ECON 6002 Problem Set 2
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Problem Set 2 Answer Key
ECON 6002
Only final answers are given, not workings. To receive full points, you must show workings.
1. Abstracting from long-run growth by setting n = g = 0 and from persistent shocks by setting ρA = ρG = 0, with t = lnAt - ln and t = lnGt - ln , and normalizing the population to N = 1, the following nine equations describe the “baseline” RBC model in Chapter 5:
|
= = = = = = =
=
= |
Ct + It + Gt Kt(a)(AtLt)1 −a Kt + It - δKt ∈ A,t ∈ G,t α(AtLt/Kt)1 −a - δ (1 - α)(Kt/AtLt)a At e −p Et - (1 + rt+1)┐ wt b |
(1) (2) (3) (4) (5) (6) (7) (8)
(9) |
(a) Find the steady state for this economy under the following calibration: α = , δ = 0.05, = 0.03, = 1, and = 0.5 and choose such that / = 0.2. In particular, you should find the remaining parameters values b and ρ that are consistent with steady state and determine steady-state values for the endogenous variables, , , I¯, , , and . (Hint: first solve for ρ using (8), then solve for using (6), then I¯ using (3), then using (7), then using (2), then using (1), then b using (9).)
First, solve for ρ using (8):
ρ = -ln ╱ ← ≈ 0.03
Next, solve for using (6):
α(/)1 −a = + δ
÷
= ≈ 4.25
Next, get I¯ using (3):
I¯ = δ ≈ 0.21
Next, get using (7):
= (1 - α)(/)a
≈ 1.36
Next, get using
Next, get using using (1):
(2):
= a ()1 −a ≈ 1.02
the calibration / = 0.2, which implies = 0.2 ≈ 0.20 . Next, get
= - I¯ - ≈ 0.60
Last, get b using (9):
(1 - )
(b) Now consider the special case of the model where δ = 1 instead of δ = 0.05 and Gt = 0 for all t (note: ρ will remain the same and b will be different, but you do not need to solve for it). Solve for Yt , Ct , It , Kt+1 , rt, and wt as analytical expressions of exogenous and predetermined variables At and Kt and constants. (Hint: with 100% depreciation, there is a constant saving rate s = αe −p and constant labour supply Lt = . Given this solution to the household optimization problem, first solve for Yt , rt, and wt from equa- tions (2), (6), and (7) and then the solutions for Ct , It, and Kt+1 are straightforward.)
We can readily obtain the solutions for Yt , rt, and wt from equations (2), (6), and (7), respectively (given constant Lt = ):
Yt = Kt(a)(At ) 1 −a
rt = α(At /Kt) 1 −a - δ
wt = (1 - α)(Kt/At)a At
To find the solution for Ct, notice that Ct = (1 - s)Yt, and hence
Ct = (1 - s)Kt(a)(At)1 −a
Next, for It:
It = Yt - Ct
= Yt - (1 - s)Yt
= sYt
÷
It = sKt(a)(At ) 1 −a
Last, for Kt+1:
Kt+1 = It + (1 - δ)Kt = It
÷
Kt+1 = sKt(a)(At)1 −a
(c) Again, for the special case of the model, what is the percentage change in output and percentage point change in the interest rate if the economy is at steady state at time t - 1, but there is a shock ∈ A,t = 0.25 (i.e., 25%) at time t? Explain the economic intuition behind the responses of output and the interest rate in terms of the marginal products of labour and capital. (Hint: note that the ∈ A,t = 0.25 shock is to lnAt, but the model solution is for the level of At . First solve for the steady-state level of output and then solve for output and the real interest rate given the shock.)
The steady-state output is ≈ 0.28, while the steady-state real interest rate is = 0.03 . Given the shock, lnAt = 0.25 s At = e0.25 ≈ 1.28 . Plugging this into the solution from part (b) for Yt we get Yt ≈ 0.34, which is approximately 18. 14% higher than steady-state . We also get rt = 0.22, which is approximately 18. 68ppt above the steady-state level. These responses are intuitive given that a positive shock to technology makes labour more productive . It also increases the marginal product of capital, thus increasing the real interest rate .
2. Consider Calvo price-setting firms with partial indexation. That is, if a firm is not visited by the Calvo tooth fairy in period t, its price in t is the previous period’s price plus γπt − 1 , 0 < γ < 1. The average price in period t is pt = αxt + (1 - α)(pt − 1 + γπt − 1 ), where α is the fraction of firms visited by the Calvo tooth fairy in any given period and xt is the price they set. The resulting Phillips curve is a hybrid one: πt = 8yπt − 1 + Etπt+1 + κy˜t , where β > 0 is the discount factor, κ = a[1 - β(1 - α)]φ > 0 determines the slope of the Phillips curve, y˜t is the output gap, and Etπt+1 is the expectation (taken at time t) of inflation at t + 1.
(a) Show that xt - pt = (πt - γπt − 1 ).
First, subtract pt − 1 from both sides . Then add and subtract αpt from the right hand side and solve for πt . Then it is easy to get the expression.
(b) Use the result in (a) and the representative firm’s optimal price under Calvo pricing with partial indexation being xt = pt + (1 - β(1 - α))φy˜t + β(1 - α)(Et(xt+1 - pt+1) + Etπt+1 - γπt) to derive the hybrid Phillips curve.
Substitute in xt - pt = (πt - γπt − 1 ) and Et(xt+1 - pt+1) = (Etπt+1 - γπt) into xt -pt = (1 - β(1 - α))φy˜t+β(1 - α)(Et(xt+1 -pt+1)+Etπt+1 - γπt) and then solve for πt .
(c) What value of γ would lead to the highest degree of inflation persistence? Why?
When γ = 1, i. e ., the highest possible degree of indexation, we obtain the highest possible degree of inflation persistence (given a fixed degree of persistence of y˜t) . This is because γ = 1 would imply the largest coefficient on the lagged inflation πt − 1 , which means that inflation has the highest possible degree of intrinsic persistence . Another way to think about this is to notice that when γ = 1, we get the lowest possible coefficient on the expected future inflation, Etπt+1 (for a given β), which means that inflation is the least forward looking, but more backward looking.
(d) Now assume that β = 0.9, γ = 0.5, and κ = 0.1. Assume that the central bank has some control of the evolution of y˜t . Suppose that the central bank announces a permanent and fully credible reduction in its target or steady-state inflation rate from 7% to 2% at t = 1 (prior to this, the economy was at the steady state with 7% inflation and zero output gap). Determine the cost of this disinflation episode. How much is the output gap reduced?
Since the disinflation is immediate and credible, agents will adjust their expectations of inflation and so π 1 = E1π2 = 2% . Also, since the economy is at steady-state at t = 0 then π0 = 7% . Hence, from the hybrid Phillips curve we have y˜1 = -24%
2023-05-18