Question

Suppose that a representative consumer has the following utility function: U(C,L)=5ln(C)+2ln(L) and she has the budget constraint w(h-L)+pie-T. Derive the consumer's rules for consumption and leisure (solve for consumption and leisure "demand functions" as a function of exogenous variables).

Graph and discuss the income and substitution effects for a consumer if her wages decreased (with labels).

Answer 1

The above scenario can be shown in the diagram as follows. The budget constraint is shown as ABh with a kink at h = pie (i.e., the non-labor income). If all the hours are alloted to labor (i.e., leisure=0), then wage income = wh and after paying tax of T, the remaining income is wh-T, which is the intercept of the budget line. Hence, the initial equilibrium point is E, where the budget line is tangent to the indifference curve. Therefore, the optiomal level of leisure = OLa or the optimal level of labor = hLa (i.e., Oh - OLa).

Now if wage falls, the wahe income falls as a result of this, the budget constraint becomes A'Bh, where the intercept is (w'h - T). Let us now assume that the new optimal leisure becomes OLc, i,e., the consumer moves from point E to point F as a result of decline in wage. If an equivalent amount of money is given to the consumer so that the consumer can attain the earlier level of utility, let he move to Es. In other words, since wage declines, some amount of leisure or labor would be substituted by consumption, which is the substitution effect. In the diagram below, the substitution effect is denoted by LbLa. That is, due to substitution effect, leisure could have decreased to OLb or labor could increase to hLb. The remianing effect LbLc (if we take away the income to bring the consumer to the new level of utility) is known as income effect.