Question

Graph the standard labor/leisure model. Assume the person has both wage earnings and unearned income. Correctly label the axes of the graph. Label the amount of unearned income, the vertical intercept of the budget constraint, and the slope of the budget line. Include a representative indifference curve. Label the amount of leisure consumed at equilibrium and also the person’s total income (consumption) at the equilibrium. Provide the formula that gives equilibrium hours of work.

Answer 1

let's say the person has total time endowment of T hours

he has to decide how many hours to have as leisure time, let's say he enjoys L hours of leisure time.

hence he works T-L hours

let say wage is w and he has a non-labour income of N (represented by ON in the graph)

hence, he has the budget constraint, consumption, C=N+(T-L)w or C=(N+Tw)-wL, now N+Tw is the level of consumption that is possible when he works, T, hours hence L=0 this is represented by the vertical intercept of the budget line. OC

hence he will choose a point where he maximises his utility and with the help of indifference curve analysis we found the point to be the tangency of the indifference curve and budget line and optimal consumption is OC* and optimal leisure hours are OL*.

we know the slope of the indifference curve is given by -MU(leisure)/MU(consumption) where Mu means marginal utility. now we know the slope of the budget line is -w. (compare the budget line equation with y=mx+c form where m is the slope).

hence the formula giving the equilibrium leisure consumption is -MU(leisure)/MU(consumption)=-w or MU(leisure)/MU(consumption)=w