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An Uncertainty-Shock-Induced Liquidity Trap and An Economy Driven by Natural-Rate Shocks
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Household Budget Constraints and Equilibrium Consumption

1. An Uncertainty-Shock-Induced Liquidity Trap

Consider a two-period sticky-price economy populated by identical households with preferences defined over consumption in period 1, C1 and consumption in period 2, C2, and described by the utility function ln C1 + βE1 ln C2,

where C1 denotes consumption in period 1, C2 denotes consumption in period 2, β = 1/1.1 is a preference parameter, and E1 denotes the expectations operator conditional on information

in period 1. In period 1, potential output, Y 1, is 10 kilos of apples, and in period 2, potential output, Y 2, is also 10 kilos of apples. Assume that nominal prices are fixed, in particular P1 = P2 = 1, and that the central bank sets the nominal interest rate, i, so as to bring about full employment at all times, that is, so that in each period all apples are sold and consumed, subject to the constraint i ≥ 0.

(a) State the household’s budget constraint in period 1 and period 2.

(b) Find the equilibrium level of consumption in period 1, C1, the nominal interest rate, i, and savings, S1. (Your answer should be 3 numbers.)

Assume now that potential output in period 1 continues to be 10 apples, but that due climate change the apple harvest in period 2 might be either very plentiful or really poor. Specifically, assume that with probability 1/3 climate-change induced increases in apple pests will lead to a 40 percent reduction of the harvest in period 2. At the same time, with probability 2/3, rainfall and sunshine conditions will be so favorable that in period 2 the apple harvest increases by 20 percent.

(c) Find the expected value of potential output in period 2, E1Y 2.

(d) Suppose the monetary authority fails to respond to this increase in uncertainty and keeps the nominal interest rate unchanged. Find consumption in period 1.

(e) Suppose now that the monetary authority does respond to the increase in uncertainty and adjusts the nominal interest rate to stimulate demand in period 1. Find the optimal monetary policy response, that is, find i, and the associated value of consumption in period 1.

2. An Economy Driven by Natural-Rate Shocks. Consider a two-period sticky-price economy populated by identical households with preferences defined over consumption in period 1, C1 and consumption in period 2, C2, and described by the utility function ln C1 + β ln C2,

where C1 denotes consumption in period 1, C2 denotes consumption in period 2, and β = 1/1.1 is the subjective discount factor. In both periods, potential output (Y ) is equal to 10 kilos 1 of apples. Let P1 and P2 denote the price levels in periods 1 and 2, respectively. Assume prices are fixed at P1 = P2 = 1 and that the economy is always in full employment in period 2 (the long run). The central bank uses the nominal interest rate, denoted i, as its monetary instrument. The nominal interest rate is subject to the zero lower bound (ZLB) constraint.

(a) Assume further that the central bank sets the nominal interest rate so as to maximize employment and minimize excess aggregate demand for goods. Denote this interest rate by i∗Find i∗.

(b) Now suppose that a financial panic causes households to become more patient. Specifically, suppose that the subjective discount factor increases to 1/0.9. Suppose that the central bank is slow to react and keeps the interest rate at i ∗ . Find the output gap, defined as (Y /Y ¯ 1 − 1)100, where Y1 denotes output in period 1.

(c) Now suppose that contrary to the assumption of the previous item, the central bank acts quickly and changes the interest rate to minimize unemployment. Denote this interest rate by i ∗∗. Find i ∗∗ and the output gap.

(d) Consider the scenario of the previous item, that is, i = i ∗∗. Suppose that the fiscal authority decides to also intervene. Let G∗ denote the lowest level of government spending that eliminates involuntary unemployment. Find G∗ . Calculate private consumption in period 1.

(e) Suppose, that the government miscalculates G∗ and instead sets government spending equal to G˜, where G˜ is 10 percent higher than G∗ . Assume further that realizing this situation, the central bank changes the interest rate to avoid excess aggregate demand, while still maintaining full employment. Find the new interest rate and private consumption. Comment.