sition
In this section we provide a description of the equilibrium for this economy, con- sidering also the time dimension of the transition. Since the transition is char- acterized by a sequence of aggregate prices and quantities, the definition of a
recursive competitive equilibrium as described in the previous chapters should be modified.
Given a sequence of interest rates and wages {rt, wt}∞t=0, a non-negative se-
quence of loan prices{qt}∞t=0and a non-negative sequence of defaults{pt}∞t=0, are-
cursive competitive equilibrium is a sequence of value functions{VtR, VtD, Vtcc, Vto}∞
t=0
and optimal policy functions {`t+1, dt, ct}∞t=0 for unconstrained households, a se-
quence of optimal policy functions {`t+1, ct}∞t=0 for credit constrained households,
optimal firm choises{Lt, Kt}∞t=0 and a set of distributions {µuct , µcct }
∞
t=0 such that
for all t:
(i) Given prices{rt, wt, qt}∞t=0, the policy functions{`t+1, dt, ct}∞t=0 solve theun-
constrained households problem and the policy functions {`t+1, ct}∞t=0 solve
the credit constrained households problem.
(ii) Given prices{rt, wt, qt}∞t=0, the firm chooses optimally its capitalKt and its labour Lt and the first order conditions imply:
wt = (1−α) K t Nt α rt=α Kt Nt α−1 −δ
(iii) The labour market clears and Nt is given by
Nt= (1−Π(u))
X
y∈Y yΠ(y)
(iv) The goods market clears
F(Kt, Lt) =Ct+Kt+1−(1−δ)Kt−γwt Z L+×Y×S ys dµcct where Ct is given by Ct= Z L×Y×S ct(`t, yt, st)dµuct + Z L+×Y×S ct(`t, yt, st)dµcct
(v) The asset markets clears and Kt+1 is given by Kt+1 = Z L×Y×S `t+1dµuct + Z L+×Y×S `t+1dµcct
(vi) For all L × Y × S the probability measureµuct+1 satisfies
µuct+1 =
Z
L×Y×S
Qt((`t, yt, st),L × Y × S)dµuct
where Qt is the transition function defined as
Qt((`t, yt, st),L×Y×S) =
X
yt+1∈YX
st+1∈S 1{`t+1(`t, yt, st)∈ L}π(st+1|st)π(yt+1|yt)(vii) For all L+× Y × S the probability measure µcc
t+1 satisfies
µcct+1 =
Z
L+×Y×S
Qt((`t, yt, st),L+× Y × S)dµcct
where Qt is the transition function defined as
Qt((`t, yt, st),L+×Y×S) =
X
yt+1∈YX
st+1∈S 1{`t+1(`t, yt, st)∈ L+}π(st+1|st)π(yt+1|yt)(viii) The first order conditions for thefinancial intermediary imply that:
qt =
(1−pt) 1 +rt−δ
3.8
Calibration
This model is characterized by inherent non-linearity which makes its analytical solution impossible. Therefore we construct a numerical algorithm to find the initial and final equilibrium of the economy and then solving backwards we can compute the transitional dynamics of our economy, when a shock to the borrowing limit is realized. We depart from other papers in the literature, for instance Guerrieri and Lorenzoni (2017), by examining two distinct cases. Firstly we examine the case of credit crunch , i.e., a tightening in the borrowing limit that households face, and secondly the case of credit easing, i.e, an increase in the borrowing limit that households face.
To analyse the model we are forced to migrate to numerical simulations. Therefore, we need to specify preferences and also choose the values for a set of parameters. Preferences are described in section 3.4 and the period utility is assumed to be a CRRA
u(c) = c
1−σ
1−σ, (3.16)
where σ is the parameter capturing the relative risk aversion.
The time period of the model is set to be one quarter. The discount factor
β is chosen to be 0.9428, targeting a yearly interest 4.15% in the initial steady state. The coefficient of relative risk aversion is set to be σ = 3. The coefficient of relative risk aversion plays a crucial role in driving the precautionary motives. However, different experiments that we have done with values between 2 and 4 do not alter qualitatively our results. Therefore, we have chosen the value that generates quantitatively better results. The earnings process is approximated, by the use of a 12-state Markov chain and similarly to the previous chapters, we have utilized the discretization method proposed byCivale et al.(2016), using an NMAR process, namely a first order autoregressive process with normal mixture innovations, in order to match the moments of Guvenen et al. (2015).
Transitions between employment and unemployment, similarly to our previous chapters were calculated inquarterly frequency following the approach of Shimer
Table 3.8.1: Calibrated Parameters
Description Parameter Value
Preferences
Households Discount Factor β 0.9428 Coefficient of Relative Risk Aversion σ 3 Employment
Transition to Unemployment πe,u 0.057 Transition to Employment πu,e 0.882
Unemployment Benefit ρ 0.4
Technology
Capital Share α 0.33
Depreciation Rate of Capital δ 0.025 Defaults
Probability of Remaining Constrained θ 0.95 Labour Earnings Garnishment γ 0.20
Notes: This table reports the calibrated parameters for our model economy.
(2005) and Krueger et al. (2016). We used CPS data for the measurement of job finding andseparations rates.6 By following this strategy we calculate the average probabilities from 2000Q1to 2014Q4,since we are interested in including the
period of the global financial crisis of 2008. The probability of a household remaining on the constrained state following default, has been chosen to match an average exclusion period of 5 years. Labour earnings garnishment is set to
γ = 0.20 so to guarantee that its value is below the upper permissible bound of 25%, as introduced by Federal Wage Garnishment Law and theConsumer Credit Protection Act III.
6We download the following series from CPS: The unemployment level (UNEMPLOY -
Thousands of Persons, Monthly, Seasonally Adjusted), theshort term unemployment level
(UEMPLT5 - Number of Civilians Unemployed for Less Than 5 Weeks, Thousands of Persons, Monthly, Seasonally Adjusted ) and theemployment level(CE16OV - Civilian Employment Level, Thousands of Persons, Monthly, Seasonally Adjusted).
Denoting by ut the unemployment rate and by ust the short term unemployment rate, we
define thejob finding rate 1−
u
t+1−ust+1
ut
and theseparation rate u
s t+1
1−ut