# Modelling the Time Allocation Effects of Basic Income

• Ana Helena Palermo Kuss and K. J. Bernhard Neumärker
From the journal Basic Income Studies

## Abstract

Most of the economic models on basic income account just for pecuniary forms of work, i. e. “time spent making money”, in employment. This restriction is a drawback of these analyses and of the standard economic labor supply model itself. If one wants to understand the potential effects of basic income on individual and social welfare, one should not restrict observation to the pecuniary uses of time. The objective of this contribution is to rethink the meaning of work usually applied in economic models, based on contributions of other social scientists. This reassessment is undertaken through the development of a microeconomic model, which discusses the effects of basic income on time use and interprets work not just as a source of income, but also of non-pecuniary benefits. Further, we disentangle the usual work-leisure dichotomy in two other ones (paid and unpaid time/unpaid work and leisure).

Funding statement: This work was supported by the Funder Name: Konrad-Adenauer-Stiftung, Funder Id: 10.13039/501100004079, Grant Number: PhD scholarship for Ana Helena Palermo Kuss.

# Appendix

## A Derivation of first-order conditions (FOC) for the first decision

Insert equation (8) in eq. (7), then you will get:

(23)I=wTwTu+G

Insert eq. (23) into eq. (6):

(24)UA=wTwTu+GIminα.TuTuminβ

Then make a monotonic transformation of this equation, taking the ln and get:

(25)uA=αlnwTwTu+GImin+βlnTuTumin

To find out which level of Tu an individual would choose to maximize utility, differentiate the equation above in relation to Tu. At the maximization level, equation (26) will be equal to 0:

(26)uATu=wαwTwTu+GImin+βTuTumin=0

Solving for Tu gives:

(27)Tu=βα+β.T+GIminw+αα+βTumin

Moreover, to find out under which circumstances basic income (G) would lead an individual to invest no time in paid work, i. e. when Tu would be equal to the total available time (T), equalize equation (27) to T:

(28)T=βα+β.T+GIminw+αα+βTumin

Then reorganize it and call this level of basic income, that would lead an individual to leave the labor market, Gˉ:

(29)Gˉ=w.αβ.TTumin+Imin

## B Derivation of first-order conditions (FOC) for the second decision

Insert eq. (14) in eq. (13), and then you will get:

(30)UB=(TueTuemin)θ.TuTueTuiminδ

Then make a monotonic transformation of (30), taking the ln and get:

(31)uB=θ.lnTueTuemin+δ.lnTuTueTuimin

To find out which level of Tue an individual would choose to maximize utility, differentiate (31) in relation to Tue. At the maximization level, (32) will be equal to 0:

(32)uBTue=θTueTueminδTuTueTuimin=0

Then, reorganize (32) so that Tueis alone on one side:

(33)Tue=θθ+δ.(TuTuimin)+δθ+δTuemin

To get the optimized value for Tui, insert (33) in eq. (14) and reorganize the values:

(34)Tui=δθ+δ.(TuTuemin)+θθ+δTuimin

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Published Online: 2018-11-24

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