Final Exam
Question 1 (40 points)
Consider the neoclassical growth model. The equilibrium conditions of the model are given by
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· Log-linearize these four equations around the steady state.
Answer to Question 1
For the left hand side (LHS) and right hand side (RHS) of the capital accumulation equation:
Take a derivative of them for each variable:
Loglinearize the LHS and RHS around the steady state:
Combine the LHS and RHS:
For the LHS and RHS of the production function:
Take a derivative of them for each variable:
Loglinearize the LHS and RHS:
Combine the LHS and RHS:
For the LHS and RHS of the Euler equation
Take a derivative of them for each variable:
Loglinearize the LHS and RHS:
Combine the LHS and RHS:
For the LHS and RHS of the production function:
Take a derivative of them for each variable:
Loglinearize the LHS and RHS:
Combine the LHS and RHS:
To sum up, we can write the equilibrium conditions of the neoclassical growth model in a loglinearized form as:
(Appendix)
For those who solved the steady state variables, we get
Then, the loglinearized neoclassical growth model becomes
Therefore, we can write the four equations as:
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
A two-period model with a static PC with the ZLB. There is a government spending shock financed by a consumption tax at time one:
with and . The payoff function for the central bank is given by the standard quadratic objective function. That is,
for each t=1,2.
Assume that the policy rate is determined by the truncated Taylor rule:
Assume also that is sufficiently small so that . Solve the model analytically.
Now, assume that the government is optimizing under discretion.
· Formulate the optimization problem(s) of the central bank.
· Define the Markov-Perfect equilibrium.
Assume that the government is optimizing under commitment.
· Formulate the optimization problem of the central bank.
· Define the Ramsey equilibrium.
Answer to Question 3
If , since ,
With , we get
Using , at t=1,
To sum up, when , the solution is
If , since ,
With , we get
Using , at t=1,
To sum up, when , the solution is
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 slves the two optimizations problem at t=1 and t=2.
The optimization problem of the central bank with commitment is
subject to
A Ramsey equilibrium is defined as a vector that slves the optimizations problem above.
Question 4
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 price-level 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
Question 5
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 Reifschneider-Williams 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 5
Using and , we get
Since , at t=1,
Using this,
Plug this into other equations:
To sum up, the solution is
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