The Agent Based Model of Macroeconomics from the Bottom Up

Financial accelerator
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The Agent Based Model of
Macroeconomics from the Bottom Up (MBU)
Part 1: Theory
Lecture #2
Domenico Delli Gatti
Domenico Delli Gatti
ABM_Lecture #2
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Financial accelerator
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Outline
The …nancial accelerator
MBU: Agents and markets
Workers/consumers
Firms
Banks
Pro…ts and losses
The sequence of events
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Financial accelerator
In the 90s several models have focused on …nancial factors as
drivers of business ‡uctuations.
These models can be lumped together under the heading of
Financial Accelerator (FA).
This literature is re-discovering Minsky’s Financial Instability
Hypothesis, i.e. the minority view in the profession in the 70s
and 80s.
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Financial accelerator (cont’d)
Three strands in this literature:
Bernanke-Gertler (1989, 1990), Bernanke-Gertler-Gilchrist
(1999)
Greenwald-Stiglitz (1993)
Kiyotaki-Moore (1997)
The frameworks di¤er in many ways but there are common
factors.
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Financial accelerator (cont’d)
Firms face a …nancing gap, which must be …lled with internal
…nance (net worth, retained pro…ts) and/or external …nance
(credit, new equities).
Due to informational frictions (asymmetric information),
external sources of …nance are not perfect substitutes.
Therefore the interest rates on di¤erent sources of funds are
di¤erent (Financing Hierarchy, FH).
Credit plays a special role in the FH.
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Financial accelerator (cont’d)
The …nancial accelerator literature is part of the Credit view:
it is labelled broad credit view. In a nutshell:
there is a wedge (the external …nance premium) between the
cost of internal and external funds,
this premium is increasing with the borrowers’…nancial
fragility, proxied by a …nancial ratio such as leverage,
the transmission mechanism of monetary policy is centered
upon the e¤ects of changes in the policy rate on agents’
…nancial conditions (balance-sheet channel or net worth
channel).
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The …nancial accelerator in a MABM: MBU
The main weakness of the FA approach is the neglect of the
dynamics of individual …nancial positions and the interaction
among them.
The FA literature should meet the AB approach.
Macroeconomics from the Bottom Up (MBU) is an attempt
in this direction.
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A look at the literature
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Agents
Households: supply labour, buy consumption goods, hold
deposits.
Firms: demand labour, produce and sell consumption goods,
demand bank loans.
Banks: receive deposits and extend loans to …rms.
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Markets
Labor
Consumption goods
Bank loans
Deposits
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Financial accelerator
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The architecture of the Macroeconomic ABM in MBU
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Individual behaviour and market processes
Behaviour (level of the agent’s control variables) is not
(necessarily) the outcome of an optimization process.
Agents adopt rules of thumb.
Generally behaviour changes adaptively.
Markets are fully decentralized and characterized by a
continuous search and matching process:
Re 1: tatonnement without auctioneer
Re 2: no need of a matching function as in Mortensen
Pissarides.
Hence transactions can occur at "false prices" i.e.
disequilibrium prices.
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Supply of labour, search
The h-th worker (h = 1, 2..., H ) supplies inelastically one unit
of labour per period (the time unit is the quarter).
If unemployed, the worker looks for a job by visiting …rms that
post vacancies.
We assume that there are transaction costs that make the
search costly.
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Supply of labour, search (cont’d)
An unemployed worker, therefore, contacts only a …nite
number Ze (in our simulations only 2) of randomly chosen
…rms to get a job, starting from the one which o¤ers the
highest wage.
If the most preferred …rm has already …lled all the vacancies,
the worker can resort to the remaining Ze 1 …rms.
If h does not succeed in …nding a job, she remains
unemployed.
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Match, separation
If h succeeds in …nding a job, she signs a labour contract
which assigns a wage for a certain time span (the duration of
the labour contract consisting of a …nite number of periods
(in our simulations: 8 quarters).
A worker whose contract has expired is unemployed. She
applies …rst to the …rm which has been her previous employer.
If the latter does not need her labour services, she starts
searching.
A worker with an active labour contract can be …red
if the …rm must reduce her desired scale of activity and
therefore her desired workforce or
if the …rm does not receive loans in the amount necessary to
implement the desired workforce (credit rationing).
Re 3: There are no …ring (and hiring) costs.
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Consumption
The worker, if employed, receives a wage.
The wage is deposited at the bank (and therefore it increases
the worker’s …nancial wealth=deposits).
A fraction (propensity to consume) of wealth is spent on
consumption goods.
The marginal propensity to consume out of wealth is a
decreasing with wealth.
If unemployed, the worker dissaves, i.e. she consumes out of
deposits.
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Capitalists
Each …rm is owned by a capitalist/rentier.
The i-th capitalist (i = 1, 2..., Fc ) receives dividends if the
…rm she owns (the i-th …rm) makes pro…ts after the payment
of interest on loans.
Dividends are deposited at the bank (and therefore they
increase the capitalist’s …nancial wealth=deposits).
A fraction (propensity to consume) of wealth is spent on
consumption goods.
The marginal propensity to consume out of wealth is
decreasing with wealth.
If the …rm does not pay out dividends, the capitalist dissaves,
i.e. she consumes out of deposits.
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Search and matching on the goods market
The consumer (worker, capitalist) looks for consumption
goods by visiting …rms that sell goods.
There are transaction costs that make the search costly
A consumer, therefore, contacts only a …nite number Zc of
randomly chosen …rms to ascertain the selling prices. Then
she ranks …rms in ascending order of price and buys …rst from
the …rm charging te lowest price.
If the most preferred …rm is in short supply, the worker can
resort to the remaining Zc 1 …rms
If the consumer does not succeed in satisfying her
consumption plan, she saves involuntarily (and deposits
involuntary savings at the bank).
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Firms
The i-th …rm (i = 1, 2..., Fc ) produces output Yit using only
labour Nit .
Technology is linear Yit = αNit .
The …rm knows the status quo in t, i.e. the point
Q (Pit , Yit )
The …rm knows the average price Pt
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Expected demand, desired supply, actual production
Assuming that the individual …rm considers herself negligible,
Pt is approximately equal to the price "of competitors".
Due to uncertainty, the …rm does not know (exactly) the
demand for her products Yitd .
Hence she does not know (exactly) the "equilibrium position":
A = Pt , Yitd .
Point A would be the Symmetric Nash Equilibrium in a
monopolistic competition setting in which …rms use the price
as strategic variable (Bertrand competition).
She forms expectations on demand and aims at ful…lling this
demand, i.e. she sets desired production at the level of
expected demand: Yit = Yite .
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Expected demand, desired supply, actual production
(cont’d)
For the …rm it is impossible to reach desired production if she
does not succeed in …lling all the vacancies and/or if she does
not get enough credit to …nance the wage bill.
In these cases the …rm will be unable to satisfy expected
demand.
Hence in these cases actual production may be smaller than
desired production (Yit < Yit ) .
In the following, however, we will assume Yit = Yit .
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Inventories
At the price Pit , given the average price Pt , demand Yitd can
be di¤erent from production Yit .
The di¤erence between production and demand is
∆it = Yit Yitd .
In general Yit = Yit = Yite and therefore
∆it = Yit Yitd = Yite Yitd
Sales are de…ned as Qit = min Yit , Yitd
If Yitd < Yit ) Qit = min Yit , Yitd = Yitd and ∆it > 0. A
positive inventory is a signal of excess supply/positive
forecasting error (demand has been overestimated).
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Inventories (cont’d)
Re 4: Goods are non-storable. The …rm therefore cannot
accumulate inventories, carry them out from today to
tomorrow and satisfy future demand. If the …rm ends up with
a positive inventory, she gets rid of the unsold (non storable)
goods (at zero costs).
If Yitd
Yit ) Qit = min Yit , Yitd = Yit and ∆it = 0. No
inventory accumulation is a signal of "equilibrium"/no
forecasting error or of excess demand/negative forecasting
error (demand has been underestimated). Due to
non-storability, in fact, there cannot be decumulation of
inventories.
Hence
∆it = max Yit Yitd , 0
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Price and quantity decisions
Re 5: The …rm is immersed in an uncertain market
environment and is exploring the price-quantity territory
around the status quo, in order to adapt price-quantity to
changing business conditions.
Imperfect information and transaction costs imply that each
…rm has a certain degree of market power on its own local
market.
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Price and quantity decisions (cont’d)
Re 6: the …rm therefore receives two signals: the price
charged by competitors Pt and the level of inventories (which
reveals forecasting errors).
These two signals capture the actual position of the …rm – i.e.
the status quo Qt = (Pit , Yit ) – relative to the equilibrium
position A = Pt , Yitd .
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Price and quantity decisions (cont’d)
On the basis of these two signals the …rm decides whether to
change the price or the expected demand and therefore the
desired quantity (but not both, see below).
This assumption is based on the evidence of survey data on
price and quantity adjustment of …rms over the business cycle
(Kawasaki et al., 1982; Bhaskar et al., 1993).
How much to change? This is uncertain (there is an element
of randomness, more on this later).
Re 7: This is a simple adjustment mechanism along two
dimensions: price and quantity
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Price adjustment
The …rm decides to change the price wrt the status quo as
follows:
Pit +1 =
Pit 1 + η it +1
Pit 1 η it +1
if
if
∆it = 0 and Pit < Pt
∆it > 0 and Pit
Pt
(1)
where η it +1 is a random positive parameter.
There is a lower bound for the price, which is the average cost
(including interest payments: Pit +1
(Wage bill + interest payments ) /output
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Quantity adjustment
The …rm decides to update the expectation of future demand
Yite+1 and therefore the desired scale of activity Yit +1 as
follows:
Yit +1 = Yite+1 =
Yit 1 + ρit +1
Yit 1 ρit +1
if
if
∆it = 0 and Pit
Pt
∆it > 0 and Pit < Pt
(2)
where ρit +1 is a random positive parameter.
The decision on the "desired" scale of production Yit +1
determines the demand for labour, i.e. desired employment
Nit +1 = Yit +1 /α.
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A digression on expectations
If expectations of demand (formed in t for t+1) changed
adaptively, then
Yite+1 = Yite + ρ Yitd
Yite
=
= Yit + ρ Yitd
Yit
=
= Yit
ρ∆it
Hence, in our case (non storability):
Yite+1 =
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Yit
Yit
if
ρ∆it
∆it = 0
if ∆it > 0
(3)
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A digression on expectations (cont’d)
In the adjustment mechanism (2), we depart from purely
adaptive expectations (3):
if ∆it = 0 expectations of demand are revised upward (instead
of being constant as in (3)) because the …rm is uncertain
whether she is in equilibrium or she has underestimated
demand; this revision upward is stochastic,
if ∆it > 0 expectations of demand are revised downward but
the magnitude of the revision is random (instead of being
determined by the size of involuntary inventories as in (3)).
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Price/quantity adjustment
There are 4 cases. Adjustment is as follows
∆it = 0
∆it > 0
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Pit < Pt
Pi "
Yi #
Pit
Pt
Yi "
Pi #
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Price/quantity adjustment in a …gure
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Banks
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Price/quantity adjustment (cont’d)
Re 9: Adjustment is partial and asymmetric. In each of the 4
cases, in fact, the …rm changes either the price or the quantity
but not both at the same time.
Why? Empirical surveys of pricing and quantity decisions of
managers point in this direction.
Moreover, this assumption makes life easier for the modeller...
...at the cost of ignoring the inventory cycle, which is an
important component of the oscillatory behaviour of GDP
Re 10: Given this adjustment mechanism, the …rm tends to
circle around the equilibrium position.
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Vacancies
The operating workforce in t+1 is equal to the employed in t
Nit less the workers whose labour contract expired in t Nit0 :
Nito +1 = Nit
Nit0
The …rms post labour vacancies if the labour requirement to
reach the desired scale of production is greater than the
operating workforce:
Vit +1 = max (Nit +1
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Nito +1 , 0)
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Vacancies (cont’d)
For instance, in the status quo Q above, the following
vacancies will be posted:
Vit +1 =
=
=
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Yit +1
Nit + Nit0 =
α
Yit
Yit
1 + ρit +1
+ Nit0 =
α
α
Yit
ρ
+ Nit0
α it +1
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Vacancies in a …gure
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Wage
There is a minimum (nominal) wage wˆ – uniform across …rms
– which is periodically revised upward in order to catch up
with in‡ation. This minimum wage can be thought of as the
result of centralized (nationwide) bargaining between trade
unions and the association of employers.
The minimum wage provides a ‡oor to the dynamics of the
actual, …rm-speci…c wage wi (i.e. the actual wage can never
be smaller than the minimum wage).
If there are no vacancies, i.e. if the …rm already operates at
the desired scale of activity, the wage she o¤ers wi is the
maximum of the contractual wage (of the previous cohort of
employed workers) and the minimum wage.
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Wage (cont’d)
If the …rm posts vacancies, i.e. if she wants to expand the
operating workforce, the contractual wage is revised upward
(by a random amount ξ it +1 )
wit +1 =
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max (wˆ t +1 , wit ) if Vit +1 = 0
max (wˆ t +1 , wit (1 + ξ it +1 )) if Vit +1 > 0
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The …nancing gap
If internal …nancial resources (net worth or equity: Eit +1 ) are
in short supply with respect to the wage bill wit +1 Nit +1 – i.e.
if there is a …nancing gap wit +1 Nit +1 Eit +1 > 0 – the …rm
asks for a bank loan Lit +1 . Otherwise, she is able to
self-…nance completely the wage bill:
Lit +1 = max (wit +1 Nit +1
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Eit +1 ; 0)
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The …nancing gap (cont’d)
For instance, in Q there will be the following demand for
loans:
Lit +1 = wit (1 + ξ it +1 )
Yit
1 + ρit +1
α
Eit +1
In t, the …rm has all the information necessary to compute
Eit +1 (see below)
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The …nancing gap in a …gure
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Search and matching on the credit market
The …rm who needs external funds contacts a …nite number
(say Zb ) of randomly chosen banks to get a loan, starting
from the one which charges the lowest interest rate
We assume that the demand for credit is divisible, so that, if
the most preferred bank is in short credit supply, the …rm can
resort to the remaining Zb 1 banks
If external funds are still not su¢ cient to pay for the wage bill
(credit rationing), the …rm will …re redundant workers and
scale down production
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Banks
The k-th bank (k = 1, 2..., K ) will extend a total amount of
credit to …rms Lkt equal to a multiple of its equity base or net
worth ("bank’s capital"):
Lkt = Ekt /ν
where 0 < ν < 1 can be interpreted as a capital requirement
coe¢ cient. Hence 1/ν is the maximum allowable leverage for
the bank.
Lkt represents the total amount of “credit vacancies” posted
by the bank on the credit market.
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The interest rate
We assume that bank k o¤ers to …rm i a standard debt
contract, which consists of an interest rate ritk and the
corresponding loan Lit .
The interest rate is set as follows:
ritk = r (1 + φkt µ (λit ))
ritk is determined by means of a "mark up" (over a baseline
rate r )
r is a proxy for the policy rate
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The interest rate (cont’d)
The mark up consists of:
φkt i.e. an idiosyncratic shock which captures random cost
factors speci…c to the bank,
µ (λit ) , µ0 > 0; i.e. the external …nance premium which
increases with the borrower’s leverage λit
The leverage of the …rm is de…ned as:
λit :=
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Lit
wit Nit
=
Eit
Eit
1
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The interest rate (cont’d)
OTBE, if Ei " ) λi # ) rik #
There is an indirect e¤ect (not shown in the …gure):
rik #) Ei " (positive feedback)
This is the root of the …nancial accelerator (at the
micro-level) in the present setting
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The interest rate (cont’d)
The notion of “external …nance premium” has been introduced
by Bernanke and Gertler (1989, 1990). See also Bernanke,
Gertler and Gilchrist (1999) on the …nancial accelerator
In the presence of ex post asymmetric information and costly
state veri…cation, the higher the borrower’s …nancial fragility,
the more frequent should be (in the optimum) the auditing
activity of the bank and the higher the interest rate charged
to the borrower
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The demand for loans and the interest rate
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Pro…ts and losses
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The demand for loans and the interest rate
Notice that the demand for loans is not elastic to the interest
rate.
An increase of borrower’s net worth impacts upon both the
demand for loans and the interest rate (which go down)
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The demand for loans and the interest rate: the e¤ect of
an increase of net worth
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The bank sorts the borrowers’applications for loans in
ascending order according to leverage, and satisfy them until
all credit supply has been exhausted.
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Pro…ts and losses: …rms
Gross pro…ts of the i-th …rm can be computed as follows:
EBIit +1 = Pit +1 Yit +1 wit +1 Nit +1
{z
}
|
Earnings Before Interest (EBI)
If pro…ts are high enough, the …rm is able to "validate debt
commitments":rit +1 (wit +1 Nit +1 Eit +1 ) where
rit +1 =
∑ xitk +1 ritk +1
k 2Λ
In words: rit +1 is the weighted average of the interest rates
charged to the …rm by the banks she gets in touch with to
fund the …nancing gap (the set of these banks is labelled Λ).
xitk +1 represents the fraction of the …nancing gap funded by
bank k.
Domenico Delli Gatti
ABM_Lecture #2
Sequence
Financial accelerator
Agents and markets
Workers
Capitalists
Firms
Banks
Pro…ts and losses
Pro…ts and losses: …rms (cont’d)
Pro…ts are π it +1 = EBIit +1
rit +1 (wit +1 Nit +1
By assumption: Divit = δπ it ; 0 < δ < 1
Net worth therefore is Eit +1 = Eit + (1
Domenico Delli Gatti
ABM_Lecture #2
δ) π it
Eit +1 )
Sequence
Financial accelerator
Agents and markets
Workers
Capitalists
Firms
Banks
Pro…ts and losses
Sequence
Pro…ts and losses: banks
The pro…ts (revenues) of the k-th bank coincide with interest
payments received:
π kt +1 =
∑+ ritk +1 (wit +1 Nit +1
Eit +1 )
i 2Θ
where Θ+ is the set of solvent borrowers whom the bank has
lent to
Net worth is Ekt +1 = Ekt + π kt
BDkt
BDkt is "bad debt" i.e. the total value of non performing
loans: ∑ (wit +1 Nit +1 Eit +1 )
Θ
Domenico Delli Gatti
ABM_Lecture #2
i 2Θ
the set of insolvent borrowers
Financial accelerator
Agents and markets
Workers
Capitalists
Firms
Banks
Pro…ts and losses
The sequence of events
1
Each …rm computes net worth in t+1 (Eit +1 ) updating net
worth in t with retained pro…ts:
Eit +1 = Eit + (1
Domenico Delli Gatti
ABM_Lecture #2
δ) π it
Sequence
Financial accelerator
Agents and markets
Workers
Capitalists
Firms
Banks
Pro…ts and losses
Sequence
Exit and entry
If Eit +1 < 0 the …rm goes bankrupt and exits.
The bankruptcy of the …rm
generates a negative demand spillover for the other …rms (due
to the loss of employment and wage income),
a¤ects the bank’s net worth (through bad debt) and therefore
will impact on the amount of credit extended in the future.
Domenico Delli Gatti
ABM_Lecture #2
Financial accelerator
Agents and markets
Workers
Capitalists
Firms
Banks
Pro…ts and losses
Exit and entry (cont’d)
We assume that each bankrupt …rm is replaced by a new
entrant.
The initial condition (size at entry) of the new …rm is set
below the average size of incumbent …rms.
This one-to-one replacement of bankrupt …rms with entrant
…rms allows us to keep the total …rms’population constant.
Domenico Delli Gatti
ABM_Lecture #2
Sequence
Financial accelerator
Agents and markets
Workers
Capitalists
Firms
Banks
Pro…ts and losses
Sequence
Price/quantity adjustment
If Eit +1 > 0 the …rm survives and proceeds to the next step.
2. Starting from the status quo, the …rm decides (adaptively) the
quantity to be produced (hence the demand for labour) and
the price to be charged.
Domenico Delli Gatti
ABM_Lecture #2
Financial accelerator
Agents and markets
Workers
Capitalists
Firms
Banks
Pro…ts and losses
The labour market opens
L1 Firms post vacancies (if any) at a certain o¤ered wage.
L2 Each unemployed worker contacts a given number of
randomly chosen …rms to get a job, starting from the one
which o¤ers the highest wage.
L3 Once the matching has been completed, the …rm computes
the wage bill and the …nancing gap, which has to be …lled by
means of loans.
Domenico Delli Gatti
ABM_Lecture #2
Sequence
Financial accelerator
Agents and markets
Workers
Capitalists
Firms
Banks
Pro…ts and losses
Sequence
The credit market opens
B1 Banks post credit vacancies (supply of loans) at a certain
o¤ered interest rate.
B2 Each …rm contacts a given number of randomly chosen banks
to get a loan, starting from the one which o¤ers the lowest
interest rate.
B3 Once the matching has been completed, the …rm assesses
whether …nancial resources are enough to pay for the wage
bill. If not – i.e. in case of credit rationing – some workers will
be …red.
B4 Production takes the whole period t.
Domenico Delli Gatti
ABM_Lecture #2
Financial accelerator
Agents and markets
Workers
Capitalists
Firms
Banks
Pro…ts and losses
Sequence
The market for the consumption goods opens
C1 Firms post their o¤er price.
C2 Each worker/consumer contacts a given number of randomly
chosen …rms to buy consumption goods, starting from the one
which o¤ers the lowest price.
C3 Once the matching has been completed, the …rm assesses
whether there is su¢ cient demand to sell her goods. If not –
i.e. in case of excess supply (inventories) – it gets rid of the
unsold goods (at zero costs).
Domenico Delli Gatti
ABM_Lecture #2
Financial accelerator
Agents and markets
Workers
Capitalists
Firms
Banks
Pro…ts and losses
Sequence
Firms: Accounting and (maybe) surviving
A1 At the end of the period, …rms collect revenues and calculate
Earnings Before Interest
EBIit +1 = Pit +1 Yit +1
wit +1 Nit +1
A2 If EBI is big enough, the …rm "validates …nancial
commitments" (i.e. makes interest payments) and gets pro…ts:
π it +1 = Pit +1 Yit +1
wit +1 Nit +1
rit +1 (wit +1 Nit +1
Eit +1 )
A3 If earnings after interest payments are positive, the …rm pays
out dividends as a fraction of pro…ts: δπ it +1
(1 δ) π it +1 are retained pro…ts, which are used to update
net worth and determine Eit +2 .
The cycle starts again.
Domenico Delli Gatti
ABM_Lecture #2
Financial accelerator
Agents and markets
Workers
Capitalists
Firms
Banks
Pro…ts and losses
Sequence
Banks: Accounting and (maybe) surviving
E1 At the end of the period, the bank collects revenues and
computes bad debt. Therefore she can compute net worth
Ekt +1 = Ekt +
∑+ ritk +1 (wit +1 Nit +1
i 2Θ
Eit +1 )
∑
(wit +1 Nit +1
i 2Θ
If Ekt +1 < 0 the bank is insolvent and is replaced by another
bank.
Otherwise, the bank survives and restart the cycle of lending.
Domenico Delli Gatti
ABM_Lecture #2
E

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