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We are left with the eﬀect of the initial CRIP on the optimal CRIP, which is large and positive. An increase in the initial CRIP translates almost one-to-one into an increase in the optimal CRIP. This is strange at ﬁrst sight. How is it possible that the initial level of one variable determines its own optimal value? If we want to understand this eﬀect, we need to go back to the calibration of the utility function to the elasticity of labour supply at the intensive margin (see Appendix A.1.2). For given (distribution and substitution) parameters of the utility function, an increase in tax progressivity that the agent faces would lead to a higher labour supply elastiThe level of tπ has no eﬀect, because it is adjusted in the simulations.
In the model of this paper, the labour input maximum coincides with the output maximum and the maximum of tax revenue from the consumption tax.
city. For this very reason the marginal wage tax rate becomes the more distortive the higher it is. However, we calibrate utility functions in all countries to the same wage elasticity of labour supply. This means that for countries with high initial progressivity (used for calibration), this eﬀect must be counterbalanced by parameters that work in the direction of a low elasticity (namely, a low elasticity of substitution between consumption and leisure). Controlled for tax progressivity, labour supply is therefore less elastic in countries with high initial tax progressivity. This in turn means that the negative eﬀects of higher tax progressivity are less severe, so that optimal tax progressivity follows actual progressivity.
The argument in the previous paragraph allows us to trace back the eﬀect to the calibration of labour supply and makes it seem less strange than it was in the beginning. Most importantly, we must not interpret the eﬀect as causal. Of course, governments cannot change the level of the optimal tax progressivity just by changing progressivity itself. It is plausible, however, to interpret the eﬀect as indicative.
If it is the case that in certain countries the same level of labour supply elasticity is reached only at higher degrees of tax progressivity, governments can exploit this fact in setting the tax schedule. Politicians ﬁnd by trial and error that making taxes more progressive is not too distorting. This leads to higher tax progressivity in these countries, which in turn drives labour supply elasticities up to a level which is close to the average of other countries. The regression results suggest that governments are relatively good in implicitly adjusting the tax rates to the speciﬁc labour market conditions they face.
However, against the background of the results in this section, we must reconsider the question of whether it is a good idea to assume identical labour supply elasticities in all countries. Section 4.1 explores an alternative way of calibrating the utility functions, which allows for diﬀerences in the labour supply elasticities across countries.
4 Sensitivity analysis
In this section, I check the model results for robustness by exploring three themes that have emerged in the discussion so far. Is it possible to decouple the actual and the optimal level of tax progressivity by changing the calibration of labour supply?
(Section 4.1) To what extent do the results depend on the choice of the (arbitrary) parameter that determines the valuation of involuntary leisure? (Section 4.2) What eﬀect has the adjustment of the proﬁt tax rate on the results? (Section 4.3)
4.1 Calibration of labour supply elasticity
The fact that initial tax progressivity plays an important role in determining optimal progressivity, is explainable, but needs a closer inspection. This was the conclusion at the end of Section 3.4. The critical step that leads to this result is calibrating the utility function in all countries to the same values of the labour supply elasticities at the intensive margin (with respect to income, ηLY, and with respect to the wage, ηHw ). These exogenous elasticity values can only be combined with diﬀerent levels of the tax rates, ta and tm, if the parameters of the utility function, θC, σ and T, are L L allowed to vary across countries. This is shown in the left hand panel of Table 4. The calibrated elasticity of substitution between consumption and leisure varies between
1.73 and 1.93. It moves in parallel with the CRIP (see Table 1).18 So countries with low initial tax progressivity have a low elasticity of substitution, which weakens labour supply responses at the intensive margin and makes tax progressivity more attractive. This is exactly what produces the large regression coeﬃcient for the CRIP in Section 3.2.
What if we instead treated σ as the deeper parameter that does not change across countries? The right hand panel of Table 4 shows this case. To arrive at the elasticity values displayed, I have taken the average OECD model as point of departure, and then – with ﬁxed σ and θC – implemented the country-speciﬁc parameters19 as a counterfactual. The resulting elasticity values are displayed in columns “ηLY ” and “ηHw ”. We see that high tax progressivity (GER) now leads to high absolute values of the elasticities, and conversely (low progressivity in JPN and USA). The values of ηLY and ηHw from Table 4 are in turn used to re-calibrate the model at the initial point. This leads to a slight adjustment of θC, because the value share of leisure is With the elasticity values chosen as exogenous, σ happens to be precisely 1 + CRIP.
The only exception is the unemployment rate, which cannot be imposed on the model as an exogenous parameter.
not constant with a CES function.20
If we repeat the exercise of Sections 3.2 and 3.3 with the alternative method of labour supply calibration, we obtain the regression coeﬃcients listed in Table 5.
The fact that the coeﬃcient of the CRIP is still positive comes unexpected, given that the set-up of this section was aimed at correcting for this eﬀect. The most probable explanation is based on the interaction between tax progressivity and unemployment. Higher initial tax progressivity induces lower initial unemployment.
Unfortunately, the clear-cut distinction between calibration and simulation breaks down at this point. The diﬀerent values of ηLY and ηHw do not only depend on ta and tm, but also on the L L deviation of H from its initial value. This, in turn, depends on the wage and is only determined in full equilibrium.
Table 5: Partial eﬀects with alternative labour supply calibration
As unemployment is a separate regressor, the total eﬀect of initial tax progressivity is decomposed into the (positive) direct eﬀect captured by the parameter in Table 5 and the (negative) indirect eﬀect via unemployment. Added up, these two eﬀects might well be zero. However, performing such a check would require a separation of tax-progressivity-induced unemployment from unemployment caused by other country-speciﬁc conditions. I have not succeeded in ﬁnding a separation method that is consistent with the calibration of the wage bargaining system (Appendix A.1.4) and leave it therefore at this qualitative discussion.
Without further empirical analysis of diﬀerences in labour supply elasticities across countries, it cannot be decided which of the two calibration variants is to be preferred. At a ﬁrst glance, having initial tax progressivity determining optimal progressivity seems dubious. However, given the results of this section, the argument from the end of Section 3.4 can be reinforced: If it is really the case that countries with higher tax progressivity have the same labour supply elasticities as countries with low progressivity, the eﬀect is not unreasonable. It must be interpreted as an indicative rather than a causal eﬀect, however. Cross-country diﬀerences in tax progressivity reﬂect diﬀerences in labour supply conditions that would surface if taxes were the same everywhere.
Table 6: Cross-country diﬀerences with alternative labour supply calibration
To my knowledge there is no example of a meta-analysis of labour supply elasticities in the literature that takes tax progressivity as a regressor.21 We thus remain agnostic about the relative performance of the two calibration methods. All we can say is that the sign of the eﬀects does not depend on the approach chosen.22
4.2 Utility from involuntary leisure
In Section 2.3, we have seen that we are left with one free parameter in the calibration, which cannot be empirically founded: the utility from involuntary leisure, δ.
The value of δ cannot be one, because then – given the speciﬁcation of utility as a CES function of consumption and leisure and given empirically plausible values of the elasticities of labour supply – the unemployed would be better oﬀ than the Evers et al. (2005), e.g., has no regressors that capture institutions of the countries the studies reviewed are about.
The only exception is the value share of labour, but this eﬀect is small, anyway.
employed.23 In the light of the discussion on a “poverty trap”, this might even be realistic at least for a subset of individuals. However, it is not consistent with the wage bargaining set-up, which is based on the assumption that trade unions bargain over a wage that gives the employed utility on top of what they would have if they were unemployed.
In Section 2.3, I apply an ad-hoc solution to this problem by assuming that involuntary leisure gives only half the utility of leisure chosen voluntarily (δ = 0.5).
Now I investigate to what extent the results are sensitive to this assumption. As an alternative, extreme choice, I set the utility-of-involuntary-leisure parameter to zero (δ = 0). We might think of an unemployment beneﬁt system that requires beneﬁt recipients to work full-time in some public services. The consequences of this change are listed in Table 7 (with the same set-up as Table 2).
Table 7: Partial eﬀects with utility of involuntary leisure set to zero
Comparing Tables 2 and 7 shows that optimal tax progressivity is higher (lower CRIP) across the board.24 This is a plausible outcome. Through the lower valuation of involuntarily leisure utility diﬀerences between the employed and the unemployed Strictly speaking, this only applies to countries with a high replacement rate.
There is a single exception from this general picture: The optimal tax progressivity for the maximum value of the replacement rate, c, goes down instead of up. The mechanism is intricate.
If the replacement rate is high, the most important mechanism is that unemployment aﬀects the become larger. Hence larger gains in expected utility are achieved by reducing unemployment. And higher tax progressivity is the means to generate this outcome.
The change in the level of optimal tax progressivity is almost uniform across all model variations. This means that the marginal eﬀects, captured by the regression coeﬃcients, remain almost the same as in the main variant of the model. The unemployment rate and initial tax progressivity remain the most important determinants of optimal tax progressivity.
4.3 No adjustment of the proﬁt tax
In the counterfactual simulations of Section 3, we have assumed that the proﬁt tax is adjusted so as to keep the consumption level of the capitalists at a constant level.
This allows us to restrict ourselves on the expected utility of workers in the welfare analysis, because these are the only households that experience welfare changes.
However, a coordinated adjustment of proﬁt and labour income taxes is not particular realistic as a policy scenario (at least not in the way assumed in the simulations:
that capitalists are compensated for the indirect eﬀects from the labour tax reform).
To put the simulations into perspective, I run them once again without the adjustment of the proﬁt tax, and disregarding the income changes of the capitalists. This can be given two diﬀerent interpretations: either all capital is owned by foreigners, whose welfare does not enter the target function of the domestic government. Or we consider the case of a “labourist” government, which only cares about the welfare of workers.