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TA Session 1 Pietro Galeone, Eugenia Menaguale A. A. 2018-2019

Plan of the day: Our goal is having a look at how to solve and learn from three types of exercises. We will first cover Ex. 1.2 and 1.3 on the Lucas model from your Workbook. Then we’ll see two problems on the intertemporal consumption/saving model (mind: one of these appeared in last year’s midterm exam!). Feel free to interrupt with questions or comments (if any).

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Exercises 1-2 (Lucas islands model)

[See Workbook, ex. 1.2 and ex. 1.3]

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Exercise 3 (C/S model)

Consider an economy with two periods and a representative consumer. The budget constraint in the first period is given by: C +S =Y −T where C is consumption, S > 0 is private savings (or borrowings if S < 0), Y is the endowment of goods and T is the lump sum tax (or transfer) imposed by the government. In period 2, the budget constraint is: C ′ = Y ′ + (1 + r)S − T ′ where (1 + r) is the real interest rate. The representative agent’s utility function is: U (C, C ′ ) = log C + β log C ′ with β ∈ (0, 1). The budget constraints of the government are given in the two periods by (respectively): G + Sg = T G′ = (1 + r)S g + T ′ where G is public expenditure and S g is public savings (or public debt ig S g < 0).

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1. Derive the intertemporal budget constraint for the consumer and for the government; also derive the intertemporal consumption condition (i.e. the Euler equation). Do taxes influence the Euler condition? Explain. 2. Assume that the consumer internalizes (i.e. includes in her optimization) the government’s intertemporal budget constraint. Derive a single intertemporal budget constraint, and combine it with the Euler condition to derive the expression for the optimal period-one consumption C ∗ for a given real interest rate r. What is the effect of a tax increase on current consumption (holding G and G′ constant)? Explain. 3. Using the optimal consumption function you just obtained, and assuming that the interest rate r is fixed (i.e. partial equilibrium), derive the value of the government expenditure multiplier on consumption. In other words, what is the value of the effect ∂C ∗ /∂G? 4. Now let’s move from the partial to the general equilibrium. In other words, let’s consider r to be endogenous (consider credit market equilibrium). First, equilibrium in the goods market requires: Y =C +G Show how you can obtain this condition starting from the credit (= savings) market equilibrium. 5. Second, we can derive the general-equilibrium multiplier by combining the Euler condition with the market equilibrum condition just derived, thus endogenizing r. With the resulting expression for consumption (call it C ∗∗ ), calculate the multiplier ∂C ∗∗ /∂G. 6. Compare this value for the multiplier just obtained with the one calculated in point 3. What are the differences? How can you make sense of them?

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Exercise 4 (C/S model)

Consider a two-period model, with real interest rate r > 0. The representative consumer has the following utility function: U = log C + β log C ′ . Write and solve the problem of the agent. Then, analyze the following scenarios: 1. a fall in r (i.e. expansionary monetary policy), in absence of a government; 2. a fall in T (i.e. expansioanry fiscal policy), keeping public expenditure G and G′ constant.

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