3 Taxes and labour supply
Labour supply considerations require the use of budget constraints relating net income to the number of hours worked, rather than the type of diagram used in the previous section. Furthermore, the net/gross income diagram does not distinguish income from earnings and other sources.
3.1 The framework of analysis
The discussion in this section assumes that the individual has a fixed gross wage (though there may be an overtime premium) in a single job. Each individual is assumed to maximise utility, which is regarded as a function of net income and leisure. An indifference map is shown in Figure 6: indifference curves that are further to the north west represent higher utility. In this simple static model, although hours of work are considered, consumption does not take time, there is no household production and leisure is simply enjoyed as time not spent working. The individual trades, at the margin, the disutility arising from working an extra hour with the utility obtained from the consumption of the resulting net income.
Utility is maximised subject to a budget constraint. The concept of the budget constraint necessarily applies to single individuals, rather than groups of individuals (though it may also apply to individuals with working or non-working partners). This is because each budget constraint depends fundamentally on the individual’s wage rate, as well as some other characteristics, and is thus unique. A basic requirement of any model designed to examine taxes in the presence of labour supply responses is therefore that it is capable of producing the budget constraints for all individuals in the data set on which the model is constructed.
3.2 The basic income – flat tax
Figure 7 shows the budget constraint for an individual under the BI/FT structure, and facing a gross wage rate per hour of 
The starting point of the constraint, where hours worked equal zero, 
, corresponds to net income at point A. This includes any benefits, along with all sources of non-wage income (and if the labour supply decision of couples is assumed to involve some kind of joint maximisation process, this may include the net income of the partner). The slope of the budget constraint measures the extra net income obtained from an extra hour of work, and is thus equal to 
, the net wage rate.
The maximisation of utility, subject to the budget constraint, may therefore lead either to a corner solution at A, where the individual does not work, or a tangency solution somewhere along AB. Within this framework, the work participation decision is associated with movement away from the corner solution. This corner is clearly more likely to be chosen, the lower is the gross wage and the higher is the tax rate, since both tend to make the constraint flatter.
Figure 8 shows the effect on labour supply of varying the wage rate. For the relatively low wage rate of 
the optimal position is at the non-participation corner A on indifference curve U
As the wage increases to 
and then to 
the individual moves to tangency positions on indifference curves U
and U
respectively. The gross wage is of course not transparent in Figure 8, because the slope of the budget line is the net wage. It is therefore useful to transfer the information to a separate diagram, the labour supply curve.
The supply curve associated with this linear budget constraint is shown in Figure 9 as AAB. As usual in economics, the ‘price’ - here the gross wage - is shown on the vertical axis while hours worked are shown on the horizontal axis (although strictly the relevant price here is that of leisure, which is reflected in the net rather than the gross wage rate). The range AA is associated with the kink point A in Figure 7 and reflects wage rates for which the individual remains at the ‘non-participation’ kink. For higher wages, labour supply expands along the range AB as this pivots about A. Depending on the individual’s preferences, the supply curve may bend ‘backwards’ as shown by the dashed line, which may be said to represent the standard elementary ‘text book’ labour supply curve.
