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


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1 Chapter 3 Credit and Business Cycles Here I present a model of the interaction between credit and business cycles. In representative agent models, remember, no lending takes place! The literature on the topic has emphasized the idea that a credit market imperfections can magnify the e ects on the real economy of given technology or monetary shocks. This happens because a positive shock increases income of owners of production technology; this rise in net worth lowers the costs associated with external nancing of investment projects, allowing for increased investment. This serves to amplify and propagate the e ects of a shock through time The modelling choices We now develop a model that illustrates the role of debt, net worth and asset price uctuations on euilibrium output. You can consider this model as an extension of the real business cycle model which allows for nancial factors to play a role in business uctuations. The model has to be as simple as possible. In fact it can be made uite simple even though it features heterogenous agents by adding to it the following features:. A constant interest rate (one less variable) 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). 3.2 A basic model The reference is the paper Credit Cycles by Kiyotaki and Moore (997), JPE, 2247, available on JSTOR. We use a slightly modi ed version. 34
2 3.2. A BASIC MODEL 35 There are two types of agents, farmers (productive) and gatherers (unproductive). They both have linear preferences over consumption, but they discount the future di erently. Farmers have a discount factor of ; gatherers have a discount factor of ; where >. They produce a nal good y t using land h t. The production functions are respectively described by: y t = A t h t y 0 t = A 0 th 0 t 3.2. Gatherers The gatherers will be the unproductive agents, that is they will earn a lower return on their asset holdings in euilibrium. They solve the following problems. max h 0 t ;b0 t E 0 X t=0 t c 0 t! The Lagrangean for this problem is: s.t c 0 t + t h 0 t + Rb 0 t = y 0 t + t h 0 t + b 0 t L = E 0 X t=0 t y 0 t + t h 0 t + b 0 t t h 0 t Rb 0 t for each t; where A 0 th 0 t. The rst order conditions are, choosing b 0 t and h 0 t respectively: R = t = E t t+ + y0 t+ h 0 t Linear preferences over consumption imply that in euilibrium the gatherers must be indifferent about any path of consumption and debt, so that the interest rate will eual their rate of time preference, and R = = Farmers Farmers maximize their ows of funds is max b t;h t E 0! X t c t t=0 c t + t h t + Rb t = y t + t h t + b t
3 36 CHAPTER 3. CREDIT AND BUSINESS CYCLES where on the righthand side we have the sources of funds, on the lefthand side we have its uses. The amount of claims farmers can issue is bound by: Rb t me t ( t+ h t ) Remark 4 Why can t the farmers borrow more than the me t ( t+ h t =R)? The idea is as follows: the Gatherer lends some goods to the Farmer, who in turn promises him to pay back at some future date. The assumption here is that the farmer s input is critical for production: once the farmer starts producing, no one can replace him/her, and the farmer cannot commit to repay his debt. If the creditor tries to extract too much from the farmer, the farmer can simply walk away from the land. Current production is lost, and the farmer only recovers the value of the land. This in turn limits his/her ability to borrow. The Lagrangean for this problem is: L = E 0 X t=0 for each t; where A t h t Euilibrium t ((y t + t h t + b t t h t Rb t ) t (Rb t m t+ h t )). The rst order conditions are, choosing b t and h t respectively: = R + t R t = E t t+ + y t+ + E t ( t m t+ ) h t There are three markets: the market for loans b t ; goods y t + y 0 t and market for h. From the Euler euation for gatherers, we know that: coupled with that of farmers t = = R = R R = > 0 which implies that the borrowing constraint will be binding in steady state. Once we collect all of them, we obtain the following euations: c 0 t + t h 0 t + Rb 0 t = y 0 t + t h 0 t + b 0 t () c t + t h t + Rb t = y t + t h t + b t (2) yt+ t = (m + ( m) ) E t t+ + E t (3) h t y 0 t = E t t+ + E t+ t (4) h 0 t Rb t = me t ( t+ h t ) (5)
4 3.2. A BASIC MODEL 37 To this group of euations, we add the market clearing conditions. Normalize the total supply of land H to. h t + h 0 t = (6) b t + b 0 t = 0 (7) y t = A t h t (8) y 0 t = A 0 th 0 t (9) Remember that if the bond market clears, so will the goods one, so we will not need to write down the market clearing condition for goods. A uick check: 9 euations, whereas the variables are c 0 t c t y t yt 0 h t h 0 t b t b 0 t t : We restrict our attention to perfectforesight euilibria, where, absent unanticipated shocks, the expectations of future variables realize themselves. An euilibrium can be de ned as follows: for given levels of h t and b t ; an euilibrium from date t onwards is characterized by paths for the endogenous variables fx t+s js 0g satisfying euations () to (9) at dates t; t + ; t + 2; ::. A transversality condition rules out exploding paths in asset prices, that is lim t+s t = 0 s! R s If the transversality condition is satis ed, there is a locally uniue perfectforesight euilibrium path starting from initial values of h t and b t in the neighborhood of the steady state Simplify Most of the euations are trivial. Also notice that c 0 t and c t only appear in and 2, so they will adjust so that and 2 hold. To simplify matters, Kiyotaki and Moore also assume that = ; so that the production technology of the farmers is linear. This implies that all the dynamics of the system can be analyzed with reference to two euations only. De ne = m + ( m) (you can think of as an average of the two discount factors, the higher m; the higher ) Steady state Let us look at the steady state rst. From 3 t = E t t+ + E t A t+ (3) A 0 t+ t = E t t+ + E t ( h t ) (4) = A
5 38 CHAPTER 3. CREDIT AND BUSINESS CYCLES from 4 which gives us h: ( ) = A0 ( h) h = A 0 ( ). A ( ) Clearly, restrictions of A 0 =A need to placed to ensure an euilibrium with 0 < h <. Remark 5 What is the marginal product of land used in the farming sector (de ne it as MP K)? Simply, it is A What is M P K in the gatherers sector? Simply from the optimality conditions of gatherers: ( ) ( ) = + MP K MP K = ( ) ( ) = ( ) A Given < < ; < ; so that the MPK in the gatherers sector is below that of the farmers sector. This implies that in the competitive euilibrium the allocation of land is ine cient since its marginal product is not euated across the two sectors. From 5 we can also derive: as for the other variables b = mh c = y (R ) b = Ah ( ) mh = A c 0 = y 0 + (R ) b m ( ) h The main idea here is that credit market imperfections might cause a misallocation of resources in euilibrium The dynamics e ects of an unexpected productivity shock To consider the e ects on the economy of a productivity shock, we linearize the model euations around the steady state and calculate the e ect of a % increase in productivity. We consider a proportional increase in productivity for both agents, that is A b t = A b0 t (where X b t Xt X is the percent deviation from steady state of a variable). Let us also X assume that once a technology shock occurs, it follows an AR() process of the form: ba t = b A t + be t From euation 3 we obtain:
6 3.2. A BASIC MODEL 39 t = E t t+ + E t A t+ d t = E t d t+ + E t da t+ d t d t+ da t+ = E t + E t A use = A! A = A b t = E t b t+ + ( ) E t b At+ (L) which can be solved forward to obtain: use E t b t+ = ( ) E tat+2 b + E t b t+2 b t = ( ) E tat+ b + ( ) E tat+2 b + E t b t+2 use E t b t+2 = ( ) E tat+3 b + E t b t+3 b t = ( ) E tat+ b + ( ) E tat+2 b + ( ) E tat+3 b + E t b t+3 b t = ( ) E t b At+ + ( ) E t b At ( ) E t b At+3 + ::: b t = ( ) E t b At+ + ( ) E t b At+ + 2 ( ) 2 E t b At+ + ::: b t = ( ) ::: E t b At+ + :::: b t = E t b A t+ = b A t (dr) where is the persistence of the technology shock. To linearize euation 4:
7 40 CHAPTER 3. CREDIT AND BUSINESS CYCLES A 0 t+ t = E t t+ + E t ( h t ) t = E t t+ + E t A 0 t+ ( h t ) take total di erential d t = E t (d t+ ) + ( h) de t A 0 t+ A 0 ( ) ( h) 2 dh t d t = E t (d t+ ) + ( h) de t A 0 t+ A 0 ( ) ( h) ( h) dh t use steady state result = + A 0 ( h)! ( ) = A 0 ( h) d t = E t (d t+ ) + ( ) de ta 0 t+ A 0 ( ) ( ) ( h) dh t d t divide everything by = E dt+ t + ( ) de ta 0 t+ ( ) ( ) A 0 h hdh t h de ne = h ( ) h d t = E dt+ t + ( ) de ta 0 t+ + ( ) dh t A 0 h b t = E t b t+ + ( ) E tat+ b + b h t (L2) where = h > 0 can be interpreted as an elasticity of asset demand with respect to ( )h the price (roughly speaking, it tells you by how much b h changes in the longrun for a given change in b; this is why we interpret it as an elasticity). Combining (dr) and (L2) together yield: A b t = 2 At b + ( ) A b t + b h t which can be solved for the response of b h t to a productivity shock, that is: b ht = ( ) ( ) ( ) ( ) A ( ) ( ) b t (dr2) (dr) and (dr2) are the two linearized decision rules of our problem (that is, they express the endogenous variables as a function of shocks only). (dr2) shows that h rises with A so long as 0 < < (the numerator can be shown to be positive because < ). Hence in this problem a productivity shocks raises asset prices which in turn raises asset demand of the most productive agents, who can then produce more and accumulate more: the e ects on aggregate activity are therefore ampli ed by the asset price e ect. The main idea here is that credit market imperfections might amplify the e ects of given shocks to the economy.
8 3.3. FORWARDLOOKINGNESS AND HISTORY DEPENDENCE 4 It is easy to show how aggregate output rises more than the increase in productivity: that is, in log deviations from the steady state: by t = y y + y by 0 t + y0 y + y 0 by0 t > A b t. 3.3 Forwardlookingness and history dependence The model here displays forwardlookingness; an increase in asset prices implies that the creditor will be able to recover more from selling the asset whenever the debtor defaults, therefore for each asset price increase he is willing to supply more credit. As credit increases, asset prices rise, aggregate investment rises (since the marginal productivity of debtors is higher than creditors ) and so on and so forth, in a cumulative fashion. However, it does not display historydependence (one way to see this is that the euilibrium decision rules do not depend on h t and b t explicitly). Kiyotaki and Moore say when a shock hits, it is too late to renegotiate, therefore repayments are the same, but current borrowing increases. If consumption is bound by some upper limit, all the new borrowing goes into investment, and this strongly magni es the response of output to a given productivity shock. The homework uestions asks you to analyze this in more detail.
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