husband has a utility function, (/^(c, hm). Then, the husband and wife try to
maximize their own utility functions either cooperatively or noncooperatively subject to the budget constraint given by (8.1). One interpretation of the wife's utility
maximization problem given by (8.1) is that the husband's labour supply is
institutionally given, and the husband cannot select his labour supply. Therefore, the wife maximizes her own utility regarding the husband's total income as a part of nonwage income.
An alternative interpretation is that the husband is the Stackelberg leader and the wife is the Stackelberg follower. In this case, the husband tries to maximize his utility taking into account the wife's reaction. Hence, his labour supply function is given by
hm
=hm(wm, mm,
w/,mf).
Once he decides the labour supply, the wife tries to maximize her utility regarding the husband's labour supply as exogenously given. Therefore, again, the wife's utility maximization problem is given by (8.1).In this chapter, we employ the former interpretation, because under the latter interpretation, the husband's response to tax reform would be too complicated. Therefore, the husband's labour supply is assumed to be fixed, even if tax system changes.
Next, let us discuss the relationship between this static model and a dynamic labour supply model. Most of the dynamic labour supply models assume that a utility function is additively separable such as,
n
= X (1 + py'uict, ht), (8.2)
r=0
where m is a utility function at each period which is strictly concave, Ct and ht are respectively consumption and labour supply at r, p is a discount rate. The individual maximizes (8.2) subject to the budget constraint.
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n n
X + '-y'PtCt = X + ry'wtht + mo, (8.3)
t=0
f=0Where r is the interest rate, pt is the price of c at t, and mo is nonwage income at the
beginning of the economic life. The future income and expenditure should be discounted by the interest rate.
This maximization problem can be solved by two steps, since the utility function is separable. The first step is to allocate the income to each period such that the
marginal utility of money is equal for all periods. Then, the second step is to solve each period's utility maximization.
Max u*{ct, ht) = {\+ pY'uict, h[),
s.t.pt*C( = wt*ht + mt*, (8.4) where = (1 + r)-'/?,, Wi* = (1 + rrit* is the allocated income which may be
negative if the individual saves at period r. This maximization problem is solved as,
ht = h{wt*,pt*,mt*). (8.5) which is quite similar to the static labour supply function. However, an important
difference between the above model and the static model is that mi* is an endogenous variable in the dynamic model. Furthermore, mi* is not the nonwage income at period
t, but the dissaving at period t. Therefore, m,* is not observed unless we have data on
expenditure, as well as on income.
2.2. Estimation Method
It is assumed that the labour supply is linear in the wage and in nonwage income. The properties of this function are explained in Chapter 6. Since the
estimation results are utilized to simulate labour supply and to measure the deadweight loss under an alternative tax system, the nonlinearity of the budget line should be taken into account when we estimate the labour supply function. Therefore, we employ the full information method explained in Chapter 3.
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the slope and intercept of the piecewise-linear budget line corresponding to each tax brackets, and the hours of work at each kink point should be calculated for all
observations. In other words, it is necessary to create the whole budget line for each wife. The necessary information is as follows: wife's wage income, wife's hours worked; wife's nonwage income, husband's total income, and wife's and husband's taxable income.
Three points should be noted. First, the market wage rate is calculated by
dividing the wage income by the hours of work. When we estimate the labour supply, however, the predicted market wage rate is used, partly because the market wage rate is not observed for nonparticipants and partly because the measurement error of the wage rate is negatively correlated with hours worked.'^
Second, total income is not generally equal to taxable income. The gap between the two incomes would stem from the deductions from taxable income, payments of provisional tax, tax evasion, and measurement errors. It is assumed that the gap is independent of hours of work, and moreover it is assumed that the gap does not change under an alternative tax system.^
Third, it is assumed that there is no fixed entry cost so that the error stnicUire is of the standard tobit type. This assumption, however, does not prohibit the existence
The way in which we estimateed the wage rate is the same as that in Chapter 7. That is, first of all, the participation equation is estimated by probit, and then the wage equation is estimated including the inverse of Mill's ratio as an explanatory variable. The participation equation does not include the wage rate, and the equation is interpreted as a reduced form of the labour supply function. Although we do not assume the fixed market entry cost in the remainder of the chapter, the probit estimation is valid even under the existence of the fixed cost. Hence the estimation method of the wage rate is not totally consistent with the model.
5 Because of this assumption, the predicted taxable income may be negative for some wives under an altemative tax system.
167 of variable entry costs which are embodied in the labour supply function, as was discussed in Chapter 4.
2.3. The Simulation Method
The purpose of the simulation is to see whether or not introducing a single rate tax system that is based on individual income would improve efficiency in terms of the wife's labour supply and the deadweight loss. We introduce two single rate tax
systems, a single rate tax system without a threshold (the proportional tax system), and a single rate tax system with a threshold of $4,595, which is the same as the first threshold under the current tax system (see Table 8.1). The tax rates of these systems are determined such that the predicted total tax revenue is equal to the predicted total tax revenue under the current tax system. The simulation procedure is as follows.
i) Using the estimated labour supply function, the labour supply and the equivalent incomes of all wives in the absence of any tax are calculated under the assumption that there is no selection error. Equivalent incomes evaluated at the sample mean of the wage rate are then calculated for all wives.
ii) All wives are ranked according to their equivalent income under the no-tax situation. The sample is divided into deciles according to this ranking, and this ranking is used simulate other tax systems.
Table 8.1. 1985-1986 Income Year Tax Schedule
Not less than 0 4595 12500 19500 28000 35000 Not more than 4595 12500 19500 28000 35000 and over Marginal tax rate 0 .25 .30 .46 .48 .60
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iii) The wife's labour supply, the husband's and wife's tax, and the
deadweight loss under the current tax system is simulated assuming that there is no selection error.
iv) A proportional tax system is created such that the predicted total tax
revenue under the hypothetical tax system is the same as the predicted total tax revenue under the current tax system.^- The labour supply, tax and the deadweight loss under the proportional tax system are compared to those under the current tax system.
v) A single rate tax system with a threshold is created such that the threshold is set to $4,595, and the tax rate is determined such that the predicted total revenue is equal to the current tax system. Again, the labour supply, tax and the deadweight loss are compared to those under the current tax system.
Thus, we will compare the current tax system to two different individual-income- based single rate tax systems.
The above simulation will show the effects of introducing single rate tax systems that are based on individual income. In order to compare these effects to the effects of introducing a single rate tax system that is based on the joint income, we will create a ^ Under the proportional tax system, the tax rate is constant regardless of the income level,
so long as taxable income is positive. If simulated taxable income is negative, which sometimes happens at low levels labour supply, it is assumed that no tax is levied on the income.
As we will see later, the predicted tax revenue under the current tax system underestimates the actual tax revenue, because we ignore the selection error. In order to see a possible bias from this underestimation, we also simulated the labour supply and tax revenue under an altemative assumption that people will make a selection error under an alternative tax system that is exactly the same as the error under the current tax system.
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