but Chetty did not at all relate to the previous EMUC literature. His aim was instead to show
that his EMUC estimates did not support an expected utility-based formulation of risk aversion.
41
However, these two studies suffered from multiple problems; the most serious is that they did not
look at responses to wage changes, but to income tax changes. On a macroeconomic level, tax
changes do not only influence labor supply responses through net wage, but again, through division
of work where higher taxes will lead to a larger degree of unpaid work and a larger informal sector.
In addition, the two studies were not peer reviewed and were based on partly the same data.
42
One positive aspect of this is that complementarity between leisure and consumption was
probably negligible at this time (leisure expenditure as the share of consumption was low around the
𝑐
year 1900 according to Vandenbroucke (2009)), which implies that 𝜀𝑐,𝐿
≈ 0 is a reasonable
approximation.
26
Household Production and the Elasticity of Marginal Utility of Consumption
bound43. This happens because substitution between consumption and leisure
cannot be distinguished from substitution between market work and household
production, which is a choice independent of EMUC. I have also derived point
estimators of EMUC for two plausible special cases from which one can be used
to estimate a lower bound of EMUC under more general conditions. I further
have offered some empirical estimates based on current evidence of the
included parameters where the lower bound of EMUC is about 0.9 and point
estimates are in the range 0.9–1.6.
An EMUC value larger than unity means that the marginal utility of consumption
diminishes fast enough for extra leisure to be more attractive than extra income
when wages increase. Even though a person who gets a raise can consume
more, she still prefers to work less because the extra leisure is worth more at
the margin. The present study has suggested a lower bound of EMUC close to
unity, which is substantially higher than some of the previous estimates
(including the mean estimate in Chetty (2006)). The mechanism behind this
result is that extra leisure can be earned without reducing the labor supply. The
reason is that a wage increase can be spent in order to devote less time to
household work. Obvious examples of options for substituting consumption for
time spent in household production include purchasing a washing machine or a
dishwasher. This means that an individual, whose EMUC value is higher than
unity, might still choose not to reduce her working time as her wage increases.
This is the basic logic for motivating why the value of EMUC should be higher
than indicated by Chetty (2006). This may have important consequences for the
appropriate way to trade off current and future consumption, as a higher EMUC
value implicates a higher discount rate as well as increased impact of risk
aversion and equity weighting when applicable.
One difficulty in estimating EMUC using the models developed in the present
study has been to find unbiased estimates of the uncompensated labor supply
elasticity w.r.t. wage. In order to obtain reliable point estimates of EMUC from
the proposed formulas, new research with the aim to study the effect of wages
on total work supply, rather than the effect of taxes on individual labor supply,
is ultimately needed.
As a last point, I want to address the possibility that the within-household
substitution between paid and unpaid work might be of increasing importance
due to increasing gender equality. As gender roles become less rigid, more tasks
are socially acceptable to handle for both sexes. As more women participate in
market work, the substitution between housework and market work becomes
more explicit (if a woman stays at home full time, it might be natural to handle
the majority of housework independent of her husband’s wage). The implication
is that bias from a model without household production, estimated with
individual data, might increase over time.
43
The upper bound of 2 is also not strongly supported by macroeconomic data; the two
macroeconomic studies included in Chetty’s study both yielded upper bound estimates of 2.3.
27
Household Production and the Elasticity of Marginal Utility of Consumption
APPENDIX 1
Setting 𝐿 = 𝜏 − 𝑀 − 𝐻, (8) can be rewritten as a maximization problem in 𝑀, 𝐻,
with the FOC:
𝜕𝑢
𝜕𝑢
𝜕𝑐
𝜕𝑢
∙ 𝑤 − 𝜕𝐿 = 0,
(A1.1)
∙ 𝑔′(𝐻) − 𝜕𝐿 = 0.
(A1.2)
𝜕𝑢
𝜕𝑐
It follows that:
𝑔′ (𝐻) = 𝑤.
(A1.3)
From this, we see that, for an internal solution, 𝐻 is dependent only on 𝑤 and
𝜕𝐻
hence independent of all other variables, for example 𝑦 or 𝑀. Thus,
= 0,
𝜕𝑦
which in turns implies:
𝜕𝐿
𝜕𝑀
= − 𝜕𝑦 .
𝜕𝑦
(A1.4)
Differentiating (A1.3) w.r.t. 𝑤 gives:
𝜕𝐻
𝑔′′ (𝐻) ∙ 𝜕𝑤 = 1,
𝜕𝐻
1
= 𝑔′′ (𝐻).
𝜕𝑤
(A1.5)
Thus, 𝑔′′ (𝐻) < 0 (from equation (8)) gives:
𝜕𝐻
< 0.
𝜕𝑤
(A1.6)
APPENDIX 2
(A1.1) is basically the same equation as equation (3) in Chetty (2006). It allows
us to follow the procedure given by equation (4) in Chetty (2006):
Differentiation of (A1.1) w.r.t. 𝑦:
𝜕2 𝑢
𝑤 𝜕2 𝑐 (1 + 𝑤 ∙
Inserting
𝜕𝐿
𝜕2 𝑢
𝜕𝑀
𝜕𝐿
) + 𝑤 𝜕𝑐𝜕𝐿 ∙ 𝜕𝑦 =
𝜕𝑦
𝜕2 𝑢
𝜕2 𝐿
𝜕𝐿
𝜕𝑀
= − 𝜕𝑦 (from A1.4) and solving for
𝜕𝑦
𝜕𝑀
𝜕𝑦
𝜕2 𝑢
=
𝜕2 𝑢
−𝑤 2 +
𝜕 𝑐 𝜕𝑐𝜕𝐿
𝜕2 𝑢 𝜕2 𝑢
𝜕2 𝑢
𝜕𝑐
=
𝜕𝑀
𝜕𝑦
𝜕𝑀
𝜕𝑦
).
𝜕2 𝑢
(A2.1)
gives:
.
(A2.2)
𝑤 2 2 + 2 −2𝑤
𝜕𝑐𝜕𝐿
𝜕 𝑐 𝜕 𝐿
Differentiation of (A1.1) w.r.t. 𝑤:
𝜕𝑢
𝜕2 𝑢
∙ 𝜕𝑦 + 𝜕𝑐𝜕𝐿 ∙ (1 + 𝑤 ∙
𝜕𝑀
𝜕𝐻
𝜕2 𝑢
𝜕𝐿
+ 𝜕2 𝑐 (𝑤 ∙ 𝜕𝑤 + 𝑀 + 𝑔′ (𝐻) ∙ 𝜕𝑤) ∙ 𝑤 + 𝜕𝑐𝜕𝐿 ∙ 𝜕𝑤 ∙ 𝑤 =
𝜕2 𝑢
𝜕2 𝐿
𝜕𝐿
𝜕2 𝑢
𝜕𝑀
𝜕𝐻
∙ 𝜕𝑤 + 𝜕𝑐𝜕𝐿 ∙ (𝑤 ∙ 𝜕𝑤 + 𝑀 + 𝑔′ (𝐻) ∙ 𝜕𝑤),
28
(A2.3)
Household Production and the Elasticity of Marginal Utility of Consumption
In the following, it will be convenient to define total production time as:
𝑃 =𝑀+𝐻.
(A2.4)
Which means that:
𝜕𝑃
𝜕𝑤
𝜕𝑀
𝜕𝐻
= 𝜕𝑤 + 𝜕𝑤.
(A2.5)
Which means that the time constraint in equation (8) gives:
𝜕𝐿
𝜕𝑃
= − 𝜕𝑤.
𝜕𝑤
(A2.6)
𝜕𝑃
Inserting (A2.5) and (A2.6) into (A2.3) and solving for 𝜕𝑤 gives:
𝜕𝑃
𝜕𝑤
=
𝜕𝑢
𝜕2 𝑢
𝜕2 𝑢
−𝑤𝑀 2 +𝑀
𝜕𝑐
𝜕𝑐𝜕𝐿
𝜕 𝑐
𝜕2 𝑢 𝜕2 𝑢
𝜕2 𝑢
𝑤 2 2 + 2 −2𝑤
𝜕𝑐𝜕𝐿
𝜕 𝑐 𝜕 𝐿
−
.
(A2.7)
Following the procedure given by equations (5) and (6) in Chetty (2006), 𝜂 can
be solved from the expression (after simplification):
𝜕𝑀
𝜕𝑦
𝜕𝑃
𝜕𝑀
−𝑀
𝜕𝑤
𝜕𝑦
𝜕2 𝑢
=
𝜕2 𝑢
−𝑤 2 +
𝜕 𝑐 𝜕𝑐𝜕𝐿
.
𝜕𝑢
−
(A2.8)
𝜕𝑐
𝜕2 𝑈
can be solved for in (A2.8), by following the procedure prior to equation (8)
in Chetty (2006):
𝜕𝑐𝜕𝐿
Consider an agent who faces the possibility of two states 1 and 2 with
probability p of state 1. Consumption can be traded freely between states at a
rate determined by p. The agent faces the following maximization problem:
max(𝑝 ∙ 𝑢(𝑐1 , 𝐿1 ) + (1 − 𝑝) ∙ 𝑢(𝑐2 , 𝐿2 )),
𝑐1 ,𝑐2
s.t.
(A2.9)
𝑝 ∙ 𝑐1 + (1 − 𝑝) ∙ 𝑐2 = 𝑝 ∙ 𝑌1 + (1 − 𝑝) ∙ 𝑌2 ,
where 𝑌𝑠 , is the total income in each state, S:
𝑌𝑠 = 𝑦𝑠 + 𝑤𝑠 𝑀𝑠 + 𝑔(𝐻𝑠 ),
where 𝐻𝑠 is the optimal level of household production time in each state and 𝑀𝑠
can be either exogenous or set at an optimal level. Hence, 𝑌𝑠 can be treated as
exogenous when optimizing the level of consumption in each state. It follows
that:
𝜕𝑢(𝑐1 ,𝐿1 )
𝜕𝑢(𝑐2 ,𝐿2 )
=
.
(A2.10)
𝜕𝑐
𝜕𝑐
29
Household Production and the Elasticity of Marginal Utility of Consumption
𝜕𝑢
A first-order Taylor expansion 𝜕𝑐 around 𝑐1 gives:
𝜕𝑢(𝑐2 ,𝐿2 )
𝜕𝑐
=
𝜕𝑢(𝑐1 ,𝐿1 )
𝜕𝑐
+
𝜕2 𝑢(𝑐1 ,𝐿1 )
𝜕2 𝑐
∙ ∆𝑐 +
where ∆𝑐 = 𝑐2 − 𝑐1 and ∆𝐿 = 𝐿2 − 𝐿1 .
Inserting (A2.10) into (A2.11) gives:
−
𝜕2 𝑢(𝑐1 ,𝐿1 )
𝜕2 𝑐
𝜕2 𝑢(𝑐1 ,𝐿1 )
𝜕𝑐𝜕𝐿
∙ ∆𝐿 + 𝑅,
𝜕2 𝑢(𝑐1 ,𝐿1 )
𝜕𝑐𝜕𝐿
∆𝑐 − 𝑅,
(A2.11)
∆𝐿 =
(A2.12)
𝜕2 𝑢(𝑐1 ,𝐿1 )
𝜕𝑐𝜕𝐿
∆𝑐
= lim (− ∆𝐿 ∙
𝜕2 𝑢(𝑐1 ,𝐿1 )
𝜕2 𝑐
∆𝐿→0
𝜕2 𝑢
=−
𝜕𝑐𝜕𝐿
𝜕𝑐 𝑐
𝜕𝐿
),
𝜕2 𝑢
∙ 𝜕2 𝑐 .
(A2.13)
(A2.14)
𝜕𝑐 𝑐
where 𝜕𝐿 is the response in consumption due to a marginal and fully
compensated change in leisure.
Inserting (A2.14) into (A2.8) and multiplying by – 𝑐 gives:
𝜕2 𝑢
−
𝑐∙ 2
𝜕 𝑐
𝜕𝑢
𝜕𝑐
∙ (𝑤 +
𝜕𝑀
𝜕𝑦
𝜂 = − 𝜕𝑃
𝑐 ∙
𝜕𝑤
𝜂=−
𝜂 =−𝑃
𝜀̃
𝑀 𝑃𝑤
𝜕𝐿
𝑐
𝑤+
𝜕𝑀 𝑦
∙
𝜕𝑦 𝑀
𝑃 𝜕𝑃𝑐 𝑤
∙
∙
𝑀 𝜕𝑤 𝑃
𝜀𝑀𝑦
𝜕𝑐 𝑐
∙
∙
𝜕𝑐𝑐
𝜕𝐿
)=−
𝜕𝑀
𝜕𝑦
𝜕𝑃𝑐
𝑐∙
,
(A2.15)
𝜕𝑤
,
𝑐
𝑦
𝑐 𝑀 𝜕𝑐𝑐 𝐿
1+
∙ ∙ ∙
𝑤𝑀 𝐿 𝜕𝐿 𝑐
(A2.16)
,
(A2.17)
𝑤𝑀
)
𝑦
𝑦 𝑀
(1+(1+
) ∙𝜀̃ )
𝑤𝑀 𝐿 𝑐𝐿
(1+
.
30
(A2.18)
Household Production and the Elasticity of Marginal Utility of Consumption
APPENDIX 3
(A2.13) can be expressed as:
𝜂=−
𝜕𝑀
𝜕𝑦
𝜕𝐻 𝜕𝑀𝑐
+
𝜕𝑤 𝜕𝑤
𝑀
𝑦
)
𝑤𝑀
𝑦 𝑀
(1+(1+
) 𝜀̃ )
𝑤𝑀 𝐿 𝑐𝐿
(1+
∙
.
(A3.1)
The corresponding equation to (A2.13) using Chetty’s original model is equation
(5), which can be expressed as:
𝜂̂ = −
𝜕𝑀
𝜕𝑦
𝜕𝑀𝑐
𝜕𝑤
𝑀
∙
𝑦
)
𝑤𝑀
𝑦
(1−(1+
)𝜀̃ )
𝑤𝑀 𝑐𝑀
(A3.2)
𝑦 𝑀
) 𝜀̃
𝑤𝑀 𝐿 𝑐𝐿
𝑦
1−(1+
)𝜀̃
𝑤𝑀 𝑐𝑀
(A3.3)
(1+
,
which means that:
𝜂
̂
𝜂
=
𝜕𝐻 𝜕𝑀𝑐
+
𝜕𝑤 𝜕𝑤
𝜕𝑀𝑐
𝜕𝑤
Inserting:
𝜀̃𝑐𝐿 =
gives that:
𝜂
̂
𝜂
𝜕𝑐 𝑐
𝜕𝐿
∙
1+(1+
𝜕𝑐 𝑐
𝐿
∙𝑐 =
= (1 +
𝜕𝐻
𝜕𝑤
𝜕𝑀𝑐
𝜕𝑤
∙
𝜕𝑀
)∙
.
𝜕𝑀 𝐿
𝜕𝐿
∙ 𝑐,
(A3.4)
𝑦 𝜕𝑐𝑐 𝑀 𝜕𝑀
) ∙ ∙
𝑤𝑀 𝜕𝑀 𝑐 𝜕𝐿
𝑦
1−(1+
)𝜀̃
𝑤𝑀 𝑐𝑀
1+(1+
,
(A3.5)
which can be expressed as:
𝜂
̂
𝜂
𝑦
𝜕𝑀
𝜕𝐿
)𝜀̃ ∙ /(− )
𝑤𝑀 𝑐𝑀 𝜕𝑤
𝜕𝑤
𝑦
1−(1+
)𝜀̃
𝑤𝑀 𝑐𝑀
1−(1+
.
= (1 +
𝜕𝐻
𝜕𝑤
𝜕𝑀𝑐
𝜕𝑤
)∙
(A3.6)
𝜂 𝐴
Let us now turn to (𝜂̂) = 1 +
𝜕𝐻
𝜕𝑤
𝜕𝑀𝑐
𝜕𝑤
from equation (16a). Because
𝜕𝑀𝑐
𝜕𝑤
> 0 and
𝜂
̂)𝐴 ∈ (0,1] holds as long as 𝜕𝑀𝑐 > |𝜕𝐻 |. The question that remains to be
<
0,
(
𝜕𝑤
𝜂
𝜕𝑤
𝜕𝑤
𝜕𝐻
𝜕𝑀𝑐
𝜕𝐻
answered is if it is possible that 𝜕𝑤 ≤ |𝜕𝑤|. To answer that question, let us
define the compensated total work supply as:
𝜕𝑃 𝑐
𝜕𝑤
𝜕𝑃
= 𝜕𝑤 − 𝑃
𝜕𝑃
𝑦
.
(A3.7)
Using simple addition and subtraction, it is straightforward to show that:
𝜕𝑃 𝑐
𝜕𝑤
=
𝜕𝑀𝑐
𝜕𝑤
𝜕𝐻
+ 𝜕𝑤.
(A3.8)
31
Household Production and the Elasticity of Marginal Utility of Consumption
Which means that:
𝜕𝑀𝑐
𝜕𝐻
𝜕𝑃 𝑐
≤
|
|
⇔
≤ 0.
𝜕𝑤
𝜕𝑤
𝜕𝑤
(A3.9)
Expanding (A3.8) gives:
𝜕𝑃 𝑐
𝜕𝑤
𝜕𝑃
𝜕𝑀
𝜕𝑢
𝜕𝑐
𝜕2 𝑢 𝜕2 𝑢
𝜕2 𝑢
𝑤 2 2 + 2 −2𝑤
𝜕𝑐𝜕𝐿
𝜕 𝑐 𝜕 𝐿
𝜕𝑃 𝑐
−
= 𝜕𝑤 − 𝑀 𝜕𝑦 =
From (A3.10) it follows that
𝜕𝑢
𝜕𝑤
.
(A3.10)
= 0 can only occur at the asymptotic limit,
lim ( ) = 0 (and only if the numerator of (A3.10) approaches zero faster than
𝑐→∞ 𝜕𝑐
𝜕𝑃 𝑐
the denominator with increasing consumption). 𝜕𝑤 = 0 will therefore be
disregarded in the following. It also follows from (A3.10) that:
𝜕𝑃 𝑐
𝜕𝑤
𝜕2 𝑢
𝜕2 𝑢
𝜕2 𝑢
< 0 ⇔ 𝑤 2 𝜕2 𝑐 + 𝜕2 𝐿 − 2𝑤 𝜕𝑐𝜕𝐿 > 0,
𝜕2 𝑢
1
𝜕2 𝑢
1 𝜕2 𝑢
< 2 (𝑤 𝜕2 𝑐 + 𝑤 𝜕2 𝐿 ).
𝜕𝑐𝜕𝐿
(A3.11)
(A3.12)
where the right hand side of (A3.12) is negative. The economic interpretation is
that there needs to be a large degree of substitutability between 𝑐 and 𝐿. In the
limiting case with perfect substitutability between 𝑐 and 𝐿, we have:
𝑢(𝑐, 𝐿) = 𝑓(𝑘𝑐 + 𝐿),
(A3.13)
where 𝑘 > 0.
The second-order conditions are:
𝜕2 𝑢
= 𝑘 2 𝑓′′,
(A3.14)
𝜕2 𝑐
𝜕2 𝑢
𝜕𝑐𝜕𝐿
𝜕2 𝑢
𝜕2 𝐿
= 𝑘𝑓′′,
= 𝑓′′,
where 𝑓′′ is negative.
Then, if
𝜕𝑃 𝑐
𝜕𝑤
< 0, (A3.13) inserted into (A3.10) gives:
𝑤 2 𝑘 2 𝑓′′ − 2𝑤𝑘𝑓′′ + 𝑓′′ > 0,
𝑤 2 𝑘 2 − 2𝑤𝑘 + 1 < 0.
(A3.15)
32
Household Production and the Elasticity of Marginal Utility of Consumption
which does not have any real roots for 𝑤𝑘, which indicates that (A3:12) would
represent an even higher degree of substitutability than for perfect substitutes.
This in turn would implicate a concave indifference curve (see Figure 3).
c
Indifference curve
c
Indifference curve
Budget
Budget
𝐿
𝐿
Case A: Convex indifference curve
Case B: Concave indifference curve
Figure 3: Two different indifference curves. Case A represents a normal, convex indifference
curve, and Case B represents a concave indifference curve.
From Figure 3 it is easy to see that when the indifference curve is concave, the
only two optimal mixes of 𝑐 and 𝐿 are the two extremes of 100% consumption
or 100% leisure, which violates either equation (9a) or (9b)44.
As a conclusion
𝜕𝑃 𝑐
𝜕𝑀𝑐
𝜕𝑤
> 0 must hold, and hence:
𝜕𝐻
> |𝜕𝑤|,
(A3.16)
𝜕𝑤
𝜂
̂ 𝐴
(𝜂) ∈ (0,1].
(A3.17)
𝜂 𝐵
For (𝜂̂) we have that:
𝐵
𝜂
|(𝜂̂) − 1| = ||
𝜂 𝐵
|(𝜂̂) − 1| = ||
1−(1+
𝜕𝑀
𝑦
𝑦
̃ 𝑐𝑀 )
)𝜀̃𝑐𝑀 ∙ 𝜕𝑤
𝜕𝐿 −(1−(1+𝑤𝑀)𝜀
𝑤𝑀
(−
)
𝜕𝑤
𝑦
1−(1+
)𝜀̃
𝑤𝑀 𝑐𝑀
𝜕𝑀
𝑦
𝑦
)𝜀̃𝑐𝑀 −(1+
)𝜀̃𝑐𝑀 ∙ 𝜕𝑤
𝜕𝐿
𝑤𝑀
𝑤𝑀
(−
)
𝜕𝑤
𝑦
1−(1+
)𝜀̃
𝑤𝑀 𝑐𝑀
||,
(1+
44
||,
Note that equation (9b) is only violated if 100% of endowment can be traded for leisure. But if that
is not the case, equation (9c) is violated instead.
33
Household Production and the Elasticity of Marginal Utility of Consumption
𝐵
𝜂
|(𝜂̂) − 1| = ||
𝜕𝑀
𝑦
)
)𝜀̃𝑐𝑀 (1− 𝜕𝑤
𝜕𝐿
𝑤𝑀
(−
)
𝜕𝑤
𝑦
1−(1+
)𝜀̃
𝑤𝑀 𝑐𝑀
(1+
|.
|
(A3.18)
APPENDIX 4
I will now derive a point estimator of EMUC for the special case
Inserting the Slutsky equation (7) into (A3.1) gives:
𝜂=
𝜂=
𝜕𝑀𝑐 𝜕𝑀
−
𝜕𝑤 𝜕𝑤
𝜕𝐻 𝜕𝑀𝑐
𝑦
)
𝑤𝑀
𝑦 𝑀
(1+(1+
) 𝜀̃ )
+
𝑤𝑀 𝐿 𝑐𝐿
𝜕𝑤 𝜕𝑤
𝑐
𝑦
𝜕𝐻 𝜕𝑀 𝜕𝑀 𝜕𝐻
(1+
)
+
− −
𝑤𝑀
𝜕𝑤 𝜕𝑤 𝜕𝑤 𝜕𝑤
𝑦 𝑀
𝜕𝐻 𝜕𝑀𝑐
(1+(1+
) 𝜀̃ )
+
𝑤𝑀 𝐿 𝑐𝐿
𝜕𝑤 𝜕𝑤
𝑦
𝜕𝑃
(1+
)
𝑤𝑀
𝜕𝑤
𝑦 𝑀
𝜕𝐻 𝜕𝑀𝑐
(1+(1+
) 𝜀̃ )
+
𝑤𝑀 𝐿 𝑐𝐿
𝜕𝑤 𝜕𝑤
(1+
∙
,
∙
𝜂 = (1 −
,
)∙
.
𝜕𝑃
𝜕𝑤
= 0.
(A4.1)
(A4.2)
(A4.3)
𝜕𝑃
When 𝜕𝑤 = 0, (A4.3) reduces to:
𝜂=
𝑦
)
𝑤𝑀
𝑦 𝑀
(1+(1+
) 𝜀̃ )
𝑤𝑀 𝐿 𝑐𝐿
(1+
.
(A4.4)
(A3.16) gives:
𝜕𝑃
𝜕𝑃
𝜕𝑤
𝜕𝐻 𝜕𝑀𝑐
+
𝜕𝑤 𝜕𝑤
𝑦
(1+
)
𝑤𝑀
𝑦 𝑀
(1+(1+
) 𝜀̃ )
𝑤𝑀 𝐿 𝑐𝐿
< 0 ⇒ (1 −
𝜕𝑤
𝜕𝑃
𝜕𝑤
<0⇒𝜂 >
) > 0,
(A4.5)
.
(A4.6)
𝜕𝐻
Also, because 𝜕𝑤 < 0:
𝜕𝑀
𝜕𝑤
𝜕𝑃
< 0 ⇒ 𝜕𝑤 < 0.
(A4.7)
34
Household Production and the Elasticity of Marginal Utility of Consumption
APPENDIX 5
I will now derive an EMUC formula given the following assumption:
𝑢(𝑐, 𝐿) ≈ 𝑓(𝑐) + 𝑘 ∙ 𝐿,
FOC:
𝜕𝑢
𝜕𝑐 𝜕𝑀
(A5.1)
𝜕𝑐 𝜕𝐻
𝜕𝐿
= (𝜕𝑀 𝜕𝑃 + 𝜕𝐻 𝜕𝑃 ) + 𝑘 𝜕𝑃 = 0,
𝜕𝑃
𝜕𝑢
𝜕𝑀
𝜕𝐻
= 𝑓′(𝑐) (𝑤 𝜕𝑃 + 𝑔′(𝐿) 𝜕𝑃 ) − 𝑘 = 0.
𝜕𝑃
(A5.2)
Inserting (A1.3) into (A5.2) gives:
𝜕𝑢
𝜕𝑃
𝜕𝑀
𝜕𝐻
= 𝑓′(𝑐) ∙ 𝑤 ( 𝜕𝑃 + 𝜕𝑃 ) − 𝑘 = 0,
𝑘 = 𝑤𝑓′(𝑐).
(A5.3)
Differentiation of (A5.3) w.r.t. 𝑤 gives:
𝜕𝑐 𝜕𝑀
𝜕𝑐 𝜕𝐻
0 = 𝑓′(𝑐) + 𝑤𝑓′′(𝑐) (𝑀 + 𝜕𝑀 𝜕𝑤 + 𝜕𝐻 𝜕𝑤),
𝜕𝑐 𝜕𝑀
𝜕𝑐 𝜕𝐻
−𝑤𝑓′′(𝑐) (𝑀 + 𝜕𝑀 𝜕𝑤 + 𝜕𝐻 𝜕𝑤) = 𝑓′(𝑐),
𝜕𝑀
𝜕𝐻
−𝑤𝑓′′(𝑐) (𝑀 + 𝑤 (𝜕𝑤 + 𝜕𝑤)) = 𝑓′(𝑐),
𝑓′′(𝑐)
− 𝑓′(𝑐) =
𝜂=
𝜂=
𝜂=
1
𝑤(𝑀+𝑤(
𝑐
𝑤(𝑀+𝑤(
,
𝜕𝑀 𝜕𝐻
+ ))
𝜕𝑤 𝜕𝑤
,
𝜕𝑀 𝜕𝐻
+ ))
𝜕𝑤 𝜕𝑤
(𝑦+𝑤𝑀)
,
𝑤 𝜕𝑀 𝜕𝐻
+ ))
𝑀 𝜕𝑤 𝜕𝑤
𝑤𝑀∙(1+ (
1+
𝑦
𝑤𝑀
𝜕𝐻
.
(A5.4)
𝜕𝑤 )
1+𝜀𝑀𝑤 (1+ 𝜕𝑀
𝜕𝑤
35
Household Production and the Elasticity of Marginal Utility of Consumption
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