## [solution]: Imagine a representative household with the following utility

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Show that the budget constraint can be rewrien in real terms (in terms of good sinstead of currency) as C + w` =wh + ? ? T. Download attachments for a (same) complete question

Imagine a representative household with the following utility function: u(C, `) = ? ln (C) + (1 ? ?) ln (`),

where 0 &lt; ? &lt; 1, and C represents consumption of a consumption good (or basket), while ` represents

hours of leisure.

(a) Find the marginal rate of substitution at some arbitrary point, (C, `).

(b) We considered three assumptions that consumer?s preferences must satisfy. Check whether this

utility function satis€es the €rst two assumptions (namely, more is be€er and a taste for diversity) or not.

(c) Does this utility function satisfy the Inada conditions? €is representative household has a budget

constraint that looks like C =wNs + ? ? T, where w represents the real wages, Ns the amount of hours

worked, T the lump-sum taxes government imposes, and ? the pro€ts of a representative €rm which the

representative household owns. Finally, the representative household has a total amount of h hours that

she can allocate to working or enjoying leisure time; that is ` + N s =h. For the rest of this problem,

suppose T &lt; ?.

(d) Show that the budget constraint can be rewri€en in real terms (in terms of goods instead of currency)

as C + w` =wh + ? ? T.

(e) Draw the budget constraint in real terms and mark every important element in it.

(f) Write the maximization problem of the household. (Do not forget to identify the domain of the utility

function, or non-negativity constraints.)

(g) Suppose that ? = 1 4 , h = 30, ? = 30, and T = 10. Solve the maximization problem of the

representative household. Make sure to argue whether the boundary points can or cannot be the

optima. What will be the optimal level of consumption, leisure, and labor, as functions of real wage, w?

(h) Are leisure and consumption good normal or inferior goods?

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