1. Designing Optimal Taxes with a Microeconometric Model of Labour Supply Evidence from Norway
Rolf Aaberge, Statistics Norway and CHILD
Ugo Colombino, Univ. of Torino, Statistics Norway and CHILD
IMA2007, August, Vienna
This paper is a joint work with Rolf Aaberge.

2. Purpose
Identify optimal (personal income) tax rules in Norway, using a structural microeconometric model
The purpose of the exercise consists in using a microeconometric model of household labour supply to identify optimal personal income taxes. The empirical application is to Norway.

3. Traditional approach to empirical optimal taxation
The typical exercise (e.g as surveyed by Tuomala):
Take some optimal tax formula derived from theory
Calibrate the parameters (preferences, distribution of characteristics, elasticities etc.) using previous empirical results
Compute optimal taxes
There is a considerable (although not enormous) literature on empirical optimal taxation. Traditionally, the exercise goes as follows. Start with some optimal tax formula (derived from the theoretical literature: Mirlees etc.). The formula typically contains parameters with an empicical counterpart (preference parameters, elasticities, parameters of the distribution of population characteristics etc.). Then choose estimates of the parameters (usually from previous empirical work) and/or calibrate them so that some criterion of fitting is satisfied. Last, substitue the numerical values into the formula and compute optimal taxes.

4. Traditional approach in empirical optimal taxation
Problem with this approach:
theoretical results typically rely on some special assumptions;
possible inconsistency between the assumptions of the theoretical model and the assumptions of the empirical analysis used to calibrate the parameters;
difficult to include household decisions, participation decisions, quantity constraints.
Recent contributions (Saez etc.) partially help in overcoming some of the problems mentioned. See an interesting application in this same session (Haan).

5. A microeconometric - computational approach
The approach adopted in this paper is different:
we do not start from a priori theoretical results;
we directly identify the optimal tax rule by running a microeconometric model of household labour supply that simulates household choices and utility for any tax rule;
the simulation searches for the tax rule that maximizes a social welfare function subject to the constraint of a constant total tax revenue.
The traditional approach is analytical. A Social Welfare Function is analytically maximized so as to obtain a closed form solution for the optimal tax rule. Afterward, the theoretical formula is fed with numbers. Clearly, in order to obtain a tractable solution, many special and simplifying assumptions have to be made.
The approach we follow here is computational, rather than analytical. We start from a complex and flexible microeconometric model of household labour supply. We then numerically maximize the Social Welfare Function by iteratively running the model.

6. The microeconometric model
It is (basically) a MNL model.
Each individual (or household) is assumed to choose within an opportunity set containing jobs.
Each job is a bundle of hours of work, net income (given a tax rule t) and unobserved characteristics e(j).
The tax rule is a function t: Gross  Net.
u(j;t) = V(j;t)e(j) = utility attained at job j, given tax rule t
The stochastic component e(j) account for the effect of unobserved job’s and household’s characteristics (effect of the job-household match).
We use a sample containing both couples and singles. Therefore we apply two different versions of the model.

7. The microeconometric model
Under suitable assumptions upon the distribution (extreme value) of the unbserved characteristics, one gets:
Prob(j is chosen) = V(j;t)/∑iBV(i;t)
The sum at the denominator goes across the opportunity set B. The opportunity set in general varies across individuals. For each individuals certain jobs are more or less easy to find, for some individual there are no jobs available, and so on and so forth.

8. The microeconometric model
Main distinctive feature of the model with respect to other MNL models used in the labour supply literature:
The job opportunity sets are different among individuals (we account for differing opportunities, differing quantity constraints etc.).
The opportunity set in this version of the MNL model is not deterministic and fixed for everyone. It is stochastic itself. Each indivisual faces a specific opportunity set generated from a distribution which in general depends on the characteristics of the individual and of her einvironment.

9. An example of the opportunity set in the (income, hours, e) space
Job J
Job K
In the actual opportunity set there is still a third dimension: the component e(j) = other characteristics.
hours

10. Estimation
V(j;t) – function of income, leisure and demographic characteristics - is given a flexible parametric specification
The 78 parameters are estimated by ML
The dataset is based on the 1995 Norwegian Survey of Level of Living
It contains 1842 couples, 309 single females and 312 single males
Only individuals with age between 20 and 62 are included
V(j;t) is a Box-Cox utility function. Besides being flexible, it still allows for a direct economic interpretation of the parameters.
Because of the stochastic nature of the opportunity set, the estimation method is actually a simulated ML.

11. Labour supply elasticities implied by the model Married couples
The elasticities are reported just as an illustration of the behavioural implication of the model. We do not use the elasticities to simulate households’ choices nor to compute optmal taxes: instead we use the full model to simulate choices and to iteratively identify optimal taxes.

12. Simulating optimal tax rules
STEP 1:
Given a tax rule f, compute for each individual 1,…,N
u = maxj V(j;f)e(j)
STEP 2:
Compute the Social Welfare Function W(u1,…,uN). The arguments u of the Social Welfare function are made interpersonally comparable by using a common utility function
STEP 3:
Iterate (on the set of tax rules) STEPS 1-2 so as to maximize W keeping constant the total net tax revenue
In principle any common utility function would do. In fact we use an estimated utility function with common parameters and characteristics across the households.
The maximization cannot be done with standard (derivative-based) algorithms. Essentially it has to be done with some more or less sophisticated version of a grid search over the tax parameter space.

13. Social Welfare Function
In general we can write the SWF as:
W = (∑iui/N)(1-I) = “Efficiency”  “Equality”.
∑iui/N = average utility (efficiency).
I = index of inequality of the distribution of utility.
In this exercise I is a rank-based index. It depends on the value of an inequality-aversion parameter.
For different values of this parameter, one gets different special cases (Utilitarian, Gini, Bonferroni etc.).
We also extend the above SWF to include a criterion of Equality of Opportunities (due to J. Roemer).
The class of SWF we use is known in the literature as Rank-Dependent SWF.
We will not say anymore about the Roemer’s EOp SWF since the simulation results are rather similar those obtained with the more usual approach (called the EO approach in the full paper).

14. 6-parameter piecewise linear tax rules
The optimal tax rule is defined by 6 parameters:
E = exemption level
Z1 = upper limit of first tax bracket
Z2 = upper limit of the second tax bracket
t1 = marginal rate of the first tax bracket
t2 = marginal rate of the second tax bracket
t3 = marginal rate of the third tax bracket
It replaces the current 1994 rule, which is also piecewise linear, with seven income brackets and a smooth sequence of marginal rates (starting with .25 and ending up with .495)
In this exercise, all transfers (social assistance, benefits etc.) are left unchanged.
The top marginal tax rate is constrained to be less than or equal to .6
We constrained the top marginal up to 60% because that is approximately the the maximum valued judged as politically sustainable in the public debate in Norway.

15. Net t3 t2 t1 E Z1 Z2
Gross

16. Net t3 t2 t1 E Z1 Z2
Gross

17. Net t2 t1 Z1 Z2
Gross

18. Actual (1994) vs Optimal tax rules according to alternative social welfare criteria
Actual (approx.)
Bonferroni
Gini
Utilitarian E 17 6 8 t1
0.17
0.20
0.22 Z1
140
172
211
264 t2
0.38
0.37
0.33 Z2
235
700
690
720 t3
0.50
0.60
The values of E and of the Zs are in thousands of 1994 NOK. One Euro is approximately equal to 8 NOK. So, for example 700 means Euros.

19. BONFERRONI-optimal rule
Average tax rates
Gross income (NOK)
Actual (1994) rule
BONFERRONI-optimal rule
50000
17.5
17.0
100000
23.9
150000
26.3
200000
28.7
19.9
400000
38.5
29.0
700000
43.2
32.8
45.1
41.0
Overall, the optimal rules imply lower average tax rates. The distance between actual and optimale rates is largest for average income levels.

21. This is a zoom on the lower part of the income distribution (below 12500 Euros).
In the actual system there is a modest non-convexity around NOK.

22. Percentage change in labour supply when the BONFERRONI-optimal tax rule is applied
Household Income Decile
Single male
Single female
Married male
Married female I
89.5
65.9
36.3
47.8 II
17.9
25.2
22.9
13.4
III - VIII
2.8
3.0
4.2
4.6 IX
0.0
1.5
-0.2 X
1.2
-0.7
-1.5
All
7.0
6.1
6.4
7.8
The change in labour supply is total, i.e. it includes both the extensive margin (participation decision) and the intensive margin (hours decision).

23. Percentage of winners when the BONFERRONI-optimal tax rule is applied
Household Income Decile
Single male
Single female
Married male
Married female I 74 62 63 II 68 55 70
III - VIII 83 69 79 82 IX 77 42 80 X 39
All 76 78
An individual is defined a winner if her utility increases with the optimal tax rule.
Note than an optimal tax rule is Pareto-efficient but in general not Pareto-improving (upon the current system).

24. Comments
Similar to the current rule, optimal tax rules imply a sequence of increasing marginal tax rates
However, optimal rules are more progressive on high income levels and less progressive on low and average income levels (somehow consistent with the pattern of labour supply elasticities)
Optimal rules imply a higher net income for almost any level of gross income  lower average tax rate: thanks to a sufficiently large labour supply response
Other things being equal, a pattern of labour supply elasticities that decrease with income should recommend – according to the optimal taxation literature – higher marginal tax rates on higher income levels.

25. Comments
Our results are partially at odds with the tax reforms that took place in many countries during the last decades.
Those reforms, with the aim of improving efficiency and incentives, embodied the idea of lowering average tax rates by lowering the top marginal rates (OECD countries: from 67% to 47% in the period ).
Our results suggest instead to lower average tax rates by lowering marginal rates on average incomes and increasing marginal rates on very high incomes: this improves both efficiency and equality.
Among major OECD countries, in the period 1980 – 2000, the top marginal tax rate on personal income has been lowered – on average – from 67% to 47%.

26. Work-in-progress Simulating tax rules with more parameters
tax reforms implemented in many developed countries during the last decades. In most cases those reforms embodied the idea of improving efficiency and labour supply incentives through a lower average tax rate and lower marginal tax rates on higher incomes.[1] Our optimal tax computations give support to the first part (lowering the average tax rate), much less to the second; on the contrary our results suggest that a lower average tax rate should be obtained by lowering the marginal tax rates particularly on low and average income brackets[2]. The optimal tax rules efficiently exploit the pattern of heterogeneous responses from different households.
Work-in-progress
Simulating tax rules with more parameters
Including transfers (social policies, lump-sum benefits or taxes etc.) – Preliminary results suggest that the current level of transfers in Norway might be close to optimal
Of course the structure of trnsfers is a much more complicated matter. Even if the level were optimal, the structure of welfare policy might be suboptimal.

27. References
Aaberge, R.., J.K. Dagsvik and S. Strøm (1995): "Labor Supply Responses and Welfare Effects of Tax Reforms", Scandinavian Journal of Economics, 97, 4,
Aaberge, R., U. Colombino and S. Strøm (1999): “Labor Supply in Italy: An Empirical Analysis of Joint Household Decisions, with Taxes and Quantity Constraints”, Journal of Applied Econometrics, 14,
Aaberge, R., U. Colombino and S. Strøm (2000): “Labour supply responses and welfare effects from replacing current tax rules by a flat tax: empirical evidence from Italy, Norway and Sweden”, Journal of Population Economics, 13,
Aaberge, R., U. Colombino and S. Strøm (2004): "Do More Equal Slices Shrink the Cake? An Empirical Investigation of Tax-Transfer Reform Proposals in Italy“, Journal of Population Economics, 17
Aaberge, R. and U. Colombino (2006): “Designing Optimal Taxes with a Microeconometric Model of Household Labour Supply“, IZA DP 2468.