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A Cash-in-advance Model
A Cash-in-advance Model
In this problem set, we will try to answer some questional issues on money and business cycle fluctuations.
More precisely, we will follow Cooley and Hansen [T.F. Cooley and G.D. Hansen (1989) “The Inflation Tax in
a Real Business Cycle Model”, The American Economic Review, 79(4), pp.733–48] and answer the following
questions:
• Does money and the form of money supply rules affect the nature and amplitude of the business cycle?
• How does anticipated inflation affect the nature and amplitude of the business cycle?
The model
Let’s consider an economy populated by a representative household seeking to maximize an intertemporal
utility function
∞
X
E0
β t (ln ct − Bht ), 0 < β < 1
t=0
depending on consumption ct and labour effort ht , and where E0 is the expectation operator with respect
to inforation available at date 0.
The model - cont’d
This household faces a budget constraint:
ct + xt + mt /pt ≤ wt ht + rt kt + (mt−1 + (gt − 1)Mt−1 )/pt
where xt is the investment, defined by kt+1 = (1 − δ)kt + xt , 0 ≤ δ ≤ 1, mt the nominal money balances,
pt a price index, wt the real wage, rt the real interest rate and kt the capital stock (k0 given.) Per capita
money supply, Mt , evolves according to Mt = gt Mt−1 , with gt+1 = gtα ḡ 1−α exp(ξt+1 ), ξt iid with variance
σξ2 . The household faces another constraint on cash good
pt ct ≤ mt−1 + (gt − 1)Mt−1
Question 1
Give a rationale for a cash-in-advance constraint. What is the basic difference with the money-in-utilityfunction paradigm?
The cash-in-advance paradigm
There are two main timing conventions
• Lucas: Households visit the financial market after seeing current period shocks but before purchasing
cash. This means households can adjust their holdings of cash in the financial market to exactly meet
their consumption needs.
• Svensson: Goods market opens before the asset market. Households have to purchase goods using cash
they carried over from the previous period. Holdings of money are chosen before households see the
current period shock. There is a precautionary motive for holding cash.
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CIA vs MIU models
The main difference
• MIU: households gain utility flows from their money holdings, no matter how they use cash.
• CIA: give a description of what people do with their money holdings.
Question 2
Write the lagrangian for the household problem and derive the first order conditions for an optimal path.
The household wants to maximize its intertemporal utility using ct , ht , mt and xt as controls. The
Lagrangian of that problem is thus:
L = E0
∞
X
β
t
ln ct − Bht
t=0
mt−1 + (gt − 1)Mt−1
+ λt wt ht + rt kt +
pt
mt
−ct − kt+1 + (1 − δ)kt −
pt
+µt
mt−1 − (gt − 1)Mt−1
− ct
pt
The first-order conditions with respect to ct , ht , mt and kt are then respectively (with the expectation
operator implicitly appended):
(ct )
(ht )
(mt )
(kt )
1
= λt + µt
ct
wt
= λt
B
pt+1
Et
β(λt+1 + µt+1 ) = λt
pt
λt+1
Et β
(rt+1 + 1 − δ) = 1
λt
The first relation equals the cost of holding one more unit of money (as a loss in consumption) to its
future benefits (larger future consumption and loosened cash constraint). The second holds the trade-off
between labour and leisure. The third is a asset pricing equation, expression the value of the liquidity services
of money. The last relation is the Euler relation.
Question 3
In which variables one can found nominal trend? Propose a method to make the problem stationary.
There is a nominal trend in the optimal level of cash holdings. The problem can be made stationary by
defining:
mt
pt
and p̂t =
m̂t =
Mt
Mt
Question 4
The firm in the economy produces output according to a Cobb-Douglas production function yt = exp(zt )ktθ h1−θ
, 0≤
t
θ ≤ 1, with zt+1 = γzt + εt+1 , 0 ≤ γ ≤ 1 and εt iid with mean 0 and variance σε2 . Give the first order
conditions for the firm’s problem.
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Optimal level of production input are given by
wt = (1 − θ)
and
rt = θ
yt
ht
yt
kt
Question 5
Compute the steady state for this economy.
Steady state
At the steady-state, we have
g = ḡ
z̄ = 0
m̄ = 1
r̄ = 1/bet − 1 + delt
Steady state - cont’d
Then, according to first order conditions, constraints, and functional forms
w̄ = (1 − θ)(θ/r̄)(θ/(1−θ))
λ̄ = B/w̄
c̄ = β/(λ̄ḡ)
p̂¯ = 1/c̄
h̄ = c̄/(w̄ + (θ/r̄)1/(1−θ) (r̄ − δ))
ȳ = w̄h̄/(1 − θ)
k̄ = θȳ/r̄
x̄ = δ k̄
µ̄ = 1/c̄ − λ̄
Simulations
We use the following calibration
β
0.99
θ
0.36
δ
0.025
B
2.86
γ
0.95
σε
0.00721
ḡ
1.015
α
0.48
σξ
0.009
Question 6
Simulate the model with 50 draws of 115 periods long. After logging and detrending, compute the standard
deviation and correlation with output for the following variables: output, consumption, investment, vapital
stock, hours, productivity, and price level.
The results of the simulation are:
Question 7
Redo the simulations with σξ = 0. Compare these results with the Hansen(1985)’s indivisible labor model
(see below) and conclude on the initial questions.
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Series
Output
Consumption
Investment
Capital Stock
Hours
Productivity
Price level
Std Dev.
1.79(0.19)
0.66(0.07)
5.88(0.58)
0.50(0.09)
1.36(0.14)
0.53(0.07)
1.94(0.27)
Corr w. Y
1.00(0.00)
0.70(0.05)
0.97(0.01)
0.07(0.05)
0.98(0.00)
0.87(0.03)
-0.26(0.17)
Table 1: Simulation 1, 50 draws, 115 periods
Standard deviations in percent (a) and
correlations with output (b) for an
economy with indivisble labor
Series
Output
Consumption
Investment
Capital stock
Hours
Productivity
(a)
1.76(0.21)
0.51(0.08)
5.71(0.70)
0.47(0.10)
1.35(0.16)
0.50(0.07)
(b)
1.00(0.00)
0.87(0.04)
0.99(0.00)
0.05(0.07)
0.98(0.01)
0.87(0.03)
With σξ = 0, the results are: According to these simulation, money does affect the business cycle through
Series
Output
Consumption
Investment
Capital Stock
Hours
Productivity
Price level
σξ 6= 0
Std Dev.
Corr w. Y
1.79(0.19)
1.00(0.00)
0.66(0.07)
0.70(0.05)
5.88(0.58)
0.97(0.01)
0.50(0.09)
0.07(0.05)
1.36(0.14)
0.98(0.00)
0.53(0.07)
0.87(0.03)
1.94(0.27)
-0.26(0.17)
σξ = 0
Std Dev.
Corr w. Y
1.76(0.24)
1.00(0.00)
0.51(0.09)
0.88(0.02)
5.72(0.81)
0.99(0.00)
0.47(0.12)
0.05(0.07)
1.34(0.17)
0.98(0.01)
0.51(0.10)
0.88(0.03)
0.50(0.08)
-0.88(0.02)
Hansen(1985)
Std Dev.
Corr w. Y
1.76(0.21)
1.00(0.00)
0.51(0.08)
0.87(0.04)
5.71(0.70)
0.99(0.00)
0.47(0.10)
0.05(0.07)
1.35(0.16)
0.98(0.01)
0.50(0.07)
0.87(0.03)
Table 2: Simulation 2, 50 draws, 115 periods, σξ = 0
the cash-in-advance constraint. In particular, consumption and price level are more volatile when money
supply is stochastic, but consumption is less correlated with output.
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