Figures and data in Combining Microsimulation and Numerical Maximization to Identify Optimal Tax-Transfer Rules

Cite this article as: U. Colombino, N. Islam; 2022; Combining Microsimulation and Numerical Maximization to Identify Optimal Tax-Transfer Rules; International Journal of Microsimulation; 15(2); 4-43. doi: 10.34196/ijm.00261

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20 figures and 15 tables

Figures

Figure 1

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Marginal Tax Rate vs. Taxable Income. France

Figure 2

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Average Tax Rate vs. Taxable Income. France

Figure 3

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Marginal Tax Rate vs. Taxable Income. Germany

Figure 4

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Average Tax Rate vs. Taxable Income. Germany

Figure 5

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Marginal Tax Rate vs. Taxable Income. Italy

Figure 6

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Average Tax Rate vs. Taxable Income. Italy

Figure 7

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Marginal Tax Rate vs. Taxable Income. Luxembourg

Figure 8

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Average Tax Rate vs. Taxable Income. Luxembourg

Figure 9

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Marginal Tax Rate vs. Taxable Income. Spain

Figure 10

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Average Tax Rate vs. Taxable Income. Spain

Figure 11

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Marginal Tax Rate vs. Taxable Income. United Kingdom

Figure 12

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Average Tax Rate vs. Taxable Income. United Kingdom

Figure 13

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France: %change of CMU by demographic group and decile, k = 0.075.

Figure 14

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Figure 15

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Italy: %change of CMU by demographic group and decile, k = 0.075.

Figure 16

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Luxembourg: %change of CMU by demographic group and decile, k = 0.075.

Figure 17

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Spain: %change of CMU b.y demographic group and decile, k =0.075

Figure 18

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The UK: %change of CMU by demographic group and decile, k = 0.075.

Figure 19

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%change of disposable income w.r.t. current values by country and Kolm’s k.

Figure 20

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Povery Gap Index by country and Optimal TTR (indexed by Kolm’s k).

Tables

Table 1

“Current” and optimal TTRs

		Current*	k = .00	k = .05	k= .075	k = .10	k =.125	k = .15
France	τ₀	603.27	61.43	181.72	265.57	367.97	453.77	592.02
	τ₁	0.52	0.93	0.88	0.84	0.80	0.76	0.70
	τ₂	3.01×10^-6	-0.0000×10^-6	0.0106×10^-6	0.013×10^-6	0.011×10^-6	0.0098×10^-6	0.0097×10^-6
	τ₃	-1.51×10^-11	0.0004×10^-11	0.0062×10^-11	0.015×10^-11	0.005×10^-11	0.0064×10^-11	0.0053×10^-11
	τ₄	0.20×10^-16	0.0009×10^-16	0.008×10^-16	0.014×10^-16	0.017×10^-16	0.0098×10^-16	0.0051×10^-16
Germany	τ₀	427.9	427.9	454.23	554.48	600.91	600.9075	701.7375
	τ₁	0.71	0.71	0.69	0.66	0.64	0.64	0.61
	τ₂	0.0016×10^-6	0.003×10^-6	0.001×10^-6	0.0001×10^-6	0.001×10^-6	0.002×10^-6	0.002×10^-6
	τ₃	0.0009×10^-11	0.005×10^-11	-0.002×10^-11	0.001×10^-11	0.001×10^-11	0.001×10^-11	0.001×10^-11
	τ₄	0.0022×10^-16	0.010×10^-16	0.004×10^-16	0.001×10^-16	0.001×10^-16	0.004×10^-16	0.002×10^-16
Italy	τ₀	217.24	98.99	177.28	236.69	270.67	350.15	417.72
	τ₁	0.745	0.752	0.698	0.66	0.63	0.58	0.53
	τ₂	-1.98×10^-6	-0.02×10^-6	-0.01×10^-6	0.002×10^-6	0.005×10^-6	0.0002×10^-6	0.0008×10^-6
	τ₃	0.69×10^-11	0.04×10^-11	0.01×10^-11	0.004×10^-11	0.004×10^-11	0.0000×10^-11	-0.0004×10^-11
	τ₄	-0.07×10^-16	-0.02×10^-16	-0.01×10^-16	-0.0000×10^-16	0.003×10^-16	0.0005×10^-16	0.0025×10^-16
Luxembourg	τ₀	1469.68	615.73	680.78	746.64	809.23	858.74	926.17
	τ₁	0.316	0.761	0.75	0.717	0.706	0.692	0.676
	τ₂	4.12×10^-6	0.228×10^-6	0.24×10^-6	0.23×10^-6	0.245×10^-6	0.227×10^-6	0.223×10^-6
	τ₃	-1.869×10^-11	0.019×10^-11	0.022×10^-11	0.014×10^-11	0.051×10^-11	0.041×10^-11	0.013×10^-11
	τ₄	0.25×10^-16	0.07×10^-16	0.017×10^-16	0.075×10^-16	0.008×10^-16	0.009×10^-16	-0.004×10^-16
Spain	τ₀	196.42	227.43	225.51	436.88	481.18	493.54	498.23
	τ₁	0.97	0.81	0.81	0.65	0.61	0.60	0.60
	τ₂	-4.24×10^-6	0.001×10^-6	0.0005×10^-6	-0.0012×10^-6	0.001×10^-6	0.0005×10^-6	-0.0001×10^-6
	τ₃	2.05×10^-11	0.001×10^-11	.0008×10^-11	-.0008×10^-11	0.0003×10^-11	-.0001×10^-11	0.0000×10^-11
	τ₄	-0.36×10^-16	0.001×10^-16	0.0008×10^-16	-0.003×10^-16	0.001×10^-16	0.0002×10^-16	0.0000×10^-16
United Kingdom	τ₀	455.51	289.08	564.61	564.82	628.61	749.57	864.62
	τ₁	0.66	0.80	0.61	0.61	0.57	0.48	0.40
	τ₂	1.717×10^-6	-0.0002×10^-6	0.0003×10^-6	0.0003×10^-6	0.001×10^-6	-0.005×10^-6	-0.001×10^-6
	τ₃	-2.008×10^-11	0.0002×10^-11	0.0001×10^-11	0.001×10^-11	-0.0006×10^-11	0.002×10^-11	-0.002×10^-11
	τ₄	0.505×10^-16	0.0001×10^-16	0.0001×10^-16	0.0004×10^-16	-0.0004×10^-16	-0.003×10^-16	-0.001×10^-16

*

In the “Current” column, the parameter τ are those of the polynomial approximation to the current TTR.

Table 2

Welfare Gains and Welfare Winners

		k = .00	k = .05	k= .075	k = .10	k =.125	k = .15
France	Social Welfare Gain	159.23	89.63	53.17	16.37	-8.6	-45.47
	Equality Gain	0	-20.31	-20.64	-12.4	-1.86	20.3
	Efficiency Gain	159.23	109.94	73.81	28.77	-6.64	-65.77
	%Winners: All	67	67	66	65	64	61
	%Winners: Couples	66	66	65	64	62	57
	%Winners: Single Males	84	80	76	71	66	62
	%Winners: Single Females	51	55	57	61	66	72
Germany	Social Welfare Gain	-7.48	-5.00	41.05	71.39	87.65	155.27
	Equality Gain	0	-2.02	8.51	23.68	39.94	78.61
	Efficiency Gain	-7.48	-2.98	32.54	47.71	47.71	76.66
	%Winners: All	46	44	46	46	46	49
	%Winners: Couples	41	41	41	41	41	41
	%Winners: Single M	37	38	38	40	40	41
	%Winners: Single F	55	56	62	63	63	71
Italy	Social Welfare Gain	72.9	45.93	28.4	18.64	-1.35	-18.54
	Equality Gain	0	-3	-3.06	-2.97	0.13	3.16
	Efficiency Gain	72.9	48.93	31.46	21.61	-1.48	-21.70
	%Winners	56	58	58	57	53	52
	%Winners: Couples	51	49	44	42	38	36
	%Winners: Single M	59	66	69	70	63	61
	%Winners: Single F	58	63	65	65	64	63
Luxembourg	Social Welfare Gain	2.06	-35.63	-68.35	-62.63	-67.54	-66.94
	Equality Gain	0	-41.82	-43.04	-46.45	-40.26	-27.94
	Efficiency Gain	2.06	6.19	-25.31	-16.18	-27.28	-39.00
	%Winners	45	48	46	48	49	51
	%Winners: Couples	54	59	54	58	59	61
	%Winners: Single M	31	32	31	30	30	30
	%Winners: Single F	44	47	48	50	52	56
Spain	Social Welfare Gain	-16.99	-13.87	10.65	13.83	21.35	27.17
	Equality Gain	0	2.17	13.88	20.25	25.47	30.29
	Efficiency Gain	-16.99	-16.04	-3.23	-6.42	-4.12	-3.12
	%Winners	73	73	67	63	63	63
	%Winners: Couples	75	75	63	56	56	55
	%Winners: Single M	70	70	72	70	70	70
	%Winners: Single F	70	70	73	73	74	74
United Kingdom	Social Welfare Gain	-18.37	37.42	37.81	48.34	65.19	77.19
	Equality Gain	0	0.81	1.14	1.24	0.64	-0.17
	Efficiency Gain	-18.37	36.61	36.67	47.10	64.55	77.36
	%Winners: All	42	76	77	79	81	82
	%Winners: Couples	23	79	82	84	88	90
	%Winners: Single M	57	64	64	65	66	68
	%Winners: Single F	64	82	83	83	83	82

Table 3

“Primitives” of the economy and characteristics of the optimal polynomial TTRs

	UBI	Leading Tax Rate
Constant	357.28 2.14	-5.04 -0.3
Kolm’s k	25.44 14.01	1.40 9.95
Productivity	0.22 11.29	0.002 1.48
Extensive Elasticity	117.35 3.05	9.90 3.31
Intensive Elasticity	253.55 8.44	1.46 0.62
Budget	-0.25 -8.08	0.002 -0.85
R²	0.83	0.88
Standard error of the estimate	53.69	0.04

t-values in italics below the estimates.
Bold estimates are significant at standard levels.

Table A1

Maximum likelihood estimates – couples (France)

Model component	Variable	Coef.	Std. Err.
Opportunity density		δ
	Employee_Man	0.4761696	0.3665576
	Self-employed_Man	0.2130577	0.3805561
	Employee_Woman	-0.3649212	0.2853927
	Self-employed_Woman	-1.426779	0.3241603
	Part-time_Employee_Man	-0.3805255	0.2414433
	Full-time_Employee_Man	2.83453	0.1249029
	Part-time_Self-employed_Man	-1.841048	0.324269
	Full-time_Self-employed_Man	0.2870089	0.1540075
	Part-time_Employee_Woman	0.6361778	0.2170085
	Full-time_Employee_Woman	2.698627	0.1676399
	Part-time_Self-employed_Woman	-1.014395	0.3149686
	Full-time_Self-employed_Woman	0.6781008	0.2277644
Income		γ
	Household_Disposable_income	0.0003342	0.0001334
	Hosuhold_Disposable_income squared	1.61E-08	6.75E-09
	Household_size X Household_disposable_income	-0.0000513	0.0000175
Leisure		λ
	Leisure_Male	0.1256514	0.0281893
	Leisure_Man squared	0.0000173	0.0001373
	Leisure_Woman	0.163189	0.0255661
	Leisure_Woman squared	-0.000107	0.0001529
	Leisure_Man X Household_disp_income	-7.88E-06	1.02E-06
	Leisure_Woman X Household_disp_income	-1.54E-07	8.04E-07
	Leisure_Man X Age_Man	-0.0059183	0.0011829
	Leisure_Woman X Age_Woman	-0.0085386	0.0009848
	Leisure_Man X Age_Man squared	0.0000742	0.0000138
	Leisure_Woman X Age_Woman squared	0.0001108	0.000012
	Leisure_Man X No. Children	-0.0026133	0.0017907
	Leisure_Woman X No. Children	0.0084813	0.0015617
	Leisure_Man X No. Children0-6	0.0027957	0.0025747
	Leisure_Man X No. Children7-10	0.006507	0.0027523
	Leisure_Woman X No. Children0-6	0.006981	0.0021144
	Leisure_Woman X No. Children7-10	0.0007436	0.002288
	Leisure_Woman X Leisure_Man	0.0001134	0.0000948

	N.observations (N. couples*49 alternatives)	195804
	N.couples	3996
	LR chi2(32)	15140.15
	Prob > chi2	0
	Pseudo R2	0.4868
	Log likelihood	-7981.6412

Table A2

Maximum likelihood estimates – singles (France)

		Male		Female
Model component	Variable	Coef.	Std. Err.	Coef.	Std. Err.
Opportunity density		δ		δ
	Employee	0.199478	0.533068	-1.19744	0.470041
	Self_employed	-0.21333	0.593638	-1.68397	0.553979
	Part-time_Employee	-0.69588	0.393698	1.256628	0.352174
	Full-time_Employee	2.213382	0.253926	2.994315	0.267448
	Part-time_Self-employed	-2.70132	0.585303	-1.80785	0.627849
	Full-time_Self-employed	-0.28412	0.336337	0.368914	0.377033
Income		λ		λ
	Disposable income	-0.00012	0.00024	7.55E-05	0.000379
	Disposable income squared	4.53E-08	2.07E-08	6.64E-08	4.13E-08
	Household size X Disp_income	-5.6E-05	4.62E-05	-6.7E-05	6.84E-05
Leisure		λ		λ
	Leisure	0.129227	0.029882	0.15447	0.034747
	Leisure2	-8.1E-05	0.000239	-9.2E-05	0.000254
	Leisure X Disposable income	1.01E-06	2.34E-06	6.43E-07	3.40E-06
	Leisure X Age	-0.00516	0.000963	-0.0075	0.001086
	Leisure X Age squared	6.36E-05	1.21E-05	0.000092	1.34E-05
	Leisure X No. Children	-0.01374	0.005121	0.006768	0.003433
	Leisure X No. Children 0-6	-0.00814	0.019875	0.015925	0.00544
	Leisure X No. Children 7-10	0.011728	0.010413	0.008727	0.004892
Other
	N.observations (N. single*7 alternatives)	9331		10465
	N.single	1333		1495
	LR chi2(17)	2318.15		2657.35
	Prob > chi2	0		0
	Pseudo R2	0.4468		0.4567
	Log likelihood	-1434.83		-1580.46

Table A3

Maximum likelihood estimates – couples (Germany)

Model component	Variable	Coef.	Std. Err.
Opportunity density		δ
	Employee_Man	-0.5269339	0.435726
	Self-employed_Man	-1.611115	0.4373339
	Employee_Woman	-1.560942	0.2543809
	Self-employed_Woman	-3.580387	0.3637122
	Part-time_Employee_Man	-0.9591195	0.2591677
	Full-time_Employee_Man	2.516702	0.1205793
	Part-time_Self-employed_Man	-1.797576	0.3673868
	Full-time_Self-employed_Man	0.940684	0.1597094
	Part-time_Employee_Woman	1.829915	0.2318468
	Full-time_Employee_Woman	2.531309	0.1913391
	Part-time_Self-employed_Woman	0.8422771	0.341443
	Full-time_Self-employed_Woman	1.067966	0.2761494
Income		γ
	Household_Disposable_income	0.001699	0.0001136
	Hosuhold_Disposable_income squared	-5.60E-08	5.58E-09
	Household_size X Household_disposable_income	0.0000479	0.0000222
Leisure		λ
	Leisure_Male	0.184064	0.0264439
	Leisure_Man squared	-0.0004684	0.0002088
	Leisure_Woman	0.2376265	0.0251642
	Leisure_Woman squared	-0.0011735	0.00014
	Leisure_Man X Household_disp_income	-1.38E-05	7.25E-07
	Leisure_Woman X Household_disp_income	-9.65E-06	5.79E-07
	Leisure_Man X Age_Man	-0.0030924	0.0009326
	Leisure_Woman X Age_Woman	-0.0046903	0.0009542
	Leisure_Man X Age_Man squared	0.0000357	0.0000106
	Leisure_Woman X Age_Woman squared	0.0000713	0.0000112
	Leisure_Man X No. Children	-0.0017621	0.0019208
	Leisure_Woman X No. Children	0.0231159	0.0016417
	Leisure_Man X No. Children0-6	0.0184762	0.0025177
	Leisure_Man X No. Children7-10	-0.0007047	0.0032015
	Leisure_Woman X No. Children0-6	0.01427	0.0025455
	Leisure_Woman X No. Children7-10	0.0037864	0.0026824
	Leisure_Woman X Leisure_Man	-0.0003689	0.0000744
Other
	N.observations (N. couples*49 alternatives)	201,243
	N.couples	4107
	LR chi2(32)	14394.76
	Prob > chi2	0
	Pseudo R2	0.4503
	Log likelihood	-8786.3249

Table A4

Maximum likelihood estimates – singles (Germany)

		Male		Female
Model component	Variable	Coef.	Std. Err.	Coef.	Std. Err.
Opportunity density		δ		δ
	Employee	-0.80193	0.57248	-1.43637	0.446197
	Self_employed	0.179376	0.597029	-8.15099	0.661612
	Part-time_Employee	-0.32109	0.406771	0.579569	0.34354
	Full-time_Employee	2.865958	0.240845	2.703306	0.249755
	Part-time_Self-employed	-2.54205	0.498362	4.688473	0.595456
	Full-time_Self-employed	0.184833	0.295293	4.011293	0.42799
Income		γ		γ
	Disposable income	0.003328	0.000448	0.003246	0.000196
	Disposable income squared	-7.89E-07	7.78E-08	-1.49E-07	1.28E-08
	Household size X Disp_income	6.42E-04	1.93E-04	8.93E-05	3.80E-05
Leisure		λ		λ
	Leisure	0.241596	0.03538	0.281701	0.033209
	Leisure2	-0.00076	0.000296	-0.00141	0.000258
	Leisure X Disposable income	-4.42E-05	2.40E-06	-3.86E-05	2.08E-06
	Leisure X Age	-0.00382	0.001035	-0.00337	0.001006
	Leisure X Age squared	5.09E-05	1.24E-05	4.29E-05	1.20E-05
	Leisure X No. Children	-0.02709	0.012505	0.014031	0.003459
	Leisure X No. Children 0-6	0.038387	0.020631	0.0287	0.007638
	Leisure X No. Children 7-10	0.016741	0.021338	0.017555	0.006452
Other
	N.observations (N. single*7 alternatives)	10,283		12,551
	N.single	1469		1793
	LR chi2(17)	2983.13		3608.3
	Prob > chi2	0		0
	Pseudo R2	0.5218		0.5171
	Log likelihood	-1366.98		-1684.87

Table A5

Maximum likelihood estimates – couples (Italy)

Model component	Variable	Coef.	Std. Err.
Opportunity density		δ
	Employee_Man	-2.227042	0.3359151
	Self-employed_Man	-1.793772	0.3327547
	Employee_Woman	-4.205803	0.3711781
	Self-employed_Woman	-3.159583	0.3091701
	Part-time_Employee_Man	1.810835	0.2235256
	Full-time_Employee_Man	3.457804	0.1466732
	Part-time_Self-employed_Man	-1.142189	0.2861769
	Full-time_Self-employed_Man	1.827801	0.1352579
	Part-time_Employee_Woman	3.522802	0.3488772
	Full-time_Employee_Woman	4.233018	0.3257372
	Part-time_Self-employed_Woman	0.2200945	0.3028192
	Full-time_Self-employed_Woman	1.989132	0.2580389
Income		γ
	Household_Disposable_income	0.0005129	0.0001534
	Hosuhold_Disposable_income squared	1.36E-08	7.25E-09
	Household_size X Household_disposable_income	-0.0001608	0.0000251
Leisure		λ
	Leisure_Male	0.0030689	0.05153
	Leisure_Man squared	-0.0000926	0.0001607
	Leisure_Woman	0.2598116	0.0365898
	Leisure_Woman squared	-0.000653	0.0001763
	Leisure_Man X Household_disp_income	4.38E-06	1.43E-06
	Leisure_Woman X Household_disp_income	-5.81E-07	1.01E-06
	Leisure_Man X Age_Man	-0.0015349	0.0025113
	Leisure_Woman X Age_Woman	-0.0097254	0.0016741
	Leisure_Man X Age_Man squared	0.0000135	0.0000318
	Leisure_Woman X Age_Woman squared	0.0001141	0.0000223
	Leisure_Man X No. Children	-0.0081218	0.0022336
	Leisure_Woman X No. Children	0.0078869	0.0017578
	Leisure_Man X No. Children0-6	0.0076125	0.0026554
	Leisure_Man X No. Children7-10	0.0002707	0.0028172
	Leisure_Woman X No. Children0-6	-0.0054445	0.0020634
	Leisure_Woman X No. Children7-10	-0.0009139	0.0020886
	Leisure_Woman X Leisure_Man	0.0003854	0.0000964
Other
	N.observations (N. couples*49 alternatives)	188405
	N.couples	3845
	LR chi2(32)	10209.91
	Prob > chi2	0
	Pseudo R2	0.3411
	Log likelihood	-9859.09

Table A6

Maximum likelihood estimates – singles (Italy)

		Male		Female
Model component	Variable	Coef.	Std. Err.	Coef.	Std. Err.
Opportunity density		δ		δ
	Employee	-1.22117	0.331639	-3.43019	0.3787
	Self_employed	-0.47643	0.315555	-2.81075	0.350903
	Part-time_Employee	1.263827	0.268794	3.554593	0.34008
	Full-time_Employee	3.310487	0.207522	4.654217	0.303264
	Part-time_Self-employed	-2.2652	0.32631	0.618142	0.341357
	Full-time_Self-employed	1.473456	0.180946	2.786139	0.266647
Income		γ		γ
	Disposable income	0.000114	0.000145	0.0003	0.000255
	Disposable income squared	5.12E-09	1.08E-08	6.55E-09	3.11E-08
	Household size X Disp_income	-5.5E-05	4.01E-05	-0.00011	4.77E-05
Leisure		λ		λ
	Leisure	0.280595	0.024332	0.312801	0.030346
	Leisure2	0.000164	0.000173	0.000428	0.000198
	Leisure X Disposable income	1.36E-06	1.55E-06	-1.91E-07	2.59E-06
	Leisure X Age	-0.01438	0.001037	-0.01841	0.001297
	Leisure X Age squared	0.000176	1.51E-05	0.000225	1.84E-05
	Leisure X No. Children	-0.0191	0.01175	0.005966	0.003381
	Leisure X No. Children 0-6	0.007813	0.020605	0.00305	0.005703
	Leisure X No. Children 7-10	0.011513	0.022161	-0.00433	0.005772
Other
	N.observations (N. single*7 alternatives)	22190		18270
	N.single	3170		2610
	LR chi2(17)	4055.02		3501.41
	Prob > chi2	0		0
	Pseudo R2	0.3287		0.3447
	Log likelihood	-4141.03		-3328.12

Table A7

Maximum likelihood estimates – couples (Luxembourg)

Model component	Variable	Coef.	Std. Err.
Opportunity density		δ
	Employee_Man	2.798179	1.230943
	Self-employed_Man	1.196799	1.218041
	Employee_Woman	-1.670879	0.4877308
	Self-employed_Woman	-3.273727	0.5811094
	Part-time_Employee_Man	-0.9321119	0.5778732
	Full-time_Employee_Man	2.740097	0.2477136
	Part-time_Self-employed_Man	-3.276221	1.176261
	Full-time_Self-employed_Man	0.3923308	0.4062014
	Part-time_Employee_Woman	2.251194	0.381928
	Full-time_Employee_Woman	3.024338	0.2864887
	Part-time_Self-employed_Woman	-0.0916981	0.6417357
	Full-time_Self-employed_Woman	0.9017009	0.4806236
Income		γ
	Household_Disposable_income	0.0001153	0.0001343
	Hosuhold_Disposable_income squared	-2.43E-09	2.07E-09
	Household_sizeÃ—Household_disposable_income	-1.63E-06	0.000023
Leisure		λ
	Leisure_Male	-0.0472945	0.0551945
	Leisure_Man squared	0.0014071	0.0004473
	Leisure_Woman	0.0416601	0.0425495
	Leisure_Woman squared	0.0003121	0.0002634
	Leisure_Man X Household_disp_income	1.64E-06	8.45E-07
	Leisure_Woman X Household_disp_income	1.18E-07	8.47E-07
	Leisure_Man X Age_Man	-0.0038039	0.0021464
	Leisure_Woman X Age_Woman	-0.0059885	0.0016256
	Leisure_Man X Age_Man squared	0.0000479	0.0000254
	Leisure_Woman X Age_Woman squared	0.0000904	0.0000201
	Leisure_Man X No. Children	-0.0067684	0.0038964
	Leisure_Woman X No. Children	0.0069002	0.0027455
	Leisure_Man X No. Children0-6	0.0085339	0.0051382
	Leisure_Man X No. Children7-10	0.002834	0.0060786
	Leisure_Woman X No. Children0-6	0.0088988	0.0034797
	Leisure_Woman X No. Children7-10	0.0022974	0.0039516
	Leisure_Woman X Leisure_Man	0.0002931	0.0001535
Other
	N.observations (N. couples*49 alternatives)	64435
	N.couples	1315
	LR chi2(32)	5058.95
	Prob > chi2	0
	Pseudo R2	0.4943
	Log likelihood	-2588.2705

Table A8

Maximum likelihood estimates – singles (Luxembourg)

		Male		Female
Model component	Variable	Coef.	Std. Err.	Coef.	Std. Err.
Opportunity density		δ		δ
	Employee	3.840025	1.467676	-3.83547	0.916069
	Self_employed	3.076828	1.44261	-6.09075	1.124339
	Part-time_Employee	-1.25405	0.727917	3.214104	0.688409
	Full-time_Employee	2.760224	0.389481	3.833989	0.508304
	Part-time_Self-employed	-17.2776	699.2763	2.533271	1.111393
	Full-time_Self-employed	0.139062	0.585945	2.068577	0.878229
Income		γ		γ
	Disposable income	3.53E-05	0.000416	0.00036	0.000262
	Disposable income squared	-8.96E-09	2.75E-08	-8.71E-09	9.30E-09
	Household size X Disp_income	0.000177	0.000082	-4.1E-05	5.76E-05
Leisure		λ		λ
	Leisure	0.083226	0.061447	0.222152	0.066109
	Leisure2	0.00187	0.000632	-0.00012	0.000504
	Leisure X Disposable income	2.07E-07	3.91E-06	3.42E-06	2.72E-06
	Leisure X Age	-0.0096	0.001664	-0.01311	0.00192
	Leisure X Age squared	0.000118	0.000021	0.00016	2.33E-05
	Leisure X No. Children	0.010493	0.008464	0.002518	0.005455
	Leisure X No. Children 0-6	0.006816	0.029223	0.00331	0.010569
	Leisure X No. Children 7-10	0.024772	0.029139	-0.0027	0.009811
Other

	N.observations (N. single*7 alternatives)	4123		3640
	N.single	589		520
	LR chi2(17)	1157.65		951.82
	Prob > chi2	0		0
	Pseudo R2	0.505		4703
	Log likelihood	-567.317		5335.965

Table A9

Maximum likelihood estimates – couples (Spain)

Model component	Variable	Coef.	Std. Err.
Opportunity density		δ
	Employee_Man	-0.2868366	0.2843099
	Self-employed_Man	-0.7698504	0.2931603
	Employee_Woman	-2.403139	0.252017
	Self-employed_Woman	-2.585398	0.2813223
	Part-time_Employee_Man	-0.1938884	0.1981381
	Full-time_Employee_Man	2.390778	0.1070763
	Part-time_Self-employed_Man	-1.089852	0.2490981
	Full-time_Self-employed_Man	0.9119172	0.1280802
	Part-time_Employee_Woman	1.511822	0.2167088
	Full-time_Employee_Woman	2.692898	0.1672168
	Part-time_Self-employed_Woman	-0.3884766	0.2749832
	Full-time_Self-employed_Woman	0.8462308	0.2033327
Income		γ
	Household_Disposable_income	-0.0001841	0.0001271
	Hosuhold_Disposable_income squared	2.51E-08	8.12E-09
	Household_sizeÃ—Household_disposable_income	-0.0000236	0.0000156
Leisure		λ
	Leisure_Male	-0.0294838	0.0239608
	Leisure_Man squared	0.0005993	0.0001227
	Leisure_Woman	0.0991669	0.0229826
	Leisure_Woman squared	-0.0003914	0.0001435
	Leisure_Man X Household_disp_income	5.62E-06	9.76E-07
	Leisure_Woman X Household_disp_income	1.22E-06	7.46E-07
	Leisure_Man X Age_Man	-0.0022498	0.0009384
	Leisure_Woman X Age_Woman	-0.0043625	0.0007961
	Leisure_Man X Age_Man squared	0.0000224	0.0000103
	Leisure_Woman X Age_Woman squared	0.0000589	9.11E-06
	Leisure_Man X No. Children	-0.0015521	0.0013717
	Leisure_Woman X No. Children	0.0036569	0.0011625
	Leisure_Man X No. Children0-6	0.0034279	0.0019022
	Leisure_Man X No. Children7-10	-0.0022426	0.0020741
	Leisure_Woman X No. Children0-6	0.0013244	0.0016922
	Leisure_Woman X No. Children7-10	0.0020022	0.0017517
	Leisure_Woman X Leisure_Man	0.00032	0.0000585
Other
	N.observations (N. couples*49 alternatives)	244755
	N.couples	4995
	LR chi2(32)	13049.94
	Prob > chi2	0
	Pseudo R2	0.3357
	Log likelihood	-12914.672

Table A10

Maximum likelihood estimates – singles (Spain)

		Male		Female
Model component	Variable	Coef.	Std. Err.	Coef.	Std. Err.
Opportunity density		δ		δ
	Employee	-1.30198	0.428921	-1.7268	0.428239
	Self_employed	-1.41084	0.456916	-1.59374	0.505891
	Part-time_Employee	0.607968	0.329195	1.895538	0.346785
	Full-time_Employee	2.496161	0.217392	3.236458	0.246681
	Part-time_Self-employed	-1.10093	0.419394	-0.78624	0.512791
	Full-time_Self-employed	0.757978	0.25528	1.336878	0.331166
Income		γ		γ
	Disposable income	0.000391	0.000202	0.000249	0.000234
	Disposable income squared	-2.86E-08	2.19E-08	-4.02E-09	2.89E-08
	Household size X Disp_income	3.35E-05	4.41E-05	0.000166	5.63E-05
Leisure		λ		λ
	Leisure	0.091073	0.027093	0.060002	0.028202
	Leisure2	7.25E-05	0.000216	0.000661	0.000213
	Leisure X Disposable income	2.92E-08	2.03E-06	2.06E-06	2.36E-06
	Leisure X Age	-0.00465	0.000802	-0.00588	0.000853
	Leisure X Age squared	5.79E-05	9.77E-06	6.94E-05	1.02E-05
	Leisure X No. Children	0.002815	0.0077	0.005045	0.002922
	Leisure X No. Children 0-6	-0.00077	0.015847	0.004662	0.006085
	Leisure X No. Children 7-10	-0.01601	0.021244	0.00168	0.005529
Other
	N.observations (N. single*7 alternatives)	12530		12194
	N.single	1790		1742
	LR chi2(17)	2335.29		2421.22
	Prob > chi2	0		0
	Pseudo R2	0.3352		0.3571
	Log likelihood	-2315.53		-2179.16

Table A11

Maximum likelihood estimates – couples (UK)

Model component	Variable	Coef.	Std. Err.
Opportunity density		δ
	Employee_Man	0.1010326	0.3378421
	Self-employed_Man	-0.6931103	0.3433464
	Employee_Woman	-2.125809	0.2756313
	Self-employed_Woman	-2.693189	0.3039796
	Part-time_Employee_Man	-1.045434	0.2048579
	Full-time_Employee_Man	2.390656	0.1114932
	Part-time_Self-employed_Man	-1.088571	0.24735
	Full-time_Self-employed_Man	1.330872	0.1382285
	Part-time_Employee_Woman	1.85452	0.2366666
	Full-time_Employee_Woman	2.886648	0.2010856
	Part-time_Self-employed_Woman	0.3428018	0.2863139
	Full-time_Self-employed_Woman	0.6378565	0.2530663
Income		γ
	Household_Disposable_income	0.0001122	0.0002534
	Hosuhold_Disposable_income squared	-3.40E-08	1.75E-08
	Household_size X Household_disposable_income	0.0000397	0.000025
Leisure		λ
	Leisure_Male	0.0426497	0.0280003
	Leisure_Man squared	0.0004884	0.0001547
	Leisure_Woman	0.1642487	0.0289328
	Leisure_Woman squared	-0.0006741	0.0001594
	Leisure_Man X Household_disp_income	9.12E-07	1.76E-06
	Leisure_Woman X Household_disp_income	-7.02E-07	1.45E-06
	Leisure_Man X Age_Man	-0.006004	0.0012687
	Leisure_Woman X Age_Woman	-0.0071774	0.0013197
	Leisure_Man X Age_Man squared	0.0000731	0.0000171
	Leisure_Woman X Age_Woman squared	0.0000904	0.0000183
	Leisure_Man X No. Children	0.0023683	0.0016979
	Leisure_Woman X No. Children	0.0169523	0.0016992
	Leisure_Man X No. Children0-6	-0.0029332	0.0017903
	Leisure_Man X No. Children7-10	0.0008209	0.001984
	Leisure_Woman X No. Children0-6	0.0142435	0.0019016
	Leisure_Woman X No. Children7-10	0.0034794	0.0020909
	Leisure_Woman X Leisure_Man	0.0006758	0.0000828
Other
	N.observations (N. couples*49 alternatives)	220843
	N.couples	4507
	LR chi2(32)	12926.1
	Prob > chi2	0
	Pseudo R2	0.3685
	Log likelihood	-11077.385

Table A12

Maximum likelihood estimates – singles (UK)

		Male		Female
Model component	Variable	Coef.	Std. Err.	Coef.	Std. Err.
Opportunity density		δ		δ
	Employee	-0.23213	0.467104	-2.73527	0.381373
	Self_employed	-0.73375	0.479139	-3.96636	0.455567
	Part-time_Employee	-0.48835	0.315075	2.234354	0.311484
	Full-time_Employee	2.423602	0.215381	3.217563	0.257497
	Part-time_Self-employed	-1.83573	0.404768	1.149457	0.429612
	Full-time_Self-employed	0.850365	0.26129	1.745809	0.371801
Income		γ		γ
	Disposable income	-8.1E-05	0.000262	0.001196	0.000448
	Disposable income squared	1.29E-08	3.04E-08	-1.33E-07	6.24E-08
	Household size X Disp_income	1.13E-05	5.03E-05	-1.9E-05	6.18E-05
Leisure		λ		λ
	Leisure	0.122021	0.027423	0.248225	0.030925
	Leisure2	0.00039	0.000239	-0.00026	0.00022
	Leisure X Disposable income	1.27E-06	2.44E-06	-9.32E-06	4.05E-06
	Leisure X Age	-0.0075	0.001001	-0.01234	0.00121
	Leisure X Age squared	0.000102	1.47E-05	0.000166	1.74E-05
	Leisure X No. Children	-0.00517	0.005611	0.015035	0.002616
	Leisure X No. Children 0-6	0.020995	0.012061	0.026352	0.003583
	Leisure X No. Children 7-10	0.013849	0.010819	0.004335	0.003571
Other
	N.observations (N. single*7 alternatives)	13937		17549
	N.single	1991		2507
	LR chi2(17)	2736.7		3775.11
	Prob > chi2	0		0
	Pseudo R2	0.3532		0.3869
	Log likelihood	-2505.96		-2990.84

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Mendeley

Marginal Tax Rate vs. Taxable Income. France

Average Tax Rate vs. Taxable Income. France

Marginal Tax Rate vs. Taxable Income. Germany

Average Tax Rate vs. Taxable Income. Germany

Marginal Tax Rate vs. Taxable Income. Italy

Average Tax Rate vs. Taxable Income. Italy

Marginal Tax Rate vs. Taxable Income. Luxembourg

Average Tax Rate vs. Taxable Income. Luxembourg

Marginal Tax Rate vs. Taxable Income. Spain

Average Tax Rate vs. Taxable Income. Spain

Marginal Tax Rate vs. Taxable Income. United Kingdom

Average Tax Rate vs. Taxable Income. United Kingdom

France: %change of CMU by demographic group and decile, k = 0.075.

France: %change of CMU by demographic group and decile, k = 0.075.

Italy: %change of CMU by demographic group and decile, k = 0.075.

Luxembourg: %change of CMU by demographic group and decile, k = 0.075.

Spain: %change of CMU b.y demographic group and decile, k =0.075

The UK: %change of CMU by demographic group and decile, k = 0.075.

%change of disposable income w.r.t. current values by country and Kolm’s k.

Povery Gap Index by country and Optimal TTR (indexed by Kolm’s k).

“Current” and optimal TTRs

Welfare Gains and Welfare Winners

“Primitives” of the economy and characteristics of the optimal polynomial TTRs

Maximum likelihood estimates – couples (France)

Maximum likelihood estimates – singles (France)

Maximum likelihood estimates – couples (Germany)

Maximum likelihood estimates – singles (Germany)

Maximum likelihood estimates – couples (Italy)

Maximum likelihood estimates – singles (Italy)

Maximum likelihood estimates – couples (Luxembourg)

Maximum likelihood estimates – singles (Luxembourg)

Maximum likelihood estimates – couples (Spain)

Maximum likelihood estimates – singles (Spain)

Maximum likelihood estimates – couples (UK)

Maximum likelihood estimates – singles (UK)

Downloads (link to download the article as PDF)

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