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These countries are characterised by seven parameters: share of labour in value added (sL ), average tax rates on consumption (tC ), labour (ta ) and capital (tπ ), L coeﬃcient of residual income progression (CRIP)9, unemployment rate (u) and replacement rate (c). These parameters are summarised in Table 1 (the exact sources are given in Appendix A.2). In addition, Table 1 reports the unweighted average of the parameters over all countries (row “AVR”), which will be used as a starting point and standard of comparison.

Apart from the country-speciﬁc paramters of Table 1, labour supply elasticities

**are an important input to calibration. These are assumed uniform across countries:**

The CRIP (coeﬃcient of residual income progression) is deﬁned as the elasticity of after-tax income with respect to pre-tax income, i.e. (1−tm )/(1−ta ). In a proportional tax regime, the CRIP L L is one, and the higher the progressivity of the tax schedule, the lower the CRIP. Jacobsson (1976) derives theoretical properties of this indicator and justiﬁes its use as a measure of tax progressivity.

Table 1: OECD parameters

elasticity of hours with respect to the after-tax wage (ηHw = 0.1), elasticity of hours with respect to income (ηLY = −0.1) and elasticity of participation with respect to the wage (ηN w = 0.2). The choice of the speciﬁc values is motivated in Appendices A.1.1 to A.1.3).

**3 Optimal tax progressivity**

In this section, I perform numerical simulations to determine the optimal degree of tax progressivity and identify its driving forces. In Section 3.1, I explain the determination of the optimal tax progressivity in the “average OECD” model. Then the driving forces are identiﬁed in two sets of numerical exercises. First, starting from the average values, I vary one parameter at a time to obtain partial eﬀects on optimal tax progressivity. Second, I run the fully speciﬁed country models and check to what extent the deviations from the average-OECD outcome can be decomposed into eﬀects of variations in the individual parameters.

3.1 “Average OECD” model As a point of reference, I calibrate the basic model of Section 2 to the unweighted OECD averages for all country-speciﬁc parameters (row “AVR” of Table 1). In this model version, I perform a number of tax variations that allow us to develop a feeling for the range of tax rates that we expect to be relevant in the model runs to follow.

They are summarised in Figure 1, which shows characteristic tax rate constellations in the (tm, ta ) space.

LL

The upper rightmost point of Figure 1, where the curves “Rev. max. tm” and “Rev. max. ta” meet, is the point of maximum tax revenue.10 We may call it “Leviathan point”, because it is the point a malevolent, exploitative dictator would choose (ta = 80%, tm = 75%). The curves “Rev. max. tm” and “Rev. max. ta” L L connect the points of partial revenue maxima. “Rev. max. tm” gives the points of maximum tax revenue when tm is varied and ta is held ﬁxed at its respective value, L L To be precise: It is the point where tax revenue is maximised through simultaneous variation of the labour tax parameters ta and tm, taking all general equilibrium interactions into account, L L compensating the capitalist by an adjustment of tπ, and treating the value of tC as given.

The iso-budget line summarises the set of choice options for the optimal taxation problem (at given levels of public goods and consumption of the capitalists). It remains to be determined which of these options should be chosen. This is captured by the “Iso-utility” line, which connects points of the same expected utility of workers, taking into account all general equilibrium interactions (i.e. adjusting wages and unemployment rates). It turns out that the iso-utility line is slightly increasing at its tangency point with the iso-budget line, which is in the upward-sloping range between the minimum for ta and the initial point. The optimal tm is 38.3%, almost L L ﬁve percentage points lower than the initial 43.1%. This gives an optimal CRIP of 0.93, compared to the initial level of 0.87.13 Finally, Figure 1 also shows the point where total labour input (product of individuals and hours) is maximised on the iso-budget line. This is considerably far from the utility maximum at a marginal tax rate of 29.0% and a CRIP of 1.08, i.e. a regressive tax. The reason for this diﬀerence is that additional labour input at the intensive margin is relatively cheap in welfare terms, while unemployment is expensive. Therefore reducing unemployment (by higher tax progressivity) is welfare-enhancing even if this does not compensate the labour volume loss caused by a decrease in hours of work.

Interestingly, these curves have a positive slope. Usually, two taxes on diﬀerent goods or factors of production result in negatively sloped curves. See the ﬁgures in Boeters (2004).

Analogously, the iso-budget curve would be vertical when it meets the “Rev. max. ta” line.

However, in the constellation of Figure 1 this point is not reached with positive levels of tm.

L This considerably deviates from the optimal CRIP level of 0.72 that Sørensen (1999) obtains in his model. Additional model runs showed that the income elasticity of labour supply (which is zero in the Sørensen (1999) model) is the most likely candidate for an explanation of this discrepancy.

3.2 Systematic parameter variation Starting from the average OECD parametrisation of Section 3.1, the parameters that lead to country heterogeneity are now varied systematically. In order to isolate the eﬀects of the diﬀerent parameters, I replace, one by one, the average parameter value with both its minimum and maximum value in all countries considered. This yields 14 model variations, two for each of the seven parameters. The results of these model exercises are summarised in Table 2.

Table 2 shows the minimum and maximum value of each of the parameters in the dataset, the respective country and the resulting optimal degree of tax progressivity.

For the interpolation of the country results, I assume that the eﬀects of the individual parameters are linear and additive. This allows us to calculate partial eﬀects, which are shown in the last column of Table 2. Take the partial eﬀect for tC (0.188) as an example. It may be interpreted in the following way: A one-percentage-point increase in the consumption tax rate leads to a value of the optimal CRIP that is about 0.2 percentage point higher. The spread between the optimal CRIP for the minimum and maximum value of a coeﬃcient is informative with respect to the relevance of the respective coeﬃcient for explaining country diﬀerences. A parameter is particularly relevant if there actually is variation between countries and its partial eﬀect is high.

Interpreted in this way, Table 2 shows that most “action” is in the CRIP and the unemployment rate.15

**3.3 Explaining cross-country diﬀerences**

In which way can the results from the systematic parameter variation be used to explain cross-country diﬀerences? Table 3 shows the actual and the optimal CRIP for all countries. Row “deviation” reports the diﬀerence between the optimal countryspeciﬁc CRIP and the optimum in the average OECD model (0.93). This diﬀerence is “explained” by the deviations of all parameters from their OECD averages, multiplied with the partial eﬀects from Table 2.

The ﬁt is not perfect, but fairly good. The R2 of this exercise (when interpreting, without any statistical implication, the partial eﬀects as regression coeﬃcients) is The surprising fact that actual initial progressivity has a strong eﬀect on optimal progressivity is discussed in Sections 3.4 and 4.1.

0.88, which means that the assumption of additive, linear eﬀects is justiﬁed. As it stands, the regression may be read in the following way: Low optimal progressivity in the Netherlands and Great Britain is mostly driven by low unemployment rates.

Reversely, high unemployment in Spain makes a high degree of tax progressivity desirable. In contrast to these countries, high optimal tax progressivity in Germany is mostly driven by the low initial CRIP. Low optimal tax progressivity in the USA and Japan is explained by a combination of relatively low unemployment and a high initial CRIP.

The decomposition results thus conﬁrm what we saw in Table 2. The largest contribution to the explanation of the eﬀects comes from the unemployment rate and the CRIP. The contributions of the replacement rate, the consumption tax and the average wage tax are smaller. The value share of labour and the proﬁt tax are negligible.

**3.4 Interpretation of the regression coeﬃcients**

Until this point, the analysis has been mainly descriptive. It can be shown by systematic parameter variation that diﬀerences in the degree of optimal tax progressivity between countries are to a large extent driven by variations in the unemployment rate and initial tax progressivity. But also the level of other taxes and the replacement rate play a role. How are these eﬀects explained economically?

Most straightforward is the eﬀect of unemployment on optimal tax progressivity. Recall that the positive welfare eﬀects of higher tax progressivity are driven by the wage-depressing and unemployment-reducing forces of higher tax progressivity in wage bargaining. The higher the initial unemployment level, the higher (in percentage points) is the reduction in unemployment through a given increase in tax progressivity, and the more people beneﬁt from this by switching into employment.

Therefore, countries with a high unemployment level have higher optimal tax progressivity. In the opposite extreme case, if there is no (or very low) unemployment, there is nothing to be gained from increasing tax progressivity, while there is still the distortionary eﬀect on labour supply.

If we want to explain the eﬀect of the existing levels of ta, and tC on optimal tax L progressivity16 (the higher these levels, the lower optimal tax progressivity), we need an intermediate step. It is important to know that choosing the utility-maximising point on the iso-budget line in Figure 1 is quite distinct from maximising the total labour input.17 The labour input maximum is considerably farther to the left of the utility-maximising point (at tm = 29.0%). In moving from here to the right, L labour input decreases, i.e., the employment of additional workers (reduction in unemployment) is over-compensated by the loss in hours per person employed. This is in fact welfare-enhancing, because the employed are much closer to their labourleisure optimum than the unemployed. Thus, an hour of work of someone formerly unemployed is worth more than an hour lost of someone who was and remains employed. However, this trade-oﬀ is altered by the existing taxes. If there are taxes, a loss in employment also means a loss in tax revenue, which must be compensated if the government’s budget is to remain balanced. Therefore, the volume of labour input carries more weight in the welfare trade-oﬀ if taxes are high. The higher the taxes, the closer we remain to the labour input maximum, i.e., the less we move towards higher tax progressivity.

The eﬀect of the replacement rate takes place via the public budget as well.

A high replacement rate means high budgetary costs of unemployment. Reducing unemployment thus becomes more attractive. We have an additional positive eﬀect on the public budget, which translates into compensatory adjustments of ta which L, modiﬁes the trade-oﬀ in the direction of higher tax progressivity.