Consider the representative consumer who decides consumption and leisure. The environment is the same as in Lecture 5. Keep the same notation. The preference is given by U (C,L) = αln C + (1 −α) ln L. Assume h = 1, i.e., the time endowment is one day. (a) Write down the utility maximization problem. (b) Derive the demand for consumption and the supply for labour. (c) Suppose the non-wage income π −T increases while the wage rate w falls at the same time. The size of the changes can be different. Determine the effects on consumption demand and labour supply (i.e., leisure demand). Use the indifference map to explain your results in terms of income and substitution effects for the following cases: (i) The increase in π −T exactly cancels out the drop in w, i.e., |∆ (π −T)|= |∆w|. (ii) The increase in π −T is greater than the drop in w, i.e., |∆ (π −T)|> |∆w|. (iii) The increase in π −T is smaller than the drop in w, i.e., |∆ (π −T)|< |∆w|. (d) Suppose the utility function is Cobb-Douglas: U (C,L) = CαL1−α. Show that the demand for consumption and the supply for labour are the same as in (b). Explain why this is the case.
Consider the representative consumer who decides consumption and leisure. The
environment is the same as in Lecture 5. Keep the same notation. The preference is given
by U (C,L) = αln C + (1 −α) ln L. Assume h = 1, i.e., the time endowment is one day.
(a) Write down the utility maximization problem.
(b) Derive the
(c) Suppose the non-wage income π −T increases while the wage rate w falls at the same
time. The size of the changes can be different. Determine the effects on consumption
demand and labour supply (i.e., leisure demand). Use the indifference map to explain
your results in terms of income and substitution effects for the following cases:
(i) The increase in π −T exactly cancels out the drop in w, i.e., |∆ (π −T)|= |∆w|.
(ii) The increase in π −T is greater than the drop in w, i.e., |∆ (π −T)|> |∆w|.
(iii) The increase in π −T is smaller than the drop in w, i.e., |∆ (π −T)|< |∆w|.
(d) Suppose the utility function is Cobb-Douglas: U (C,L) = CαL1−α. Show that
the demand for consumption and the supply for labour are the same as in (b). Explain
why this is the case.
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