4.2  A non-convex budget set

An example of a budget constraint with a means-tested benefit is given in Figure 14, as ABC. The benefit is gradually withdrawn at a relatively high rate until it is exhausted at B, when the individual only pays income tax (where integration of the benefit and tax systems avoids a discontinuity). Here the budget set is said to be non-convex. This raises the possibility of an indifference curve being simultaneously tangential to the two sections of the constraint, for a particular wage rate; this is shown in Figure 14 by the two tangencies at J and K. A small increase in the wage rate would therefore produce a discrete jump in hours worked from J to K.

Figure 14 – Means-testing: A non-convex budget set

Figure 15 – Gross earnings and the wage rate

A discrete jump of this kind is reflected in the relationship between gross earnings and the wage rate, shown in Figure 15. When the wage rate is increased, earnings become positive after a particular wage has been passed and the individual increases labour supply along the range AB. But at a particular wage, the ‘switching’ wage, the individual jumps to point K along the range BC in Figure 15. Further increases in the wage produce a gradual increase in labour supply along KC. The associated labour supply curve is shown in Figure 16, where the vertical jump in Figure 15 translates into a horizontal range of the supply curve. There is a range where the individual is working and receiving benefits along part of AJ; but this is just a proportion of the hours range over which the individual is actually eligible for benefits.

Figure 16 – Labour supply curve

These properties of the relationship between the shape of the budget constrain and the labour supply curve are completely general: a kink in the budget constraint associated with a rise in the marginal tax rate is associated with a ‘backward bending’ range along a rectangular hyperbola (where gross earnings are constant), and a kink associated with a fall in the effective marginal tax rate produces a horizontal range in the supply curve. In the first case the kink produces a degree of rigidity (no movement from the ‘travelling’ kink as the wage changes), while in the second case the kink produces a discrete jump in labour supply. The phenomenon of a jump may suggest that the earnings distribution may have a ‘gap’ or antimode - but (as with the other type of kink) the distribution of earnings is generated by substantial heterogeneity in wages and preferences, and such a gap may not necessarily be observed.

Figure 17 – Corner and tangency solutions

Figure 18 – Gross earnings and the wage rate

The existence of means-testing may actually rule out a complete section of the budget line from the point of view of labour supply. Such a possibility is shown in Figure 17, where instead of two tangency positions, the highest indifference curve is shown to give rise simultaneously to a corner solution at A and a tangency at K. Hence the whole range of AB is ruled out, as well as the range BK of BC. The relationship between gross earnings and the wage is shown in Figure 18 where the ‘switching’ wage produces a jump from zero labour supply and thus zero earnings to the point K. This individual would not be observed both working and receiving the means-tested benefit, that is, would not be among the ‘working poor’

Figure 19 – Fixed costs of working

In addition to means testing, a non-convex range of the budget set may arise from the existence of fixed costs of working. This is shown in Figure 19, where fixed costs are shown as the length AB. Their existence substantially raises the minimum wage above which the individual works, and implies that low hours levels are not chosen at all.