8.2  A numerical example

This section presents a small tax policy simulation using the example from subsection 6, in order to illustrate the procedure described above. The utility for all individuals is the estimated utility function In the simulation, a linear benefit and tax system is introduced. Individuals without income receive 15 units of income and gross income (excluding this basic income of 15) is taxed at 20 per cent.[44] Table 4 presents the income and utility at the discrete hours points for all three individuals before and after the reform.

Table 4– Utility pre- and post-reform

Pre-reform
0	0	0	0	0	0	0
20	80	-153.8	160	0.6	200	77.8
40	160	-307.6	320	1.2	400	155.6
Post-reform
0	15	28.9	15	28.9	15	28.9
20	79	-155.7	143	-32.2	175	29.5
40	143	-340.4	271	-93.4	335	30.2

From the table it is clear that the introduction of the tax system has made work much less attractive. Adding draws from the extreme value distribution to the estimated utility function, in order to obtain the s, results in different utility levels for each draw. Table 5 presents, for each individual, ten sets of draws from the extreme value distribution which result in the observed hours being the optimal choice for each individual. The corresponding utility levels are presented below each value of , where indicates utility pre-reform.

Calculation of the utility conditional on this draw, after the reform has been introduced, results in utility levels post-reform, indicated by . From the utility levels in Table 4, it is clear that individuals 1 and 2 are most likely not to participate whereas individual 3 has utility levels at 0, 20 and 40 hours of work which are relatively close to each other. In Table 5 it can be seen that in draw 9 the utility of individual 3 is highest for 20 hours of work and in draw 4 it is highest at zero hours of work, whereas in the other draws the utility is highest when the person is working full time. For the other two individuals, non-participation always results in the highest utility.

Table 5 – Utility pre-reform and post-reform for ten sets of accepted draws from the extreme value distribution

	Person 1			Person 2			Person 3
	0	20	40	0	20	40	0	20	40
Note:  indicates the error term for draw i, which is added to the calculated utility level  before and after the reform
	-1.070	1.361	0.178	2.997	3.491	0.217	1.176	1.026	2.426
	-1.070	-152.439	-307.422	2.997	4.091	1.417	1.176	78.826	158.026
	27.880	-154.369	-340.232	31.947	-28.719	-93.153	30.126	30.576	32.576
	-0.805	0.437	-0.012	0.777	0.241	-1.416	0.907	-0.781	5.678
	-0.805	-153.363	-307.612	0.777	0.841	-0.216	0.907	77.019	161.278
	28.145	-155.293	-340.422	29.727	-31.969	-94.786	29.857	28.769	35.828
	0.233	-0.742	0.037	-0.168	1.285	0.080	0.801	1.232	0.992
	0.233	-154.542	-307.563	-0.168	1.885	1.280	0.801	79.032	156.592
	29.183	-156.472	-340.373	28.782	-30.925	-93.290	29.751	30.782	31.142
	2.554	2.402	-0.022	-0.638	1.635	0.522	2.069	1.249	0.456
	2.554	-151.398	-307.622	-0.638	2.235	1.722	2.069	79.049	156.056
	31.504	-153.328	-340.432	28.312	-30.575	-92.848	31.019	30.799	30.606
	-0.019	0.656	1.257	0.712	3.741	2.412	-0.715	-0.400	-0.329
	-0.019	-153.144	-306.343	0.712	4.341	3.612	-0.715	77.400	155.271
	28.931	-155.074	-339.153	29.662	-28.469	-90.958	28.235	29.150	29.821
	0.062	1.628	-1.269	0.113	2.428	-0.412	-1.243	-0.673	-0.535
	0.062	-152.172	-308.869	0.113	3.028	0.788	-1.243	77.127	155.065
	29.012	-154.102	-341.679	29.063	-29.782	-93.782	27.707	28.877	29.615
	-0.626	1.079	-0.550	1.196	0.844	-1.501	1.771	1.518	2.311
	-0.626	-152.721	-308.150	1.196	1.444	-0.301	1.771	79.318	157.911
	28.324	-154.651	-340.960	30.146	-31.366	-94.871	30.721	31.068	32.461
	0.136	1.233	0.174	-0.507	1.855	1.036	-1.346	-0.555	1.123
	0.136	-152.567	-307.426	-0.507	2.455	2.236	-1.346	77.244	156.723
	29.086	-154.497	-340.236	28.443	-30.355	-92.334	27.604	28.995	31.273
	2.745	-0.530	0.363	0.163	1.044	-0.216	0.633	0.433	-0.695
	2.745	-154.330	-307.237	0.163	1.644	0.984	0.633	78.233	154.905
	31.695	-156.260	-340.047	29.113	-31.166	-93.586	29.583	29.983	29.455
	1.730	-0.330	-1.190	-1.240	1.479	-0.861	-0.497	0.187	0.229
	1.730	-154.130	-308.790	-1.240	2.079	0.339	-0.497	77.987	155.829
	30.680	-156.060	-341.600	27.710	-30.731	-94.231	28.453	29.737	30.379

The results from these ten draws can be summarised in a transition table. Table 6 presents such a matrix for this example. The last column presents the distribution of labour supply before the reform and the last row presents this distribution after the reform. The distribution before the reform consists of the percentages of individuals observed in each of the hours points. The distribution after the reform is constructed from the individual probabilities of being at each of the discrete hours points. After the reform an individual cannot be assigned to one of the discrete hours points, but has a positive probability of being at each of the hours points. However, some of these probabilities may be extremely close to zero. All these probabilities for an individual add up to one. The numbers inside the matrix are row percentages indicating the probability of individuals moving from one discrete hours point to another. Thus, the probability of moving from zero hours is nil, the probability of moving from 20 hours to zero hours is 100 per cent and the probability of remaining at 40 hours is 80 per cent. There is a probability of 10 per cent of moving out of the labour force and the probability of reducing labour supply to 20 hours is also 10 per cent.

The predicted probability of person 3 being in zero hours, 20 hours and 40 hours is 15.4 per cent, 28.1 per cent and 56.5 per cent respectively.[45] These are unconditional probabilities, but given the large difference between utility at the different hours levels in the starting situation and the observed hours being the optimal hours, there should not be much difference between the conditional and unconditional probabilities in this case, because most draws from the extreme value distribution would be accepted. The simulation method using draws from the extreme value distribution provides results that are different from these expected probabilities. Table 6 shows that these were 10, 10 and 80 per cent respectively for 0, 20 and 40 hours of work. However, by increasing the number of draws the approximation becomes more accurate.[46]

Using a similar simulation approach, wage elasticities can be calculated for the three individuals in the example. These can be computed with and without calibration. Table 7 show the results of using the alternative methods for each individual. At the wage levels of persons 1 and 3 a small change does not have any effect on the relative utility levels at each of the hours points. Therefore no change in labour supply is expected. However, for person 2 the utility levels of the three hours points are closer to each other. As a result, a small change in the wage level has a large effect on expected labour supply. It is only for person 2 that calibration has an effect on the outcomes, because for the other two persons nearly all possible draws of the error term result in the correct labour supply choice, whereas for person 2 the error term can shift the optimal outcome from one point to another. Here it is shown that calibration can make a difference to the result. Using calibration in this example, the expected wage elasticity is about twice as large as without calibration.