4 Piecewise-linear budget constraints
This section extends the previous analysis to deal with budget constraints made up of several linear segments, each associated with a given marginal tax rate and earnings threshold. Such multi-segment constraints are referred to as piecewise-linear budget constraints.
4.1 A convex budget set
The effect of an increase in the gross wage on a more complex budget constraint is shown in Figure 10. The constraint ABC has a kink at B, reflecting the presence of an earnings threshold where the marginal effective tax rate increases. The budget set is said to be convex: a straight line joining any two points is associated with a feasible position, that is a net income less than or equal to the net income along the budget line.
The earnings threshold is not evident from the budget constraint, unlike the net/gross income diagram, since the hours level at which it occurs depends on the wage rate. For a higher wage rate, the budget constraint pivots to AB′C′, and the kink point B′ is to the left of B. This is because a lower hours level is required, at the higher wage, to reach the earnings threshold where the marginal rate increases; gross and thus net income remain constant at the kink.
A corner solution is shown in Figure 11, where the highest indifference curve just touches the budget constraint at point B. However, the optimal position for the individual can nevertheless be represented as if it were a tangency solution. The artificial budget line can be drawn which is tangential to the indifference curve at the kink, B. This generates the important concepts of the virtual wage and virtual income. These concepts are taken from the theory of rationing, where corner solutions are important, and are particularly useful when considering the welfare changes arising from tax reforms.
Before considering the labour supply function implied by the budget constraint in Figure 10, it is useful first to examine the relationship between gross earnings (the product of hours worked and the gross wage, that is, 
) and the wage rate. This is shown in Figure 12. At very low wages, utility maximisation gives rise to the corner solution at A. When the wage rate exceeds some level (as the section AB of the constraint pivots about A), the individual moves to a tangency position. Increases in the wage induce higher labour supply until the gross earnings threshold is reached at which the marginal effective tax rate increases. A characteristic of this kind of kink in the budget constraint - where there is no tangency solution but a position where the highest indifference curve touches the relevant corner - is that the individual ‘sticks’ at the corner for a range of wage rates. In this case the gross earnings remain constant as the wage rate rises over a range, although of course the associated hours level falls. Eventually, for a sufficiently high wage rate, the individual moves to a tangency along the range BC of the constraint. The lengths of the flat sections AA and BB, and the nature of the rising sections AB and BC, depend on the individual’s preferences. A higher preference for consumption over leisure (flatter indifference curves) gives rise to shorter sections AA and BB in Figure 12.
Individuals may face the same tax rates and thresholds, but the variation in non-wage incomes, wage rates and preferences means that each budget constraint and gross earnings/wage schedule is unique. However, the existence of the kink may suggest that some ‘bunching’ of individuals around the threshold in the distribution of gross earnings. This kind of phenomenon is nevertheless only observed in particular cases - a tax threshold need not produce a ‘spike’ in the earnings distribution, and modes may in practice correspond to tangency solutions. Hence, the distribution of earnings need not necessarily provide any information about the extent of the labour supply effects of taxation.
The range BB in Figure 12 is associated with a fixed level of earnings, so that is constant. This implies that, in a graph of hours worked plotted against the wage rate, the hours of work would follow a rectangular hyperbola over the relevant range. The labour supply function is shown in Figure 13. This property, that the labour supply curve turns sharply backwards, following a rectangular hyperbola, as the wage rate increases over a range, is entirely general and applies to any kink in the budget constraint associated with an increase in the marginal effective tax rate at a threshold level of earnings.
Two important general lessons emerge from this discussion. First, the number of hours of work supplied and the net wage are jointly determined. Second, it makes little sense to attempt to describe the labour supply function (hours supplied as a function of the gross wage) in terms of a single elasticity. Even if the ranges AB and BC have a constant elasticity, large variations occur at the kink points, and of course the elasticity changes sign twice.
