Consider a two-period economy with a government. Let preferences of the representative household be described by the utility function: U(C₁,C₂) = log c₁ + log c₂, where c₁ denotes consumption in period one and c₂ denotes consumption in period two. The parameter is the subjective discount factor and measures the consumers degrees of impatience in the sense that the smaller is , 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 household 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 expenditures.

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 intertemporal 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 t₁ < 0, g₁ = 0 and g₂ = 0(where the symbol denotes change). 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 changes 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 g₁ > 0, while keeping the same expenditure in the second period so g₂ = 0. The change in government spending is fully financed with current taxes, so that g₁ = t₁ > 0. How are household savings impacted in this case? What about national savings?