Exercises 1. Suppose that we have an economy with many identical households. There is a government that exogenously consumes some output and pays for it with lump sum taxes. Lifetime utility for a household is: U = lnCt +ln0t+1 The household faces two within period budget constraints given by: Ct + St = Y: T: Ct+1 = Yt+1 _ n+1 + (1+ Vast (a) Combine the two budget constraints into one intertemporal budget constraint. (b) Use this to find the Euler equation. Is the Euler equation at all affected by the presence of taxes, Ti and Tt+1? 308 (c) Use the Euler equation and intertemporal budget constraint to derive an expression for the consumption function. The government faces two within period budget constraints: Gt + SC = Tt Gt+1 = Tt+1 + (1 + re)SCThe government faces two within period budget constraints: Gt + SG = 1} Gt+1 : Tt+1 + (1+ \")8? In equilibrium, what must be true about St and 8?? Combine the two period budget constraints for the government into one intertemporal budget constraint. Suppose that the representative household knows that the government's intertemporal budget constraint must hold. Combine this information with the household's consumption function you derived above. What happens to Tt and Tull? What is your intuition for this? Equilibrium requires that Y; = Ct + Gt. Plug in your expression for the consumption function (assuming that the household knows the government's intertemporal budget constraint must hold) to derive an expression for Y}. Derive an expression for the \"xed interest rate multiplier,\" i.e. & th ldrt :0- Assuming that Y} is exogenous, what must happen to rt after an increase in G3? 0) Now, assume the same setup but suppose that the household does not anticipate that the government's intertemporal budget constraint will hold 7 in other words, do not combine the government's intertemporal budget constraint With the household's consumption function as you did on part (f). Repeat part (h), deriving an expression for the \"xed interest rate multiplier\" While not assuming that the household anticipates the government's budget constraint holding. Is it bigger or smaller than you found in (h)? Since Y, is exogenous, What must happen to 7", after an increase in G, in this setup? Will the change in rt be bigger or smaller here than what you found in part (i)? For the setup in which the household does not anticipate that the government's intertemporal budget constraint must hold, What will be the \"xed interest rate tax multiplier\than What the tax multiplier would be if the household were to anticipate that the government's intertemporal budget constraint must bind? Is it smaller or larger than the xed interest rate multiplier for government spending (assuming that the household does not anticipate that the government's intertemporal budget constraint will hold)