Final Exam
Question 1 (30 points)
Consider the following neoclassical growth model. The equilibrium conditions of the model are given by
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· Log-linearize the second, third, and fourth equations around the steady state. Note that “L” is a constant, not a variable.
Answer to Question 1
Here, we use the formulas of log-linearization introduced in the practice session because this is an easier way. You can also derive the same results by following procedures in “Slides_NK_LL.”
Recall the following formulas:
For the production function, it follows from (1) and (2) that
For the LHS and RHS of the Euler equation,
we first use (1) and (2):
where . Then, by using (3), we obtain
Substituting these results into (4) yields
Here, we use
For the market clearing condition, we use (3) as follows:
Question 2 (40 points)
Consider a version of the NK model with the following preference. The model is identical to the baseline model discussed in the lecture, except that the household’s per period utility is given by
· Derive the private-sector equilibrium conditions of the model.
· Assuming that (i) the policy rate is determined by the standard Taylor rule, (ii) the inflation target is zero (that is, ), and (iii) , analytically compute the standard steady state of the model.
Answer to Question 2
The household’s maximization problem can be written as
subject to the budget constraint
Its Lagrange function is given as
Take a derivative of the Lagrange function for each variable:
Multiply the second equation with :
Combine the first equation and the third equation:
Therefore, the new private-sector equilibrium conditions of the model are
Now, we want to find the steady state of the model. We are assuming that (i) the policy rate is determined by the standard Taylor rule, (ii) the inflation target is zero (that is, ), and (iii) .
Then, the standard steady state of the model satisfies:
Manipulating the second equation, we get
Because , comparing this equation with the Taylor rule, we get
From the fourth equation,
Using and ,
Because , the fourth equation becomes
Because and , the third equation becomes
To sum up, the solution is
Question 3 (60 points)
Consider the following two-period model with a static PC with the ELB. There is a labor income tax that is distributed to households as a lump-sum transfer at time one. Key equilibrium conditions of this model are given by the following equations.
The utility function for the government at each time is given by the standard quadratic objective function. That is,
for each t=1,2.
Now, assume that the government is optimizing under discretion.
Assume also that the value of is such that but .
Assume that the government can optimally choose labor income taxes at time one and time two.
· Formulate the optimization problem(s) of the central bank.
· Define the Markov-Perfect equilibrium.
· Solve the model analytically.
Assume that the government is optimizing under commitment.
Assume also that the value of is such that but .
Assume that the government can optimally choose labor income taxes at time one and time two.
· Formulate the optimization problem of the central bank.
· Define the Ramsey equilibrium.
· Solve the model analytically.
Answer to Question 3
The optimization problem of the central bank under discretion at t = 2,
subject to
The optimization problem of the central bank under discretion at t = 1,
subject to
taking and as given.
A Markov-Perfect equilibrium is defined as a vector that solves the two optimization problems at t=1 and t=2.
First, we set up a Lagrange function to solve the optimization problem at t=2:
Take a derivative of the Lagrange function for each variable:
Since we assume , should hold. By (10) and (12), holds. Moreover, substituting and into (9) yields Therefore, and are implied by (7) and (8), respectively.
Next, we set up a Lagrange function to solve the optimization problem at t=1:
Take a derivative of the Lagrange function for each variable:
By assumption, we have . By (14) and (16), holds. Moreover, it follows from (5) that holds. Therefore, implies .
In summary, a Markov-Perfect equilibrium is characterized by
The optimization problem of the central bank with commitment is
subject to
A Ramsey equilibrium is defined as a vector that solves the optimization problem above.
We set up a Lagrange function to solve the optimization problem:
Take a derivative of the Lagrange function for each variable:
By assumption, we have . Moreover, since we assume , should hold. By (20), (23), and (24), Thus, (18) implies . Now, (17), (21), and (22) imply , , and , respectively. Substituting them into (5) yields
which implies
Then, by (6), (7), and (8), we have
In summary, a Ramsey equilibrium is characterized by
Question 4 (20 points)
Consider the following two-period loglinearized model with a static Phillips curve and with a time-one demand shock ( and ). The policy rule is given by a nominal-income targeting rule.
· Assuming that (i) and and (ii) the shock size is such that the policy rate is zero and positive at time one and two, respectively, solve the model analytically.
Answer to Question 4
Since and , at t=2,
Since , at t=1,
Then, at t=2,
To sum up, the solution is
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