Parallel Session IARIW 2014

1. The views expressed herein are those of the author and should not necessarily be attributed to the IMF,
its Executive Board, or its management
IARIW 33rd General Conference, Rotterdam
Session 2A
August 25
On the Valuation of Leisure, Labour
Supply and Household Production
Erwin Diewert and Paul Schreyer authors
Marshall Reinsdorf, discussant
8/28/2014
1
IMF Statistics Department
The views expressed are those of the authors and should not be attributed to the IMF, its Executive Board or its
management, nor to the US Department of Commerce or the Bureau of Economic Analysis

2. Main Innovation of the Paper
 In Becker model of time use, consumers divide time
between market work and leisure
 Full Income is FB = Y + wLT, where wL is after tax wage.
 In this paper, non-market time is divided between leisure
activities and household production
 Household production can be do-it-yourself or can pay a
household production worker wS per hour to do the work
 To increase leisure time, can either supply less market
work time or use more household production worker time
8/28/2014IMF Statistics Department
2

3. Maximization with 2 Constraints
8/28/2014IMF Statistics Department
3
F(.,.) = output of leisure activities; leisure time input is tF
H(.,.) = home production, but time input can be purchased for wS
Time spent on home production and time spent in market
production have negative marginal utility

4. Maximization problem with w* as
opportunity cost of time
8/28/2014IMF Statistics Department
4
Max U[F(qF,tF),H(qH, tH+qS),tH,tL] subject to:
pFqF + w*tF + pHqH + w*tH + wSqS  (wL  w*) tL  Y + w*T
Let w* be defined as */*
, the “shadow wage”.
Then maximization problem can be written as:
Diewert & Schreyer’s imputed full income FI = Y + w*T
Becker’s full income FB = Y + wLT

5. Case 1: consumer who both buys and
“sells” time
8/28/2014IMF Statistics Department
5
U4[F(qF
*,tF
*),H(qH
*,tH
*+qS
*),tH
*,tL
*] = * (wL  w*)
Together these conditions imply that:
0 < w*  min {wS,wL}

6. consumer who both buys and “sells” time
8/28/2014IMF Statistics Department
6
 Time doing household work, tH
*, should be valued at the opportunity cost of hiring
external staff wS to do this work provided that some staff is actually hired.
PH
*QH
* = pHqH
* + wS(qS
* + tH
*)
 Time in leisure activities, tF
*, should be valued at the household’s price of leisure time
w* which is probably less than both wS and the household’s after tax wage rate wL.
 Full consumption PF
*QF
* + PH
*QH
* less disutility of work equals full income:
PF
*QF
* + PH
*QH
*  (wS  w*)tH
*  (wL  w*)tL
* = Y + w*T

7. Consumer who does not buy time
8/28/2014IMF Statistics Department
7
there are now two unobserved imputed prices of time, w* (imputed
price of household leisure time) and wH
* (imputed price of
household working time)
0 < w*  wH
*  wS w*  min {wH
*,wL}
pFqF
* + w*tF
* + pHqH
* + wSI
*tH
*  (wH
*  w*)tH
*  (wL  w*)tL
* =
Y + w*T

8. Consumer who does not sell time
8/28/2014IMF Statistics Department
8
Set tL to 0. All we know is 0 < w*  wS

9. Consumer who doesn’t buy or sell time
8/28/2014IMF Statistics Department
9
0 < w*  wH
*  wS
pFqF
* + w*tF
* + pHqH
* + wH
*tH
*  (wH
*  w*)tH
* = Y + w*T
we need knowledge of two imputed prices of time, w* and wH
*.

10. Econometric estimation for case 1
8/28/2014IMF Statistics Department
10
pF
qF
/FB

wL
tF
/FB

pH
qH
/FB

wS
qS
/FB

wL
tH
/FB

Full income FB  Y + wLT
Equations for the following shares as functions of marginal utilities are derived:
Period  price of leisure w is complicated function of utility function parameters
Leisure goods
Leisure time
Household prod. goods
Household production labor
Household production time

11. Econometric estimation for other cases
 Strategies are explained for estimating utility function
parameters in cases where no labor is purchased for
household production and/or no market labor is supplied
 These cases involve corner solutions and are likely to be
common.
8/28/2014IMF Statistics Department
11

12. Recap
 Approach brings together Becker’s approach to the valuation
of household time approach, which uses the household’s
market wage, and the approach used by national income
accountants, which values time doing household chores at
the wage rates applicable for hired household help
 Paper relaxes assumptions made by Becker, including:
 No corner solutions (i.e. everyone works)
 No disutility of work (but authors recognize that allowing for
positive utility would be a useful extension)
 Add possibility of paying for household labor in addition to working
8/28/2014IMF Statistics Department
12

13. Comments
 Adding household production to Becker model and treating
corner solution cases are useful innovations
 Allowing for disutility (utility) of types work becomes necessary
when households can simultaneously sell and buy time
 Value of time depends on what it’s being used for
 Disutility of work is like a tax wedge
8/28/2014IMF Statistics Department
13

14. Comments (2)
 It’s impressive that the authors could even identify an
estimation strategy. Nonetheless estimation doesn’t seem
so straightforward. It would be nice to demonstrate
feasibility with an empirical illustration
 Does the result that w*  min {wS,wL} imply that time spent in
household production in satellite national accounts could be valued at
less than the hourly cost of employing household workers?
8/28/2014IMF Statistics Department
14