Disa Asplund

# Abstract

The popular approach to estimating the elasticity of marginal utility of consumption (EMUC) (Chetty 2006, “A New Method of Estimating Risk Aversion.” American Economic Review 96 (5):1821–1834. doi:10.1257/aer.96.5.1821. http://www.aeaweb.org/articles.php?doi=10.1257/aer.96.5.1821) is here extended by including household production. It is shown that this generalization of the model is important as omission of household production may lead to bias, as demonstrated in a numerical sensitivity analysis. An extended model with household production is used to derive new EMUC formulas. Empirical estimates based on current evidence of the included parameters suggest a lower bound for EMUC of about 0.9.

JEL Classification: D10; D60; J22

Funding statement: Center for Transport Studies, Stockholm (grant/award number: 353)

# Acknowledgments

I am most grateful to Prof. Raj Chetty, Stanford University, and Prof. Thomas Aronsson, Umeå University, for valuable comments. I would also like to send special thanks to my two outstanding PhD supervisors, Prof. Lars Hultkrantz, Örebro University, and Prof. Jan-Eric Nilsson, the Swedish National Road and Transport Research Institute (VTI). I thank Debbie Axlid for proof reading.

# Appendix

## A Theory

### Lemma 1

Setting L=τMH, (8) can be rewritten as a maximization problem in M,H, with the FOC:

(27)ucwuL=0
(28)ucg(H)uL=0.

It follows that:

(29)g(H)=w.

From this, we see that, for an internal solution, H is dependent only on w and hence independent of all other variables, for example y or M. Thus, Hy=0, which in turns implies:

(30)Ly=My.

Differentiating (28) w.r.t. w gives:

(31)g(H)Hw=1,Hw=1g(H).

Thus, g(H)\lt0 (from eq. [9]) gives:

(32)Hw\lt0.

### Proposition 1

Equation. (27) is basically the same equation as eq. (3) in Chetty (2006). It allows us to follow the procedure given by eq. (4) in Chetty (2006) :

Differentiating (27) w.r.t. y yields:

(33)w2uc2(1+wMy)+w2ucLLy=2uL2Ly+2ucL(1+wMy).

Inserting Ly=My (from 30) and solving for My gives:

(34)My=w2u2c+2ucLw22uc2+2uL22w2ucL.

Differentiating (27) w.r.t. w gives:

(35)uc+2uc2(wMw+M+g(H)Hw)w+2ucLLww=2uL2Lw+2ucL(wMw+M+g(H)Hw).

In the following, it will be convenient to define total production time as:

(36)P=M+H

which means that:

(37)Pw=Mw+Hw

which in turn means that the time constraint in eq. (8) gives:

(38)Lw=Pw.

Inserting (37) and (38) into (35) and solving for Pw gives:

(39)Pw=ucwM2uc2+M2ucLw22uc2+2uL22w2ucL.

Following the procedure given by eqs (5) and (6) in Chetty (2006), η can be solved from the following expression (after simplification):

(40)MyPwMMy=w2uc2+2ucLuc.

2UcL can be solved for in (40) by following the procedure prior to eq. (8) in Chetty (2006) :

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:

(41)maxc1,c2(pu(c1,L1)+(1p)u(c2,L2))

s.t.

pc1+(1p)c2=pY1+(1p)Y2

,

where Ys, is the total income in each state, S:

Ys=ys+wsMs+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:

(42)u(c1,L1)c=u(c2,L2)c.

A first-order Taylor expansion uc around c1 gives:

(43)u(c2,L2)c=u(c1,L1)c+2u(c1,L1)c2Δc+2u(c1,L1)cLΔL+R

where

Δc=c2c1

and

ΔL=L2L1

.

Inserting (42) into (43) gives:

(44)2u(c1,L1)cLΔL=2u(c1,L1)c2ΔcR
(45)2u(c1,L1)cL=limΔL0(ΔcΔL2u(c1,L1)c2)
(46)2ucL=ccL2uc2

where ccL is the response in consumption due to a marginal and fully compensated change in leisure.

Inserting (46) into (40) and multiplying by c gives:

(47)c2uc2uc(w+ccL)=cMyPcw
(48)η=MyPcwcw+ccL
(49)η=MyyMPMPcwwPcy1+cwMMLccLLc
(50)η=εMyPMε~Pw(1+wMy)(1+(1+ywM)MLε~cL).

### Proposition 2

Equation (49) can be expressed as:

(51)η=MMyHw+Mcw(1+ywM)(1+(1+ywM)MLε~cL).

The equation corresponding to (45) using Chetty’s original model is eq. (4), which can be expressed as:

(52)η^=MMyMcw(1+ywM)(1(1+ywM)ε~cM)

which means that:

(53)ηη^=Hw+McwMcw1+(1+ywM)MLε~cL1(1+ywM)ε~cM.

Inserting:

(54)ε~cL=ccLLc=ccMMLLc

gives that:

(55)ηη^=(1+HwMcw)1+(1+ywM)ccMMcML1(1+ywM)ε~cM

which can be expressed as:

(56)ηη^=(1+HwMcw)1(1+ywM)ε~cMMw/(Lw)1(1+ywM)ε~cM.

Let us now turn to (ηη^)A=1+HwMcw from eq. (21). Because Mcw\gt0 and Hw\lt0, (ηη^)A(0,1] holds as long as Mcw\gt|Hw|. The question that remains to be answered is whether it is possible that Mcw|Hw|. To answer this question, let us define the compensated total work supply as:

(57)Pwc=PwPPy.

Using simple addition and subtraction, it is straightforward to show that:

(58)Pwc=Mcw+Hw

which means that:

(59)Mcw|Hw|Pcw0.

Expanding (58) gives:

(60)Pcw=PwMMy=ucw22u2c+2u2L2w2ucL.

From (59) it follows that Pcw=0 can only occur at the asymptotic limit, limc(uc)=0 (and only if the numerator of (59) approaches zero faster than the denominator with increasing consumption). Pcw=0 will therefore be disregarded in the following. It also follows from (59) that:

(61)Pcw0w22uc2+2uL22w2ucL0
(62)2ucL\lt12(w2uc2+1w2uL2)

where the right-hand side of (62) 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:

(63)u(c,L)=f(kc+L)

where

k\gt0

.

The second-order conditions are:

(64)2uc2=k2f
2ucL=kf

,

2uL2=f

,

where f is negative.

Then, if Pcw\lt0, (64) inserted into (61) gives:

w2k2f2wkf+f\gt0

,

(65)w2k22wk+1\lt0

which does not have any real roots for wk. This indicates that (62) would represent an even higher degree of substitutability than for perfect substitutes. This in turn would implicate a concave indifference curve (see Figure 3).

### 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 eq. (10) or (11).[27]

As a conclusion Pcw\gt0 must hold, and hence:

(66)Mcw\gt|Hw|
(67)(ηη^)A(0,1].

For (ηη^)B we have that:

|(ηη^)B1|=|1(1+ywM)ε~cMMw(Lw)(1(1+ywM)ε~cM)1(1+ywM)ε~cM|

,

|(ηη^)B1|=|(1+ywM)ε~cM(1+ywM)ε~cMMw(Lw)1(1+ywM)ε~cM|

,

(68)|(ηη^)B1|=|(1+ywM)ε~cM(1Mw(Lw))1(1+ywM)ε~cM|.

### Proposition 3

I will now derive a point estimator of EMUC for the special case when Pw=0. Inserting the Slutsky equation (Footnote 4) into (50) gives:

(69)η=McwMwHw+Mcw(1+ywM)(1+(1+ywM)MLε~cL)
(70)η=Hw+McwMwHwHw+Mcw(1+ywM)(1+(1+ywM)MLε~cL)
(71)η=(1PwHw+Mcw)(1+ywM)(1+(1+ywM)MLε~cL).

When Pw=0, (70) reduces to:

(72)η=(1+ywM)(1+(1+ywM)MLε~cL).

(66) gives:

(73)Pw0(1PwHw+Mcw)0
(74)Pw0η(1+ywM)(1+(1+ywM)MLε~cL).

Also, because Hw\lt0:

(75)Mw\lt0Pw\lt0.

### Proposition 4

I will now derive an EMUC formula given the following assumption:

(76)u(c,L)f(c)+kL

FOC:

(77)uP=(cMMP+cHHP)+kLP=0,uP=f(c)(wMP+g(H)HP)k=0.

Inserting (29) into (77) gives:

(78)uP=f(c)w(MP+HP)k=0,k=wf(c).

Differentiating (76) w.r.t. w gives:

(79)0=f'(c)+wf''(c)(M+cMMw+cHHw),wf''(c)(M+cMMw+cHHw)=f'(c),wf''(c)(M+w(Mw+Hw))=f'(c),f''(c)f'(c)=1w(M+w(Mw+Hw)),η=cw(M+w(Mw+Hw)),η=(y+wM)wM(1+wM(Mw+Hw)).η=1+ywM1+εMw(1+HwMw)

## B Empirical evidence

### Uncompensated labor supply elasticity

Evers, De Mooij, and Van Vuuren (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, i. e., −0.08 to 2.8[28] for women and −0.24 to 0.45 for men. Moreover, as Blundell and MaCurdy (1999) and Evers, De Mooij, and Van Vuuren (2008) demonstrated, the estimation of such elasticities is a demanding task involving numerous potential pitfalls. For example, Evers, De Mooij, and Van Vuuren (2008) explained that “the standard model of labor supply does not distinguish between the effect of wages and taxes on the decision to participate (the extensive margin) and the 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).

At the long-term macroeconomic 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, e. g., Figure 1 in Morley et al. 2010). This observation is not new. In fact, already in 1957 it led Lewis to the conclusion that uncompensated wage elasticity must be negative. Burgoon and Baxandall (2004) and Huberman and Minns (2007) presented regression results of working time using mean income[29] as an independent variable for 14–18 OECD countries and time series data for 22 and 31 years, respectively[30] (with a difference of about a century between the two studies).[31] Both studies showed a significant negative relationship. Huberman and Minns (2007) provided estimates of uncompensated labor supply elasticity in the −0.11 to −0.16 range for the full sample.

However, none of these studies controlled for the diminishing productivity of workers with long working hours. Pencavel (2015) suggested that marginal productivity is rather constant up to about 50 hours/week for factory workers. According to; Huberman and Minns (2007); this threshold was passed on average (from above) around 1920 for all the OECD countries considered, which means that later studies of the uncompensated wage elasticity are more reliable in this respect.

### 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. The studies concerning the development of household production and leisure are based on data compiled from various survey studies. A complicating factor is that sometimes the division between household production and leisure is not straightforward, e. g., 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 (in both household production and market work, according to Ramey and Francis 2009) and longer average education.

Two recent studies of the development of ΔH over time (as wages have increased) in the United States arrived at partly different conclusions, despite using similar methods.[32]Aguiar and Hurst (2007), whose analysis was restricted to 21–64 year-old respondents who were neither students nor retirees, found substantial decreases in both market and non-market work and increases in leisure from 1965 to 2003.[33]Ramey and Francis (2009), who estimated allocation of time for all demographic ages for the years 1900–2005, found the same trends for the total population as Aguiar and Hurst found for 21–65 age group . But contrary to common expectations,[34] Ramey and Francis found that the allocation of time spent on market work, household production, and leisure averaged over both sexes in the 25–54 age group was basically the same in 2005 as a century ago.[35]

Table 3 shows the discrepancy[36] in leisure development between the two studies 1965–2003.

### Table 3:

Comparison of results between Aguiar and Hurst (2007) and Ramey and Francis (2009) .

 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) 18–24 25–54 55–64 Increase in leisure (hours/week) 3.66 −2.5 1.12

### The compensated consumption elasticity w.r.t. leisure

No studies that estimate the compensated consumption elasticity w.r.t. leisure, i. e., ε~cL, have been found. When it comes to the compensated consumption elasticity w.r.t. market work, i. e., ε~cM, the studies cited in Chetty (2006) showed very small if any signs of a relationship (i. e., they indicated that ε~cM0). However, ε~cM can be used as a proxy of MLε~cL only if Hw0. Otherwise, the compensated consumption elasticity w.r.t market work needs to be adjusted by the ratio between the response in market work and the response in leisure,[37] i. e., Mw/(Lw), which in turn is hard to find reliable data on.

One crude approach to approximating ε~cL is to assume that consumption responses to income-compensated leisure shocks are proportional to the share of leisure-related expenditures (out of total consumption). When using this approach, it is also necessary to estimate what proportion of out of total leisure time gives rise to extra leisure expenditures and to insert only that part of the leisure into the ratio ML in expression (18) or (26).

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 (2004) reported shares of expenditures spent on leisure of 8.1–8.6 % for near retired and retired people in the United States based on data from 1995.[38] In line with these statistics, e. g., Pawlowskia and Breuera (2012)[39]; Thompson and Tinsley (1979) concluded that leisure expenditure is mostly a luxury good, and this might also indicate that complementarity between leisure and consumption might increase with increased income.

### Ratio between market work and leisure

The ratio between market work and leisure, i. e., ML, in the twentieth century in the 25–54 age group in the United States is calculated using Table 2, Table 4, and Table 5 in; Ramey and Francis (2009). The mean ratio for the century is ML=0.25 if all non-production time (including educational time, commuting time, and personal care time, e. g., sleep) is considered leisure and ML=0.85 if we include only core (recreational) leisure.

### Ratio between unearned income and earned income

In the present study, the ratio of unearned income to earned income, ywM, is calculated with more precision than in Chetty (2006), by using official data from the United States and Sweden. For the United States, the mediany is divided by the median wM (data from the U.S. Census Bureau) for each gender and year 1960–1968 and restricted to full-time workers. ywM turns out to be in the 0.00–0.08 range, with a tendency of higher values later in the period.[40]

For Sweden, meany is divided by mean wM (data from Statistics Sweden, Table) for five different worker categories and for each year from 2003 to 2013.[41] For white- and blue-collar workers, ywM ranges from 0.08 to 0.18, and there is a tendency of ywM decreasing in salary for a given year.[42]

The Swedish numbers are notably higher than the U.S. range. One reason for this may 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 child.[43]

## C Results

### Sensitivity of Chetty’s result

Table 4 presents the values of the complementarity error factor, (ηη^)B=1(1+ywM)ε~cMMw/(Lw)1(1+ywM)ε~cM, for different values of the input parameters Mw/(Lw), ywM, and ε~cM.

### Table 4:

The value of the complementarity error factor,(ηη^)B, for different parameter values.

Response proportions

(x units)
Mw/(Lw)Complementarity error factor, (ηη^)B
ywM00.5
MwHwLwεcc,M−0.10.10.010.01
1−1+0++
1−1.0150.015~−66.67−5.158.520.002.03
1−1.10.1−100.002.220.841.17
1−21−10.821.220.971.03
0−1100.911.110.991.02
−1−120.50.951.060.991.01
10−111.001.001.001.00
1−0.5−0.521.090.891.010.98
1~−0.67~−0.3331.180.781.030.97
1−0.9−0.1101.820.001.130.86
1−0.985−0.015~66.676.97−6.301.970.00
1−1−0+++

The first three columns show some (plausible) time allocation responses to a wage change that result in a particular value of Mw/(Lw), which in turn will determine the value of the complementarity error factor[44](ηη^)B. This error factor is calculated for two type cases. One is a reference case where ε~cM=±0.1 and there is no unearned income. The other type case is constructed to match the findings in Chetty (2006) that ε~cM is small in absolute terms and unearned income is half that of earned income.

Table 5 shows point estimates of η for the special case of additive utilities between leisure and consumption, i. e., 2ucL=0, based on Proposition 4.

### Table 5:

Estimates of η for the special case when 2ucL=0, restricted to observed values of the uncompensated labor supply elasticity w.r.t. wage, i. e., εMw, and the ratio between unearned and earned income, ywM, based on Proposition 4.

Hw/(Mw)εM,wy/(wM)0.000.030.08
0−0.1061.121.151.21
−0.1241.141.181.23
−0.1601.191.231.29
0.5−0.1061.191.221.28
−0.1241.231.271.33
−0.1601.321.361.42
1−0.1061.271.311.37
−0.1241.331.371.44
−0.1601.471.511.59

Estimates of εM,w are from Huberman and Minns (2007) and are based on firm-level data for 14 OECD countries, predominantly the United States, in the period 1870–1900 (restricted to full sample estimates with different model specifications[45]). The ratio of unearned income to earned income is based on median values of earned and unearned income for each gender and year 1960–2014 in the United States.[46]Hw/Mw=0 is not an empirical value but is still included for comparison. The other two figures of Hw/Mw correspond roughly to the parallel trends in household production and market work 1965–2003 in the United States, controlled for demographic changes, given by Aguiar and Hurst (2007). Hw/Mw=0.5 corresponds to the case where child care is defined as household production, and Hw/Mw=1 corresponds to the case where it is defined as leisure.

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1. a

Maiden name Thureson

Published Online: 2017-9-1

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