Question 5: A two-period model economy with a government.

Consider a two-period model economy with a government. Let preferences of the representative household be described by the utility function: u(C1,C2) = logC1+Blogc2, where C1 denotes consumption in period one and C2 denotes consumption in period two. The parameter B is the subjective discount factor and measures the consumer's degree of impatience in the sense that the smaller is B, the higher the weight the consumer assigns to present consumption relative to future consumption. The representative household enters the economy with zero real financial wealth a0=0. The house hold earns Y1 units of goods in period one and y2 units in period two. They also pay lump-sum taxes t1 and t2 in periods one and two, respectively , in order to finance government expenditure. The real interest rate paid on assets (a1) held from period one to period two is denoted by r. The government starts period one with no outstanding assets or liabilities and spends g1 and g2 in each period. Like the household, the government has access to financial markets where it can borrow or lend between periods at the interest rate r. Therefore the inter temporal government budget constraint is given by: g1+ g2/(1+r) = t1 + t2/(1+r).

a. Solve the household problem and derive optimal consumption and savings in each period.

b. Suppose the government reduces taxes in the first period but keeps government expenditure the same, that is dt1<0, dg1=0 and dg2=0( where d denotes change/derivative). If the intertemporal government budget constraint should remain balanced, show how consumption in period one and household savings between period one and two are influenced by the change in the timing of taxation. How are national savings (government plus household savings) influenced by this change?

c. Suppose the government increases its expenditure in the first period dg1>0 while keeping the same expenditure in the second period so dg2=0. The change in government spending is fully financed with current taxes, so that dg = dt1>0. How are household savings impacted in this case? What about national savings?