QUESTION 4.4 and 4.5 Consider a household that maximizes utility from consumptio

Question

QUESTION 4.4 and 4.5

Consider a household that maximizes utility from consumption over two periods max u (G) + «G) subject to the intertemporal budget constraint C2 C1+ Question 4.1 What is the economic reason for why we assume that the utility function is concave? Question 4.2 Derive the Euler equation for the household. The Euler equation relates the marginal utility of consumption in periods 1 and 2 under the optimal consumption choice Make sure you provide the derivation so that it is clear how you did it (just the final answer is not enough Question 4.3 Assume that the household can leave bequests in the amount of B 2 0 at the end of period 2, maybe for the children or for charity. Modify the intertemporal budget constraint appropriately to allow for this possibility. Question 4.4 Assume that the household still maximizes utility given by equation (1) above. Will it leave a strictly positive amount of bequests? If yes, determine this amount. If no, propose a modification of the utility function in equation (1) that would make the household leave behind a positive amount of bequests 4 Question 4.5 Imagine that the government needs to spend G in period 1. This spending must be paid for by taxing the houschold. The government can either tax the houschold in period 1, or borrow the necessary amount through international financial market at interest rate Rc and repay the loan (including interest) by taxing the household in period 2 Assume RG R. Is it better for the household if the government taxes in period 1, or period 2? Show it mathematically Does the Ricardian equivalence hold? Explain why or why not

Explanation / Answer

The utility fucntion is given as

[U=u(C_1)+eta u(C_2)]

Therefore, at equilibrium

[rac{rac{partial U}{partial C_1}}{rac{partial U}{partial C_2}}=1+r\\ herefore rac{u'(C_1)}{eta u'(C_2)}=1+r]

[ herefore u'(C_1)=(1+r)eta u'(C_2)]

From the budget constraint

[C_1+rac{C_2+B}{1+r}=W]

[ herefore C_1+rac{C_2}{1+r}=W-rac{B}{1+r}]

[ herefore C_1+rac{eta u'(C_2)C_2}{u'(C_1)}=W-rac{B}{1+r}]

[ herefore B=(1+r)left [W-C_1-rac{eta u'(C_2)C_2}{u'(C_1)} ight ]]

[ herefore B=(1+r)(W-C_1)-(1+r)left [rac{eta u'(C_2)C_2}{u'(C_1)} ight ]]

Let the utility fucntion be

[U=ln (C_1)+eta ln (C_2)]

[ herefore u'(C_1)=rac{1}{C_1}]

[ herefore u'(C_2)=rac{eta }{C_2}]

Therefore,

[ herefore u'(C_1)=(1+r)eta u'(C_2)]

[ herefore rac{1}{C_1}=rac{eta(1+r) }{C_2}]

Therefore,

[ herefore C_1+rac{C_2}{1+r}=W-rac{B}{1+r}]

[ herefore C_1+eta C_1=W-rac{B}{1+r}]

[ herefore (1+eta) C_1=W-rac{B}{1+r}]

[ herefore C_1=rac{W}{(1+eta)}-rac{B}{(1+eta)(1+r)}]

[ herefore C_2=rac{Weta (1+r)}{(1+eta)}-rac{Beta }{(1+eta)}]

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