Full employment equilibrium-optimal level of investment

Question

In a 2 period model, suppose that people's preferences are such that they want complete smoothing of consumption, i.e., C₁ = C₂.  The government has a larger horizon than households, so it has a certain amount of debt by the end of the second period. Thus, its intertemporal budget constraint is:

G₁ + G₂/(1+r) = T₁ + T₂/(1+r) + D^g₂/(1+r)

The structure of the economy is the following:

Ld = 50-2w       G₁= 50        K₁ = 20
Ls = 3w            G₂ = 110
Q = LK/3           T₁ = 40      T₂ = 55

Where K₁ is the stock of capital in period 1.

Question 1: Find out the full employment output. If the economy is at full employment, what is the income level at time t = 1.

Question 2: If the supply of labor increases to Ls = 25 + 3w in period 2, determine the new full employment equilibrium.

Question 3: Determine the optimal level of investment for a household when i = 10%, and there is no depreciation. Is the economy at an optimal level of investment in period 1? Would an introduction of an investment tax credit facilitate efficiency in investment?

Question 4: Based on 2 and assuming an investment of 10 in the second period, find the full equilibrium output at t = 2.

Question 5: Based on 1-4, what is the household permanent income?

Question 6: Determine total savings in each period.  Is the Investment/Saving market at equilibrium in each period?

Question 7: What is the present value of the government spending?

Question 8: Is the balance of payment in equilibrium in each period? Show your answer graphically and formally.

Question 9: What is the value of the Government debt at the end of the second period? 

Question 10: Assume that the government modifies taxes so that T₁ = 50 and T₂= 44 but leaves expenditures unchanged. Did the present value of taxes change? What does your answer tell you about Ricardian equivalence?

Question 11: How would your answer in 10 change if the new taxes were T₁ = 30 and T₂ = 44.  What would be the impact on consumption expenditures?

Question 12: If the government cannot finance its deficit domestically and it is a small economy, what would be the impact on the current account?

Question 13: If this country is a small economy, wants to avoid a default on their debt, and does not have any further access to foreign borrowing, what type of restriction can it impose to obtain the desire funds?  How would this affect the country’s level of investment and aggregate demand?  Explain.

Question 14: If instead of pursuing the alternative proposed on n.13, the country wants to seek an increase of foreign reserves while achieving a current account balance, what would the government need to do? Assuming a marginal propensity to consume of 0.75 and a marginal tax of 10%, what would be the require change in government expenditures in period 2?