4.3  A multi-rate tax structure

Figure 20 contains a budget constraint having four marginal tax rates. Means-testing of benefits involves an increase in the marginal rate after point B, and a subsequent reduction in the marginal rate once benefits have been exhausted at point C. The range DE reflects an increase in the marginal income tax rate beyond D. This degree of nonlinearity of the budget constraint generates the possibility of multiple local optima on different ranges of the constraint. Figure 21 shows a situation where indifference curve U₁ is tangential to the budget constraint along CD, while curve U₂ gives a corner solution at point B where higher taper or benefit withdrawal rates start to apply. This does not introduce any new principles, but raises complications when modelling labour supply, as care must be taken to select the global optimum.

Figure 20 – A multi-rate system

Figure 21 – Multiple local optima

The labour supply curve resulting from this four-rate tax structure is shown in Figure 22. This reflects the general properties obtained above. Hence the two corners B and D of the constraint (and the earnings thresholds associated with them) in Figure 20 generate the two ranges BB and DD of Figure 22 which involve reductions in labour supply along rectangular hyperbolae. The flat section of the supply curve JK is associated with the jump from a tangency along BC to a tangency along DC of the budget constraint. The rising sections of the supply curve are associated with tangency solutions.

Figure 22 – Labour supply curve with multi-rate tax system

The supply curve need not necessarily take the precise form shown in Figure 22. Consider again the budget constraint shown in Figure 20. The supply curve may involve, for a small increase in the wage rate, a jump directly from a corner solution at point B on the budget constraint to a tangency along the length CD of the constraint.[3] This is more likely to occur, the higher is the benefit withdrawal rate (the flatter the segment AB in Figure 20). This alternative is shown in Figure 23.

Figure 23 – Alternative labour supply curve

Furthermore, it is quite possible that a jump could occur from the kink in the budget constraint of Figure 20 at B directly to the kink at D, thereby leaving out the whole of the range of the constraint from B to D. The resulting labour supply curve is illustrated in Figure 24.

Figure 24 – Another alternative labour supply curve