Consider the household model of child labour and schooling. The household preferences by the utility function U(c, h)=lnc+h , where is a preference parameter indicating the relative value of a child's human capital h relative to current consumption c. A household's consumption is derived from parental labour income, wplp , and child labour income,wclc . A child devotes their time tc either to schooling ec or to child labour lc. A unit of time allocated to schooling contributes to human capital accumulation according to the function h=ec . wp and wc are the market wages of adult (parental) labour and child labour respectively. Assume that tc=1, =4, =1,tp= lp=1 and wc=2 .

(3 pts.) Write down the household's optimization problem.

(3 pts.) Substitute the budget constraint, the time constraint and the human accumulation function into the objective function.

(2 pts.) Write down the first-order condition.

(2 pts.) Calculate the interior solution of labour supply is given by

(3 pts.) Calculate the supply function of child labour, including its interior solution, .

(2 pts.) Briefly interpret the implications of your answer in part d) for the labour supply curve.

(3 pts.) Suppose that there are two children whose parents earn and respectively. Given that parental wages are increasing in human capital, shows that a lower parental wage implies a qualitatively different labour supply for their children.