Handout Class 4

Alessandro Di Nola

November 2012

Kiyotaki and Moore

1 The modelling choices

I presented you in class a simplied version of the Kyotachi-Moore model that introduces credit

market imperfections into the picture of real business cycle models. The reference is the paper

Credit Cyclesby Kiyotaki and Moore (1997), JPE, 211-248 that you nd in Prof. Giavazzis

reading list.

The model can be made quite simple even though it features heterogenous agents by adding to

it the following features:

1. A constant interest rate (one less variable), it follows from households having linear preferences

over conumption (see below).

2. No labor supply decision.

3. No variable capital.

4. Only one asset that can be used for production, and is available in xed supply in the aggregate

(call it land). It also serves as a collateral constraint for the borrowers.

2 A Basic Model

There are two types of agents, farmers (productive) and gatherers (unproductive). They both

have linear preferences over consumption, but they discount the future dierently. Farmers have a

discount factor of ; gatherers have a discount factor of ; where > :

1

They produce a nal good yt using land ht. The production functions are respectively described

by:

yt = Athvt1;

y0t = A0th0t1:

2.1 Gatherers

The gatherers will be the unproductiveagents, and they will lend to other agents. They solve the

following problems

maxc0t;h

0t;b

0t

E0

Xt

tc0t

!s.t.

c0t + qth0t +Rb

0t1 = y

0t + qth

0t1 + b

0t:

where b0t denotes the amount of new debt issued in period t. Notice that in equilibrium b0t is actually

negative, reecting the fact that gatherers are creditors to the farmers.

By sustituting the budget constraint into the objective function we get the following equivalent

unconstrained problem1:

maxh0t;b

0t

E0Xt

ty0t + qth

0t1 + b

0t qth0t Rb0t1

where y0t = A

0th0t1 of course. The rst order conditions are, choosing b

0t and h

0t respectively:

R = 1;

qt = Et

qt+1 +

y0t+1h0t

:

Linear preferences over consumption imply that in equilibrium the gatherers must be indierent

about any path of consumption and debt, so that the interest rate will equal their rate of time

preference, and R = 1=.

2.2 Farmers

Farmers maximize

maxct;ht;bt

E0

Xt

tct

!their ows of funds is

ct + qtht +Rbt1 = yt + qtht1 + bt;

1 If you prefer you can use the Lagrangian, but here the substitution method is faster.

2

where on the right-hand side we have the sources of funds, on the left-hand side we have its uses.

The amount of claims farmers can issue is bound by:

Rbt mEt(qt+1ht): (1)

The credit constraint arises due to limited enforcement of contracts. It is assumed in fact that

lenders cannot force borrowers to repay their debts unless the debts are secured by land. The

lenders therefore take care never to allow the size of the debt (gross of interest, which is the LHS

of 1) to exceed the values of the collateral (which is the RHS of 1). Specically, if at date t the

farmer/borrower has land ht then he can borrow bt in total, as long as the repayment does not

exceed the market value of his land at date t+ 1. The parameter m indexes the amount of credit

frictions that aect the economy.

The Lagrangean for this problem is:

L = E0

1Xt=0

t (yt + qtht1 + bt qtht Rbt1) + tt (mqt+1ht Rbt)

where yt = Athvt1 and

tt is the non-negative multiplier attached to the inequality constraint

(1).

The Kuhn-Tucker necessary conditions are:

1 = R+ tR; (2)

qt = Et

qt+1 +

yt+1ht

+ Et (tmqt+1) ; (3)

t (mqt+1ht Rbt) = 0; (4)

t 0; (5)

mqt+1ht Rbt 0: (6)

From the rst equation we nd

t = =1 RR

= > 0;

which implies that the multiplier is constant over time and strictly positive. By (4) the borrowing

constraint must be binding at each date: mqt+1ht Rbt = 0 8t:

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2.3 Equilibrium

Collecting all the equilibrium equations and eliminating the multiplier gives:

c0t + qth0t +Rb

0t1 = y

0t + qth

0t1 + b

0t (7)

ct + qtht +Rbt1 = yt + qtht1 + bt (8)

qt = Et

qt+1 +

y0t+1h0t

(9)

qt = (m + (1m))Etqt+1 + Etyt+1ht

(10)

Rbt = mEt(qt+1ht) (11)

ht + h0t = 1 (12)

bt + b0t = 0 (13)

yt = Athvt1 (14)

y0t = A0th0t1 (15)

There are three markets: the market for loans b, goods y and market for land h. Equation

(12) is the market clearing condition for land2, (13) is the market clearing condition for bonds.

Remember that by WalrasLaw if the bond market and the land market clear, so will the goods

one, and hence we dont need to write down the market clearing condition for goods. A quick check:

9 equations, whereas the variables are c0t, ct, yt, y0t, ht, h

0t, bt, b

0t, qt.

2 I normalized to 1 total supply of land.

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