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Relative Absence Concerns, Positional Consumption Preferences and Working Hours

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Positional income or consumption preferences can induce individuals to work too much to enhance their relative standing. Empirical evidence suggests that people are also characterised by relative considerations with respect to sickness-related absence from work. We analyse theoretically how relative consumption and absence concerns interact. Although relative absence concerns may mitigate the consequences of relative consumption preferences, the market outcome will never be efficient. Hence, we derive the income tax rate and the level of sick pay which induce efficient behaviour as the market outcome.


  • Externality
  • Income tax
  • Market outcome
  • Pareto-efficiency
  • Relative absence
  • Relative consumption
  • Sick pay

Manfred Holler’s list of publications (in Google Scholar) contains more than 400 entries. Few of them explicitly deal with labour issues—although he has written two textbooks in German on labour economics (Goerke & Holler, 1997; Holler, 1986). My academic interests, which developed during the work as Manfred’s chair in Hamburg, attest to the climate of intellectual openness and curiosity, which he created. Braham and Steffen (2008, p. vi) write in their introduction to the Festschrift for Manfred’s 60th birthday: ‘(H)e has always made every effort to free his staff from unnecessary administrative burdens and he never burdened anyone with his own work’. This implied that staff members were free to pursue their own research projects, also if only modestly linked to or even without any relationship to Manfred’s work. While, to the best of my knowledge, Manfred has not worked on positional concerns, the present analysis was certainly inspired by his attitude of openness and tolerance.

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  1. 1.

    See, for example, Seidman (1988), Choudhary and Levine (2006) and Arrow and Dasgupta (2009). Gómez (2008) presents a growth model in which the market equilibrium is efficient if the consumption and leisure externality have the same intensity. In the set-up by Aronsson and Johansson-Stenman (2013), there is asymmetric information with respect to ability, such that the two externalities would not balance out, even if they were equally strong. Alpizar et al. (2005) and Carlsson et al. (2007) present evidence based on hypothetical choice experiments that positional leisure preferences are less pronounced than relative income considerations.

  2. 2.

    See, inter alia, Duesenberry (1949), Boskin and Sheshinski (1978), Persson (1995), Ireland (1998), Corneo (2002), Gómez (2008), Dodds (2012), Aronsson and Johansson-Stenman (2013), Eckerstorfer (2014), Wendner (2014) and Goerke and Neugart (2021).

  3. 3.

    There are further contributions which point into the same direction. Aronsson et al. (1999) analyse the implications of interdependent labour supply behaviour for estimated labour supply elasticities, using repeated cross-sectional data from Sweden. They find that average working hours in a reference group raise individual labour supply. Pingle and Mitchell (2002) set up a hypothetical choice experiment. They present individuals with combinations of working time and income and report that the average level of hours worked affect individuals’ choices.

  4. 4.

    Palme and Persson (2020, Sect. 4) concisely review pertinent empirical studies. Miraglia and Johns (2021) provide a much broader survey of the literature on social determinants of absence behaviour, also including contributions from economics. To the best of our knowledge, the implications of relative absence concerns have not yet been analysed in a theoretical model. Somewhat related to our analysis, Skåtun and Skåtun (2004) analyse an efficiency wage model in which individuals can choose hours of work. The authors interpret this choice as a decision about absence behaviour. They assume that fewer hours worked by colleagues raise the workload of individuals and, hence, reduce the individual's working hours as well. The main prediction of the model is that, in contrast to traditional shirking frameworks, employment may be higher in the presence of efficiency wages than in their absence.

  5. 5.

    Bradley et al. (2014) investigate the impact of a move from temporary to permanent employment on absenteeism for public sector employees in Australia. In some of their specifications they include an indicator of the average absence level at the employee's workplace. The estimated coefficients are consistently positive and significant.

  6. 6.

    Absence can also have detrimental effects on future wages and employment (see, e.g., Hansen, 2000; Hesselius, 2007; Markussen, 2012; Scoppa & Vuri, 2014). While we do not model such consequences explicitly, one feasible short-cut in order to incorporate them into the model is the above assumption that utility from absence is distinct from that due to leisure.

  7. 7.

    Carrieri (2012) found that a higher sickness level of a reference group reduces well-being. If (1) higher sickness induces people to be absent more and (2) utility from absence can be approximated by subjective well-being, Carrieri’s (2012) result suggests v2 < 0. However, this line of argument may be problematic, given survey results that positional concerns with regard to health are relatively weak (cf. Solnick & Hemenway, 2005; Grolleau & Saïd, 2008; Wouters et al., 2015). These findings from surveys contrast with evidence from panel data for Australia (cf. Mujcic & Frijters, 2015) according to which the self-assessed health status of a peer group is consistently and strongly associated with a reduction in life satisfaction.

  8. 8.

    The subsequent findings are unaffected by the assumption that sick pay is untaxed, unless noted below (cf. Proposition 3). To focus on relative absence and consumption concerns, the model developed below is static. As mentioned above, there is substantial evidence that sickness-related absence has detrimental long-term labour market effects (Hansen, 2000; Hesselius, 2007; Markussen, 2012; Scoppa & Vuri, 2014). An alternative or additional way of including this empirical observation in the present static setting is the assumption that sick pay is less than the net wage, i.e. s < w[1 − t].

  9. 9.

    Note that terms in square brackets describe multiplicative components, while parentheses indicate a functional dependence.

  10. 10.

    It could be argued that absence has a distinct, positive utility effect. If, therefore, preferences were given by u(c, \(\overline{c}\)) − H(h, a) + v(a, γ\(\overline{a}\)), where the partial derivatives are \(\tilde{H}_{1}\) < 0 < \(\tilde{H}_{2}\), the nature of the first-order conditions for individual choices and for the characterisation of Pareto-efficiency would not be altered under mild additional restrictions (see Appendix 4). Hence, the findings derived below are unlikely to be affected.

  11. 11.

    Since effort is too low in market equilibrium and declines with sick pay, such payments can be argued to reduce presenteeism (see Pichler and Ziebarth (2017) for according empirical evidence).

  12. 12.

    Since the Pareto-efficient consumption level may be higher or lower if there are absence externalities than in a setting without such externalities, the magnitude of t*(c*, γ), relative to \(\hat{t}\), cannot be determined. An exception arises if consumption levels are the same, as it will be true for an iso-elastic utility function u. Since the Pareto-efficient consumption level does not vary with γ in such a setting, tax rates are also the same, i.e. t*(c*, γ) = \(\hat{t}\).

  13. 13.

    If sick pay were taxed, the level inducing efficient behaviour would have to be higher in absolute terms in order to counteract the mitigating impact of taxes and given by s*(c*, a*, γ) = γv2(a*, γ)/(u1(c*, c*, γ) + u2(c*, c*, γ)).


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1. Utility Maximum

From an individual’s perspective, the reference levels of consumption and absence, profits, the tax rate and the lump-sum transfer are constant. Therefore, the second-order conditions for a maximum are given by Zhh, Zaa < 0 < ZhhZaa − (Zah)2, where Zh and Za are defined in Eqs. (3a) and (3b) and the net wage equals \(w^{{\text{N}}}\) = w[1 − t].

$$Z_{{{\text{hh}}}} = u_{11} \left[ {w^{{\text{N}}} } \right]^{2} - H^{\prime\prime} < 0$$
$$Z_{{{\text{aa}}}} = u_{11} \left[ {s - w^{{\text{N}}} } \right]^{2} + v_{11} < 0$$
$$Z_{{{\text{ha}}}} = Z_{{{\text{ah}}}} = u_{11} w^{{\text{N}}} \left[ {s - w^{{\text{N}}} } \right] \ge 0$$

Hence, we have

$$Z_{{{\text{hh}}}} Z_{{{\text{aa}}}} - \mathop {Z_{{{\text{ah}}}} }\nolimits^{2} = u_{11} \left[ {v_{11} \left[ {w^{{\text{N}}} } \right]^{2} - H^{\prime\prime}\left[ {s - w^{{\text{N}}} } \right]^{2} } \right] - H^{\prime\prime}v_{11} > 0$$

2. Stability of Market Equilibrium

In equilibrium, lump-sum payments, T, are determined endogenously in order to balance the budget. Thus, T = wt[h − a]. Moreover, profits as defined in (4) are paid out to individuals and affect their consumption. Hence, the equilibrium level of consumption equals production:

$$\begin{aligned} c^{{\text{m}}} = & w\left[ {1 {-} t} \right]\left[ {h^{{\text{m}}} {-}a^{{\text{m}}} } \right] + sa^{{\text{m}}} + wt\left[ {h^{{\text{m}}} {-}a^{m}} \right] + f\left( {h^{{\text{m}}} {-}a^{{\text{m}}} } \right) - w\left[ {h^{{\text{m}}} {-}a^{{\text{m}}} } \right] - sa^{{\text{m}}} \\ = & \,f\left( {h^{{\text{m}}} {-}a^{{\text{m}}} } \right) \\ \end{aligned}$$

To ascertain whether the market equilibrium is stable, we calculate the Jacobian determinant \(\left| J \right|\) of the system defined by Eqs. (3a), (3b) and (5), taking into account (17). Moreover, all individuals behave identically. Hence, changes in consumption, c, and the reference level, \(\overline{c}\), are the same. Similarly, the variations in a and \(\overline{a}\) coincide. Thus, the derivatives of (3a), (3b) and (5) with respect to contractual hours, h, absence, a, and wages, w, incorporating (17), are given by πhh = f″ = −πha < 0, πhw = −1 and:

$$Z_{{{\text{hh}}}}^{{\text{e}}} = \left[ {u_{11} + u_{12} } \right]f^{\prime}w^{{\text{N}}} - H^{\prime\prime} < 0$$
$$Z_{{{\text{aa}}}}^{{\text{e}}} = - \left[ {u_{11} + u_{12} } \right]f^{\prime}\left[ {s - w^{{\text{N}}} } \right] + v_{11} + \gamma v_{12} < 0$$
$$Z_{{{\text{ha}}}}^{{\text{e}}} = - \left[ {u_{11} + u_{12} } \right]w^{{\text{N}}} f^{\prime} > 0$$
$$Z_{{{\text{ah}}}}^{{\text{e}}} = \left[ {u_{11} + u_{12} } \right]\left[ {s - w^{{\text{N}}} } \right]f^{\prime} > 0$$
$$Z_{{{\text{hw}}}}^{{\text{e}}} = u_{1} \left[ {1 - t} \right] = - Z_{{{\text{aw}}}}^{{\text{e}}} > 0$$

In (18a) to (18e), we use the superscript e to indicate that equilibrium repercussions via lump-sum payments T and profits are incorporated. The Jacobian determinant \(\left| J \right|\) of the system defined by the modified Eqs. (3a), (3b) and (5) is negative.

$$\begin{aligned} \left| J \right| = & - \left[ {Z_{{{\text{hh}}}}^\text{e} Z_{{{\text{aa}}}}^\text{e} - Z_{{{\text{ha}}}}^\text{e} Z_{{{\text{ah}}}}^\text{e} } \right] - f^{\prime\prime}Z_{{{\text{hw}}}}^\text{e} \left[ {Z_{{{\text{ha}}}}^\text{e} + Z_{{{\text{aa}}}}^\text{e} + Z_{{{\text{hh}}}}^\text{e} + Z_{{{\text{ah}}}}^\text{e} } \right] \\ = & \, - H^{\prime\prime}\left[ {\left[ {u_{11} + u_{12} } \right]\left[ {s - w^{{\text{N}}} } \right]f^{\prime} - \left[ {v_{11} + \gamma v_{12} } \right]} \right] \\ & + \,f^{\prime\prime}u_{1} \left[ {1 - t} \right]\left[ {H^{\prime\prime} - \left[ {v_{11} + \gamma v_{12} } \right]} \right] < 0 \\ \end{aligned}$$

3. Pareto-Efficient Allocation

The second-order conditions for a maximum of Γ are

$$\Gamma_{{{\text{hh}}}} = \left[ {u_{11} + 2u_{12} + u_{22} } \right]\left[ {f^{\prime}} \right]^{2} + \left[ {u_{1} + u_{2} } \right]f^{\prime\prime} - H^{\prime\prime} < 0$$
$$\Gamma_{{{\text{aa}}}} = \left[ {u_{11} + 2u_{12} + u_{22} } \right]\left[ {f^{\prime}} \right]^{2} + \left[ {u_{1} + u_{2} } \right]f^{\prime\prime} + v_{11} + 2\gamma v_{12} + \gamma^{2} v_{22} < 0$$

and \({\text{Det}} = \Gamma_{{{\text{hh}}}} \Gamma_{{{\text{aa}}}} - \mathop {\Gamma_{{{\text{ha}}}} }\nolimits^{2} > 0\). Using

$$\Gamma_{{{\text{ha}}}} = - \underbrace {{\left[ {u_{11} + 2u_{12} + u_{22} } \right]}}_{\left( - \right)}\left[ {f^{\prime}} \right]^{2} - \underbrace {{\left[ {u_{1} + u_{2} } \right]f^{\prime\prime}}}_{\left( - \right)} = - \Gamma_{{{\text{hh}}}} - H^{\prime\prime} > 0,$$

the determinant of the system of Eqs. (8a) and (8b) is found to be positive.

$$\begin{aligned} {\text{Det}} = & \underbrace {{\left[ {\left[ {u_{11} + 2u_{12} + u_{22} } \right]\left[ {f^{\prime}} \right]^{2} + \left[ {u_{1} + u_{2} } \right]f^{\prime\prime}} \right]}}_{\left( - \right)}\underbrace {{\left[ {v_{11} + 2\gamma v_{12} + \gamma^{2} v_{22} - H^{\prime\prime}} \right]}}_{\left( - \right)} \\ & - \,H^{\prime\prime}\underbrace {{\left[ {v_{11} + 2\gamma v_{12} + \gamma^{2} v_{22} } \right]}}_{\left( - \right)} > 0 \\ \end{aligned}$$

4. Alternative Specification of Preferences

Suppose preferences are given by

$$\tilde{Z}\left( {h,a} \right) = u\left( {c,\overline{c}} \right) - \tilde{H}\left( {h,a} \right) + v\left( {a,\gamma \overline{a}} \right),$$

where \(\tilde{H}_{1}\) < 0 < \(\tilde{H}_{2}\). Pareto-efficiency can then be characterised by maximising:

$$\tilde{\Gamma }\left( {h,a} \right) = u\left( {c\left( {h,a} \right),\overline{c}\left( {\overline{h},\overline{a}} \right)} \right) - \tilde{H}\left( {h,a} \right) + v\left( {a,\gamma \overline{a}} \right)$$

The first-order conditions for individually optimal choices and describing Pareto-efficiency are

$$\tilde{Z}_{h} = u_{1} w\left[ {1 - t} \right] - \tilde{H}_{1} \left( {h,a} \right) = 0$$
$$\tilde{Z}_{{\text{a}}} = u_{1} \left[ {s - w^{{\text{N}}}} \right] - \tilde{H}_{2} \left( {h,a} \right) + v_{1} = 0$$
$$\tilde{\Gamma }_{{\text{h}}} = \left[ {u_{1} + u_{2} } \right]f^{\prime}\left( {h - a} \right) - \tilde{H}_{1} \left( {h,a} \right) = 0$$
$$\tilde{\Gamma }_{{\text{a}}} = - \left[ {u_{1} + u_{2} } \right]f^{\prime}\left( {h - a} \right) - \tilde{H}_{2} \left( {h,a} \right) + v_{1} + \gamma v_{2} = 0$$

The properties of the model will be unaffected if (1) \(\tilde{H}_{2}\) is not too large in absolute value such that (25b) and (26b) define interior solutions for absence choices, and (2) v1(a, \(\gamma \overline{a}\)) − \(\tilde{H}_{2} \left( {h,a} \right)\) exhibits the same qualitative features as v1(a, \(\gamma \overline{a}\)) with respect to a.

5. Proof of Proposition 2

Part (a): If income is untaxed, working hours will be excessive, there will be too little absence from work and, hence, effort and consumption will also be excessive.

The combination of (11a), (11b), and (12a), (12b) shows that

$$H^{\prime}\left( {h^{{\text{m}}} } \right) = v_{1} \left( {a^{{\text{m}}} } \right)$$


$$H^{\prime}\left( {h^{*}} \right) = v_{1} \left( {a^{*}} \right).$$

Given the assumptions on the derivatives (H′, H″, v1 > 0 > v11), there are three possible combinations of market outcomes relative to the efficient combination:

Case (1): hm = h* and am = a*,

Case (2): hm < h* such that H′(hm) < H′(h*) and v1(am) < v1(a*), which implies am > a*,

Case (3): hm > h* and am < a*, according to the same line of argument as in Case (2).

In Case (1), Eqs. (27a) and (27b) hold, but (11a) and (12a), respectively (11b) and (12b), cannot be satisfied simultaneously. Therefore, hm = h* and am = a* do not guarantee that the conditions which characterise the market equilibrium and the efficient outcome are both fulfilled.

In Case (2), hm − am < h* − a* results, which implies that cm = f(hm − am) < c* = f(h* − a*) holds. This, in turn, indicates that f′(hm − am) > f′(h* − a*) and u1(cm) > u1(c*) due to the strict concavity of f and u. Furthermore, deducting (12a) from (11a) yields

$$\begin{aligned} u_{1} \left( {c^{{\text{m}}} ,\overline{c}} \right)f^{\prime}\left( {h^{{\text{m}}} - a^{{\text{m}}} } \right) - & H^{\prime}\left( {h^{{\text{m}}} } \right) - \left[ {\left[ {u_{1} \left( {c^{*},c^{*}} \right) + u_{2} \left( {c^{*},c^{*}} \right)} \right]f^{\prime}\left( {h^{*} - a^{*}} \right) - H^{\prime}\left( {h^{*}} \right)} \right] \\ = & \,\underbrace {{u_{1} \left( {c^{{\text{m}}} ,\overline{c}} \right)f^{\prime}\left( {h^{{\text{m}}} - a^{{\text{m}}} } \right) - u_{1} \left( {c^{*},c^{*}} \right)f^{\prime}\left( {h^{*} - a^{*}} \right)}}_{{ = {\text{A}}1}} \\ & + \,\underbrace {{H^{\prime}\left( {h^{*}} \right) - H^{\prime}\left( {h^{{\text{m}}} } \right)}}_{{ = {\text{A}}2}} - \underbrace {{u_{2} \left( {c^{*},c^{*}} \right)f^{\prime}\left( {h^{*} - a^{*}} \right)}}_{\left( - \right)} = 0 \\ \end{aligned}$$

In Case (2), the terms A1 and A2 are positive. Therefore, equality (28) cannot hold and hm < h*, am > a* do not describe the market outcome relative to the efficient situation.

In consequence, the only constellation of working hours and absence which simultaneously guarantees the conditions which describe the market equilibrium and the Pareto-efficient allocation is described by Case (3). If hm > h* and am < a*, cm > c* must also hold. ∎

Part (b): The tax rate \(\hat{t}\):  = t(s = γ = 0) which ensures that individuals choose the optimal number of working hours and the optimal duration of absence is given by

$$0 < \hat{t} = - \frac{{u_{2} \left( {c^{*},c^{*}} \right)}}{{u_{1} \left( {c^{*},c^{*}} \right)}} < 1.$$

This part can be demonstrated by substituting \(\hat{t}\):  = t(s = γ = 0) = −u2(c*, c*)/u1(c*, c*) in Eqs. (11a) and (11b). Given a unique market equilibrium, it can only be characterised by the values of h and a which fulfil Eqs. (12a) and (12b), i.e. the Pareto-efficient combination. As tax receipts are returned to individuals and they obtain all profit income, consumption will be the same as in the Pareto-efficient allocation, given the same levels of working hours and absence. ∎

6. Sick Pay

In market equilibrium, consumption equals production, cm = f(hm − am). Therefore, the derivatives of the first-order conditions (3a), (3b) and (5) with regard to sick pay, s, are \(Z_{{{\text{hs}}}}^{{\text{e}}}\) = πhs = 0 and \(Z_{{{\text{as}}}}^{{\text{e}}} = u_{1}\). Also taking into account (18a) to (18e), the changes in contractual hours, absence and effort due to a rise in sick pay, s, are found to be

$$\frac{{{\text{d}}h^{{\text{m}}} }}{{{\text{d}}s}} = \frac{{u_{1} }}{\left| J \right|}\left[ {f^{\prime\prime}Z_{{{\text{hw}}}}^{{\text{e}}} - Z_{{{\text{ah}}}}^{{\text{e}}} } \right] > 0$$
$$\frac{{{\text{d}}a^{{\text{m}}} }}{{{\text{d}}s}} = \frac{{u_{1} }}{\left| J \right|}\left[ {Z_{{{\text{hh}}}}^{{\text{e}}} + f^{\prime\prime}Z_{{{\text{hw}}}}^{{\text{e}}} } \right] > 0$$
$$\frac{{{\text{d}}\left( {h^{{\text{m}}} - a^{{\text{m}}} } \right)}}{{{\text{d}}s}} = - \frac{{u_{1} \left[ {u_{11} + u_{12} } \right]f^{\prime}s}}{\left| J \right|} < 0$$

7. Proof of Proposition 3

Notation: Market outcomes in a world with absence externalities are denoted by hm(am, γ), am(hm, γ) and cm(hm, am) = cm(hm(am, γ), am(hm, γ)), while the Pareto-efficient allocation is characterised by h*(a*, γ), a*(h*, γ), and c*(γ).

Part (a): If the tax rate is zero (t = 0), sick pay is non-negative (s ≥ 0), and higher absence by the reference group does not raise utility from absence (v2 ≤ 0), working hours in market equilibrium will be excessive, while the differences between Pareto-efficient and market outcomes with respect to absence and consumption are indeterminate.

The comparison of the first-order conditions characterising the market equilibrium and the Pareto-efficient outcome or of a combination of them does not provide insights with respect to the relative levels of working hours and absence. However, it can be shown that only a number of combinations of h, a and h − a are feasible. Basically, the differences [hm(am, γ) − am(hm, γ)] − [h*(a*, γ) − a*(h*, γ)], hm(am, γ) − h*(a*, γ) and am(hm, γ) − a*(h*, γ) could be positive, zero or negative. Hence, the theoretically maximal number of outcomes is 27. To simplify the subsequent argument, note that imposing a sign on the term Diff 1: = hm(am, γ) − am(hm, γ) − [h*(a*, γ) − a*(h*, γ)] = hm(am, γ) − h*(a*, γ) − [am(hm, γ) − a*(h*, γ)] implies that the same sign applies to the difference Diff 2: = cm(hm, am) − c*(γ) because c = f(h − a).

Some of the 27 feasible combinations are logically impossible. If Diff 2 > (<) 0 holds, hm(am, γ) − h*(a*, γ) ≤ (≥) 0 and am(hm, γ) − a*(h*, γ) ≥ (≤) 0 cannot occur simultaneously. This argument rules out 4 (and another 4) of the 27 combinations. Additionally, if Diff 2 = 0 holds, hm(am, γ) − h*(a*, γ) and am(hm, γ) − a*(h*, γ) must have the same signs. Hence, another six combinations cannot describe the market outcome relative to the Pareto-efficient allocation (cf. Table 1).

Table 1 Feasible and impossible combinations of working hours, absence and effort

We next consider the case of Diff 2 ≤ 0 again. This implies that u1(cm) ≥ u1(c*), given u11 + u12 < 0 and f′(hm − am) = w ≥ f′(h* − a*). As a result, u1(cm)w > [u1(c*) + u2(c*)]f′(h* − a*), since u2 < 0. The comparison of (3a) and (8a), assuming t = 0, clarifies that H′(hm) > H′(h*) < 0 must hold, because otherwise the equations cannot be fulfilled simultaneously. Given the convexity of H in h, H′(hm) > H′(h*) implies that hm(am, γ) > h*(a*, γ) holds. Accordingly, all theoretically feasible cases for which Diff 2 ≤ 0 is assumed are only compatible with hm(am, γ) > h*(a*, γ), ruling out a further 6 of the remaining 13 (27 − 4 − 4 − 6) combinations as incompatible with hm(am, γ), am(hm, γ) characterising the market equilibrium and h*(a*, γ), a*(h*, γ) the Pareto-efficient allocation (argument A).

Note that thus far the proof has required no restrictions with respect to sick pay and the sign of v2. Suppose, next, that hm(am, γ) ≤ h*(a*, γ) holds. In accordance with the above line of argument, this implies that H′(hm) < H′(h*) is true. Combining (3a), (3b) and (8a), (8b) yields

$$u_{1} \left( {c^{{\text{m}}} } \right)s - H^{\prime}\left( {h^{{\text{m}}} } \right) + v_{1} \left( {a^{{\text{m}}} } \right) = 0$$
$$\gamma v_{2} - H^{\prime}\left( {h^{*}} \right) + v_{1} \left( {a^{*}} \right) = 0$$

For γv2 < 0 and s ≥ 0 or γv2 = 0 and s > 0, Eqs. (31a) and (31b) can only hold at the same time if v1(am) < v1(a*), that is for am(hm, γ) > a*(h*, γ), and given v11 < 0 (argument B). Hence, two further combinations have been ruled out. Because no further incompatibilities of the first-order conditions, or combinations thereof, can be discerned, the above considerations leave 5 of the 27 permutations (see Table 1). All of them are characterised by hm(am, γ) > h*(a*, γ). ∎

The proof that hm(am, γ) > h*(a*, γ) is the only feasible outcome, assumes either a positive level of sick pay (s > 0) and γv2 ≥ 0, or non-negative sick pay (s ≥ 0) and envy with respect to absence (γv2 < 0); cf. argument B. Therefore, it also covers the case of positive sick pay and no absence externality. Hence, the above argument constitutes an alternative to the proof provided in Appendix 6 establishing that working hours in market equilibrium will be excessive if sick pay is positive.

Part (b): If a higher absence level by the reference group increases the level of absence chosen individually (v12 > 0), a greater strength of relative absence concerns, as captured by an increase in the parameter γ, raises the number of working hours and the duration of absence in market equilibrium, while effort declines.

Since \(Z_{{{\text{h}}\gamma }}^{e} = \pi_{{{\text{h}}\gamma }} = 0\) and \(Z_{{{\text{a}}\gamma }}^{e} = v_{12} \overline{a}\), the changes in working hours, absence and effort are

$$\frac{{{\text{d}}h^{{\text{m}}} }}{{{\text{d}}\gamma }} = - Z_{{{\text{a}}\gamma }}^{{\text{e}}} \underbrace {{\frac{{Z_{{{\text{ah}}}}^{{\text{e}}} + f^{\prime\prime}Z_{{{\text{aw}}}}^{{\text{e}}} }}{\left| J \right|}}}_{\left( - \right)}$$
$$\frac{{{\text{d}}a^{{\text{m}}} }}{{{\text{d}}\gamma }} = Z_{{{\text{a}}\gamma }}^{{\text{e}}} \underbrace {{\frac{{Z_{{{\text{hh}}}}^{{\text{e}}} + Z_{{{\text{hw}}}}^{{\text{e}}} f^{\prime\prime}}}{\left| J \right|}}}_{\left( + \right)}$$
$$\frac{{{\text{d}}\left( {h^{{\text{m}}} - a^{{\text{m}}} } \right)}}{{{\text{d}}\gamma }} = - Z_{{{\text{a}}\gamma }}^{{\text{e}}} \frac{{Z_{{{\text{hh}}}}^{{\text{e}}} + Z_{{{\text{ha}}}}^{{\text{e}}} }}{\left| J \right|} = Z_{{{\text{a}}\gamma }}^{{\text{e}}} \underbrace {{\frac{{H^{\prime\prime}}}{\left| J \right|}}}_{\left( - \right)}$$

Part (c): A greater strength of relative absence concerns has ambiguous consequences for the Pareto-efficient allocation.

The partial derivatives of Eqs. (8a) and (8b) with respect to γ are given by Γhγ = 0 and Γaγ = v2 + (v12 + γv22)\(\overline{a}\). Since Γaγ cannot be signed without specifying the utility function v, the changes in working hours, absence and effort in the Pareto-efficient allocation are ambiguous. ∎

Part (d): The tax rate and level of sick pay which induce a Pareto-efficient allocation as market outcomes are given by \(0 < t^{*}\left( {c^{*},\gamma } \right) = - \frac{{u_{2} \left( {c^{*},c^{*},\gamma } \right)}}{{u_{1} \left( {c^{*},c^{*},\gamma } \right)}} < 1\;{\text{and}}\;s^{*}\left( {c^{*},a^{*},\gamma } \right) = \frac{{\gamma v_{2} \left( {a^{*},\gamma } \right)}}{{u_{1} \left( {c^{*},c^{*},\gamma } \right)}}.\)

Replacing t and s in Eqs. (3a) and (3b) by −u2(c*, c*, γ)/u1(c*, c*, γ) and γv2(a*)/u1(c*, c*, γ) and using w = f′(h − a) from (5) shows that Eqs. (3a) and (3b) will hold for those values of working hours and absence which characterise the Pareto-efficient allocation described by Eqs. (8a) and (8b). All tax payments are returned to individuals via lump-sum payments. Moreover, individuals obtain the entire profit income. Consequently, income and consumption will be the same as in the Pareto-efficient allocation, given hm(am, γ) = h*(a*, γ) and am = a*. ∎

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Goerke, L. (2023). Relative Absence Concerns, Positional Consumption Preferences and Working Hours. In: Leroch, M.A., Rupp, F. (eds) Power and Responsibility. Springer, Cham.

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