EC 502: Problem Set 5
1 Real Business Cycle
Consider a two-period real business cycle model that is similar to the one covered in the class.
Households have the following utility function:
Households face the following budget constraints:
where L1 is exogenously given. All the notations are the same as in the lecture notes. The firms maximize the profit:
where
and the capital accumulation equation is
Here, we assumed there is no adjustment costs.
The market clearing conditions are
1. Derive the optimality conditions of the household’s problem. Interpret the optimality condi-tions in words.
2. Considering the optimality conditions of the firm’s problem, show that the wages and interest rates satisfy must satisfy
3. Using the goods market clearing condition at t = 1, your answers to 1.2 and housheold’s
Using your answers to 1.1 and 1.2, show that the relationship between C0 and L0 can be written as
4. Using (14), (15) and the goods market clearing condition at t = 0, show that the L0 is given by the solution to the following equation:
Show that the solution is unique.
5. Using (16), explain how the labor supply L0 responds to an increase in following parameters: A0 , A1 , , and β .
6. Explain how consumption C0 and Investment I0 respond to an increase in following param- eters: A0 , A1 , , and β . Explain which of the shock produces responses that resemble business cycles.
2 Monetary Policy
Consider a New Keynseian economy similar to the one described in the lecture note with two periods.
Households have the following preferences:
Households face the following budget constraints:
where L1 is exogenously given, and Dt denote the sum of profits of wholesale firms and retailers. All the notations are the same as in the lecture notes.
Wholesale firms solve the following problem in each period:
where pt is the wholesale price at time t.
Retailers purchase goods from the wholesale firms and sell them to the households. The retailers maximize the following profitat time 0:
The labor market clearing conditions are
The goods market clearing conditions are
Throughout, we fix the price level at t = 1 to be
1. Derive the optimality conditions of the household’s problem.
2. Explain why it must be the case that Wt = Atp t inequilirbrium.
3. Derive the optimality conditions of the retailer’s problem.
4. Using the household Euler equation and market clearing conditinos, show that the labor demand can be expressed as
5. Show that the labor supply at t = 0 can be expressed as
6. Suppose the central bank increases i. What happens to L0? What happens to P0?
7. Now suppose prices are sticky. Only a fraction 1 - λ of the firms can adjust their prices in response to a shock. The remaining firms keep the prices that they would set before the shock, which we denote as Note that the aggregate price level is the average price in the economy: Show that the aggregate price level at t = 0, P0 and labor at t = 0, L0, satisfy the following equation
8. Equation (24) is what we call the aggregate demand curve. Equation (26) is what we call the Phillips curve (aggregate supply curve). Explain how the aggregate demand curve and the Phillips curve can be used to analyze the impact of monetary policy. What happens if the central bank increases the interest rate i? You can assume the denominator of (26) is positive.
3 Fiscal Policy
We use the modle in the previous question to analyze the impact of fiscal policy. Households’ budget constraints are modified as
where Tt represents lump-sum taxes on households.
The government sets spending Gt, taxes, Tt, and defits B t that satisfy the following government budget constraint:
where we assumed G1 = 0. The goods market clearing conditions are modified as
The rest of the model remains unchanged.
1. Re-do the analysis in the previous question to derive the aggregate demand curve.
2. Re-do the analysis in the previous question to derive the Phillips curve.
3. Using the aggregate demand curve and the Phillips curve, explain how an increase in G0 impacts employment L0 .
4. Prove that an increase in G0 always lower C0 .
5. Using the aggregate demand curve and the Phillips curve, explain how a decrease in T0 impacts P0 and L0 .
6. Now we introduce hand-to-mouth households who consume all their income. A fraction θ of hand-to-mouth households have the following budget constraints:
where the last equalities use the national income accounting. We assume L0 is pinned down by the Phillips curve that you derived in 3.2, rather than each household choosing l0 . The remaining households solve the same problem as before, which we call permanent income households. Their budget constraints are, after imposing national income accounting,
The goods market clearing conditions are
Show that the consumption of permanent income households at t = 0 is given by
7. Using (32), (37), and the goods market clearing condition at t = 0, show that the aggregate demand curve can be expressed as
8. Note that the Phillips curve does not change even after introducing the hand-to-mouth households and is still given by the same equation that you derived in question 2. As- sume the balanced budget, T0 = P0G0, and analyze the impact of an increase in G0 on L0 and C0 using the aggregate demand and the Phillips curve.
9. Now suppose the government runs defits by setting T0 = 0. Show that the fiscal multiplier can be above one.
10. Now suppose G0 = 0. Analyze the impact of reducing T0 .