Kiyotaki–Moore model

The Kiyotaki–Moore model of credit cycles is an economic model developed by Nobuhiro Kiyotaki and John H. Moore. A mathematical inappropriate (see details below) solution of the model's equations of motion (EOMs) has resulted in a long-living incorrect statement that the model shows how small shocks to the economy might be amplified by credit restrictions, giving rise to large output fluctuations.

The model assumes that borrowers cannot be forced to repay their debts. Therefore, in equilibrium, lending occurs only if it is collateralized. That is, borrowers must own a sufficient quantity of capital that can be confiscated in case they fail to repay. The old and incorrect narrative of the model was, that this collateral requirement amplifies business cycle fluctuations because in a recession, the income from capital falls, causing the price of capital to fall, which makes capital less valuable as collateral, which limits firms' investment by forcing them to reduce their borrowing, and thereby worsens the recession.

Kiyotaki (a macroeconomist) and Moore (a contract theorist) originally described their model in a 1997 paper in the Journal of Political Economy.

The paper also analyzes cases where debt contracts are set only in nominal terms or where contracts can be set in real terms, and considers the differences between the cases.

Structure of the model

In their model economy, Kiyotaki and Moore assume two types of decision makers, with different time preference rates: "patient" and "impatient." The "patient" agents are called "gatherers" in the original paper, but should be interpreted as households that wish to save. The "impatient" agents are called "farmers" in the original paper, but should be interpreted as entrepreneurs or firms that wish to borrow in order to finance their investment projects.

Two key assumptions limit the effectiveness of the credit market in the model. First, the knowledge of the "farmers" is an essential input to their own investment projects—that is, a project becomes worthless if the farmer who made the investment chooses to abandon it. Second, farmers cannot be forced to work, and therefore they cannot sell off their future labor to guarantee their debts. Together, these assumptions imply that even though farmers' investment projects are potentially very valuable, lenders have no way to confiscate this value if farmers choose not to pay back their debts.

Therefore, loans will only be made if they are backed by some other form of capital which can be confiscated in case of default. In other words, loans must be backed by collateral. Kiyotaki and Moore's paper considers land as an example of a collateralizable asset. Thus land plays two distinct roles in the model: (i) it is a productive input, and (ii) it also serves as collateral for debt.

Hence, impatient agents must provide real estate as collateral if they wish to borrow. If for any reason the value of real estate declines, so does the amount of debt they can acquire. This feeds back into the real estate market, driving the price of land down further (thus, the borrowing decisions of the impatient agents are strategic complements). The model exhibits a positive feedback, but this feedback does not lead to amplifications of economic fluctuations due to a small temporary shock. The model's aggregate EOMs for land $K_{t}$ and borrowing $B_{t}$ have the form:

$K_{t}={\frac {1}{u_{t}}}\left[(a+q_{t})K_{t-1}-RB_{t-1}\right]{\text{ and }}B_{t}={\frac {1}{R}}q_{t+1}K_{t},$

where $u_{t}\equiv q_{t}-q_{t+1}/R$ can be interpreted as the down payment for one unit of land. The technology $a$ is considered time invariant. $q_{t}$ is the land price and $R$ the interest rate.

The system is assumed to be in equilibrium at $t-1$ and a temporary shock $\Delta$ is applied to the productivity $a$ at $t$, leading to:

${\begin{aligned}\underbrace {u(K_{t})K_{t}} _{\text{non linearity}}&=(a\underbrace {\cdot (1+\Delta )} _{\text{shock to productivity}}+q_{t}-q^{*})K^{*}{\text{ at }}t,\\\underbrace {u(K_{t+s})K_{t+s}} _{\text{non linearity}}&=aK_{t+s-1}{\text{ at }}t+s{\text{ with }}s\geq 0.\end{aligned}}$

Kiyotaki and Moore suggest to linearize the EOMs. It is important to notice, that the EOMs are not linearized or Taylor expanded in $\Delta$, as the model is already linear in $\Delta$. The equations are expanded in powers of $K_{t}$ or more precise in powers of relative deviations from the equilibrium land holding $(K_{t}-K^{*})/K^{*}={\hat {K}}_{t}$ and non-linear terms in ${\hat {K}}_{t}$ are ignored.

Such a linearization in ${\hat {K}}_{t}$ is correct, but only valid if ${\hat {K}}_{t}$ remains small, otherwise higher powers ${\hat {K}}_{t}^{n}$ for $n>1$ are potentially not negligible and even more important.

Hence, the finding of Kiyotaki and Moore in the 1997 paper in the Journal of Political Economy that large output variations, i.e. large ${\hat {K}}_{t}$, are found after a small temporary shock, is clearly mathematically inconsistent with their solution technique (linearization or first order Taylor expansion).

Correction of the original results

The recent paper Credit Cycles Revisited by Jörg Urban has shown that no large output variations are found if the model is solved correctly. The solution based on linearized EOMs (presented by Kiyotaki and Moore) is only valid for the uninteresting situation of small output variations (small ${\hat {K}}_{t}$). The paper Credit Cycles Revisited contains several further findings: First, it shows that the singularity for $R=0$ (zero interest rate environment) is not present anymore if the model is solved correctly. Second, the extension of the model to a two sector model features no large spillovers from the shocked to the un-shocked sector. Furthermore, the model exhibits a persistence of output fluctuations due to the temporary shock, i.e. ${\hat {K}}_{t}$ asymptotically approaches zero. This finding is, however, of no relevance, because the predicted output fluctuations are small.

Extensions

The original paper of Kiyotaki and Moore was theoretical in nature, and made little attempt to evaluate the quantitative relevance of their mechanism for actual economies. In 2005, Kiyotaki's student Matteo Iacoviello embedded the Kiyotaki-Moore mechanism inside a standard New Keynesian general equilibrium macroeconomic model. For realism, Iacoviello assumed that the collateralizable form of capital corresponds to real estate, and compared the predictions of his model with observed fluctuations of real housing prices in the United States.