Suppose we have a household with the following lifetime utility function U(e,) Ine+lne In each period...

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Suppose we have a household with the following lifetime utility function U(e,) Ine+lne In each period they recieve an endowment y and y2. Now suppose that the household has access to a credit market in period 1. They will choose to save or borrow which is given by the variable s. Where s > 0 signifies savings and 82 < 0 signifies borrowing. In period 2 the household is not permitted to borrow and so will not engage with the credit market (This is the same as saying 82 = 0). Any savings (or borrowing) the household does will earn the interest rate r. Finally the government will also have government spending. They will raise taxes via a lump sum tax in period 1 denoted as T. In period 2 they will raise taxes via a lump sum tax denoted as T2 and also proportional tax on savings in period 2 denoted p2 (this is rho but lower case p works as well - just don't want you to confuse with price P). This means that savings income in period 2 for the household will be (1+r)(1 p2)81. Finally the government operates a balanced budget constraint over the two periods so that their lifetime budget constraint is given by T2 P(1+r)s G = T+ + (1+r) (1+r) Where p2(1+r)s is the value of tax revenue from private savings in period 2. (a) (5 pts) First write out the per period budget constraint of the households, and combine to make the lifetime budget constraint (e) (5 pts) Suppose Government spending today increases by 1 dollar and it's funded by an increase in lump sum taxes today by 1 dollar. How do aggregate savings change? (f) (5 pts) Suppose instead Government spending today increases by 1 dollar and it's funded by an increase in lump sum taxes tomorrow by 1 dollar times (1+r). How do aggregate savings change? Comment on this result relative to part (e). (g) (5 pts) Let's plug in some numbers. Let's suppose that y = 13.75, y2 = 2, B = 0.8, r = 0.25, T = 0, T = 0, p 0 and initially government spending is equal to 0. Suppose they wish to increase G to be equal to 1. If ini- tially they fund this through T2, what will be lump sum taxes in period 2 and how do aggregate savings change? 1 via (h) (5 pts) Using the same numbers from part (h). Let's instead suppose they decide to fund the G proportional taxation. What will be the value of p2 so the government will fully meet it's obligations? (Hint: Use the government budget constraint and remember you need to plug in the value of sfrom the consumer problem, Note there will be two solutions use the smallest one as we will actually have something called a laffer curve, you are permitted to use an equation solver like wolfram alpha if done correctly it will be a nice round number). How do aggregate savings change? (The algebra for this is really intense and I don't want to break you so make sure to plug in those numbers from above to figure it out) Note: This is an example why micro- foundations are so important. Government policy impacts consumer decisions and therefore also impacts the revenue the government receives the policy maker needs to take into account how their change in policy changes behavior of the household! (i) (5 pts) For your answer to part (i) and (h) does Ricardian equivalence hold? I.e. do we see the same decrease in consumption as if I had just increased lump sum taxes in period 1 by enough to fund the government. Explain why or why not? (You can probably look at how aggregate savings changes to get an idea of how consumption is changing since government savings are changing by the same amount in part (h) and (i)). Suppose we have a household with the following lifetime utility function U(e,) Ine+lne In each period they recieve an endowment y and y2. Now suppose that the household has access to a credit market in period 1. They will choose to save or borrow which is given by the variable s. Where s > 0 signifies savings and 82 < 0 signifies borrowing. In period 2 the household is not permitted to borrow and so will not engage with the credit market (This is the same as saying 82 = 0). Any savings (or borrowing) the household does will earn the interest rate r. Finally the government will also have government spending. They will raise taxes via a lump sum tax in period 1 denoted as T. In period 2 they will raise taxes via a lump sum tax denoted as T2 and also proportional tax on savings in period 2 denoted p2 (this is rho but lower case p works as well - just don't want you to confuse with price P). This means that savings income in period 2 for the household will be (1+r)(1 p2)81. Finally the government operates a balanced budget constraint over the two periods so that their lifetime budget constraint is given by T2 P(1+r)s G = T+ + (1+r) (1+r) Where p2(1+r)s is the value of tax revenue from private savings in period 2. (a) (5 pts) First write out the per period budget constraint of the households, and combine to make the lifetime budget constraint (e) (5 pts) Suppose Government spending today increases by 1 dollar and it's funded by an increase in lump sum taxes today by 1 dollar. How do aggregate savings change? (f) (5 pts) Suppose instead Government spending today increases by 1 dollar and it's funded by an increase in lump sum taxes tomorrow by 1 dollar times (1+r). How do aggregate savings change? Comment on this result relative to part (e). (g) (5 pts) Let's plug in some numbers. Let's suppose that y = 13.75, y2 = 2, B = 0.8, r = 0.25, T = 0, T = 0, p 0 and initially government spending is equal to 0. Suppose they wish to increase G to be equal to 1. If ini- tially they fund this through T2, what will be lump sum taxes in period 2 and how do aggregate savings change? 1 via (h) (5 pts) Using the same numbers from part (h). Let's instead suppose they decide to fund the G proportional taxation. What will be the value of p2 so the government will fully meet it's obligations? (Hint: Use the government budget constraint and remember you need to plug in the value of sfrom the consumer problem, Note there will be two solutions use the smallest one as we will actually have something called a laffer curve, you are permitted to use an equation solver like wolfram alpha if done correctly it will be a nice round number). How do aggregate savings change? (The algebra for this is really intense and I don't want to break you so make sure to plug in those numbers from above to figure it out) Note: This is an example why micro- foundations are so important. Government policy impacts consumer decisions and therefore also impacts the revenue the government receives the policy maker needs to take into account how their change in policy changes behavior of the household! (i) (5 pts) For your answer to part (i) and (h) does Ricardian equivalence hold? I.e. do we see the same decrease in consumption as if I had just increased lump sum taxes in period 1 by enough to fund the government. Explain why or why not? (You can probably look at how aggregate savings changes to get an idea of how consumption is changing since government savings are changing by the same amount in part (h) and (i)).

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Answer a The per period budget constraint of the household can be written as c1 y1 s1 T1 c2 y2 1r1ho2s1 T2 To obtain the lifetime budget constraint we ... View the full answer