A Mathematical Model of Foreign Capital Inflow
AA Mathematical Model of Foreign Capital Inflow
Gopal K. Basak ∗ , Pranab Kumar Das † , Allena Rohit ‡ Abstract
The paper models foreign capital inflow from the developed to the developing countries ina stochastic dynamic programming (SDP) framework. Under some regularity conditions, theexistence of the solutions to the SDP problem is proved and they are then obtained by numer-ical technique because of the non-linearity of the related functions. A number of comparativedynamic analyses explore the impact of parameters of the model on dynamic paths of capitalinflow, interest rate in the international loan market and the exchange rate.
JEL Classification:
C61, E44, E47, F32, F34, F42, G11
Mathematics Subject Classification:
Keywords and Phrases:
Capital Inflow, Interest Rate Determination, Dynamic Programming Principle,Portfolio Theory, E-M Algorithm, Dynamic Terms of Trade
The paper develops a simple model of foreign capital inflow in a multi period dynamic programmingframe work. Typically capital flows from the developed to the developing countries owing to a higherrate of return in the latter because of scarcity of capital and thus a higher marginal product ofcapital than in the developed countries which are capital rich with a lower marginal product ofcapital. The foreign capital supplements the domestic resources of the developing countries forachieving a higher rate of capital formation and hence higher rate of economic growth. Generallydeveloping countries are characterized by consistent current account deficit. This together witha higher interest rate creates a natural condition for flow of capital from the developed to thedeveloping countries. The inflow to the borrowing country helps appreciate the domestic currency ∗ Corresponding author: Stat-Math Unit, Indian Statistical Institute, Kolkata 700108, Email: [email protected] of this author is partially supported by the Institute’s CPDA grant for 2014-2017. † Centre for Studies in Social Sciences, Calcutta, R1, B.P. Township, Kolkata 700094, India, Email:[email protected] ‡ Emory University, Atlanta, USA, Email: [email protected] Marginal product of capital is directly related to interest rate in general and exactly equal in a competitive marketframework. a r X i v : . [ q -f i n . E C ] M a y ollowed by rise in asset prices and local goods prices which via favourable fiscal conditions furtherencourages domestic credit creation. Repayment of loans in the next period has an adverse effecton exchange rate resulting into depreciation of exchange rate. This does not matter as long asinflow is sufficient. The crisis situation occurs particularly when the rate of capital inflow reachesa plateau or the global investors find a new country with even a higher rate of return then thenew paradigm looks shop worn; a sudden stop a la Calvo (1998) to capital inflow or a reversalof current account balance leading to asset price contraction, decreased domestic investment andthe economy adjusts backwards. In an extreme situation depreciation of domestic currency makesdebt servicing difficult which can lead to a foreign exchange crisis or banking crisis or both (seeReinhart and Reinhart, 2008; Reinhart and Rogoff, 2009; Wolf, 2008). The crisis may furtherget aggravated by bad macroeconomic management on fiscal and monetary fronts. The problemof debt servicing arises because capital inflow to developing countries are typically pegged in hardcurrency which has been termed as ’original sin’ (Eichengreen and Hausman, 1999; Eichengreen,Hausman and Panizza, 2005).In a simple two period model the potential of financial crisis emanating from the depreciation ofexchange rate at the time of repayment of foreign currency loans was modeled by Marjit, Dasand Bardhan (2007). The argument is based on simple text book explanation of terms of tradeeffect based on trade theoretic argument as in Caves et al 1993; Helpman and Krugman, 1989.When all the producers sell in the international market at a given world price the combined supplyreduces the world price though no individual seller or even a country can affect world price. Thesame argument applies in the context of capital inflow in a two period framework. It is arguedthat the terms of trade effect may dominate leading to a financial crisis. Basak, Das and Marjit(2012) extended the model by introducing repayment of previous period borrowing. It adds multiperiod dynamics in the structure for a better understanding of the possibility of crisis. Costinotet al. (2014) has argued for capital control on the basis of dynamic terms of trade in the samespirit as in this paper. The present paper is a formalization of the process of capital inflow inan infinite horizon set up with micro foundations of the choice problem of the agents - borrowersand lenders. The model investigates the effect of an increase in the perception of risk of defaultof international loan in the lending country, shift in the expectation of the future exchange rateor decreased productivity in the domestic sector in the borrowing country on capital inflow. Theissue pertaining to the origin of the financial crisis in the process of inflow is also discussed.With this short introduction the paper proceeds as follows. Section 2 formulates a model of capitalinflow in a two country setup, a borrowing country and a lending country. From the individualoptimization exercise we derive equilibria in the two markets, viz. international loan market andthe foreign exchange market which are then solved for the interest rate in the international loanmarket and exchange rate by numerical method. Proofs of the existence of the solutions are given inthe Appendices. In section 3, we undertake comparative dynamic exercises of changes in parametervalues. The final section concludes. General empirical evidence can be found in Bordo and Meissner (2005); Bussi`ere et al (2004) while country specificevidence can be found in Dominguez and Tesar (2005), Goldfajn and Minella (2005), Noland (2005). East Asian crisisof 1997 has been extensively discussed by Allen and Gale (2007), Krugman (1998), Gorton (2008), Rakshit (2002). The Model
We consider a two-country model with one borrowing (developing) and another lending (developed)country. The two country framework is an abstraction from the reality where the two countriesrepresent two country groups. The borrowing country group is capital scarce, thus has a highermarginal product of capital and hence a higher domestic interest rates whereas the lending countryhas lower domestic interest rates as this group is capital rich. Actual borrowing-lending takesplace through banks or financial intermediaries, there is no direct lending. A typical bank in thedeveloping country borrows from both the domestic and international markets and lends to domesticproduction sector while a typical bank in the developed country borrows from domestic sector andlends to both the domestic and international borrowers. The developing country banks have torepay the foreign debt along with the interest rate in the next period denominated in the lendingcountry’s currency, i.e., hard currency. International loan market is assumed to be competitive sothat the international loan rate is given to both lending country and borrowing country banks. Wedo not consider the determination of domestic borrowing or lending rates of interest in the model.It is only the interest rate on foreign borrowing/ lending and the exchange rate that are determinedin this model. Each bank in both the developed and the developing countries maximize discountedexpected utility over infinite horizon, specifically an intertemporal mean-variance utility function,by choice of their portfolio of borrowing and lending.
Competitive banking in the developed country ensures that all banks are symmetric so that eachone of them raise the same amount of deposit, assumed to be K Dt in each period. Number of banksis assumed to be m D . We define, at time period t r Dt = interest rate paid on deposits, R Dt = interestrate on domestic loans, R ∗ t = interest rate on international loan, F t = total funds at the disposalof the bank. The bank lends µ t fraction in the international market the rest in the internationalmarket with a risk of default (cid:15) t for the foreign loan defined as (cid:15) t = p t ,(cid:15) with probability 1 − p t , (cid:15) ∈ [0 , . Domestic loan has no risk of default. Total funds for lending F t is the sum of accumulated profitfrom the previous period and the amount of new deposits and is given by: F t = F t − [(1 − µ t − )(1 + R Dt − ) + ( µ t − )(1 + R ∗ t − ) (cid:15) t − ] − (1 + r Dt − ) K Dt − + K Dt . (2.1) The borrowing country group resembles the emerging market economies among the developing countries withhigh growth potential such as China, India, Brazil, South Africa etc. These economies are attractive location forinternational capital compared to the least developed countries of Africa or Asia. In recent times a few emerging market economy firms have been borrowing directly in the international market,but major part of international borrowing takes place via banks or consortium of banks. External CommercialBorrowing, American Depository Receipts are two prominent examples of such direct borrowing by Indian firms inthe international market. E [ π Dt ] = F t [(1 − µ t )(1 + R Dt ) + µ t (1 + R ∗ t ) E ( (cid:15) t )] − (1 + r Dt ) K Dt , and the variance by, V t ( π Dt ) = F t µ t (1 + R ∗ t ) V t ( (cid:15) t ) . A typical bank chooses its portfolio to maximize its intertemporal expected profit adjusted for risk(measured by variance). The instantaneous utility (mean-variance utility function) of a bank inthe developed country at time period t isΩ Dt = E t ( π Dt ) − γ V t ( π Dt ) . (2.2)The bank maximizes[Ω Dt + B D [ E t ( π Dt +1 ) − γ V t ( π Dt +1 )] + B D [ E t ( π Dt +2 ) − γ V t ( π Dt +2 )]+ B D [ E t ( π Dt +3 ) − γ V t ( π Dt +3 )] + . . . , by choice of µ t subject to the funds constraint, where B D is the discount factor assumed same forall time periods.The above optimality problem is solved by Bellman optimization technique.[5] The problem of thebank can be stated in the Bellman framework as V D ( F t ) = max µ t { Ω Dt + B D E t ( V D ( F t +1 )) } , (2.3)subject to fund constraint (2.1) where V D ( F t ) denotes the value function assumed to be stationary.Since (i) the utility function is concave and the constraint function is linear, (ii) the utility functionis time separable in contemporaneous control and state variables, and (iii) current decision affectcurrent and future utilities but not the past ones, Bellman principle can be applied here to solve thedynamic programming problem. In particular, solution would be of the form of quadric functionsince the utility function is quadratic. So we let, V D ( F t ) = a + bF t + cF t , (2.4) a , b , c are the coefficients of the value function to be determined. It is to be noted that the valuefunction needs to be concave to have a unique maximum which is satisfied if c is negative. Theorem Under the linear quadratic framework, for the existence of unique positive solution µ t for all t it is necessary that R Dt ) < B D < min(1 , ( Z t +(1+ R ∗ t ) V t ( (cid:15) t )) (1+ R Dt ) [(3 / Z t (1+ R ∗ t ) V t ( (cid:15) t )+((1+ R ∗ t ) V t ( (cid:15) t )) ] ) whenever Z t = (1 + R ∗ t ) E t ( (cid:15) t ) − (1 + R Dt ) > . Further, for γ ∈ ( γ , γ ) , for some < γ < γ , thesolution µ t ∈ (0 , exists for all t and is given by µ ∗ t = Z t [(1 + B D b ) + 2 B D cF t (1 + R Dt ) − B D cr Dtk ] F t [ γ (1 + R ∗ t ) V t ( (cid:15) t ) − B D c ( Z t + (1 + R ∗ t ) V t ( (cid:15) t ))] (2.5) with r Dtk = (1 + r t ) K Dt − K Dt +1 , where b and c are given in (5.11) and (5.8) . Value function can be approximated by Kalman filtering techniques, but for simplicity a quadratic value functionis assumed to solve the Bellman equation. µ increases from zero to positive values utility from out of lending increases but themarginal utility of lending falls with rising γ (i.e., rising perception of risk) and at a high enough γ the maxima of the utility function achieves at some µ <
1. Thus for a high enough γ , µ is boundedat unity from above. We also assume competitive banking in the developing country with total number of banks m U each of which is symmetric. Number of firms which borrow from each bank are normalized atunity. A typical bank raises a fraction λ t from the international market at an interest rate R ∗ t andthe rest from the domestic market at an interest rate r ut . These funds are lent to domestic sectorat an interest rates R ut with a risk of default arising out of productivity shock denoted by η t . Atypical bank is assumed to hedge against currency fluctuations by buying a currency forward. Letus define e t = spot exchange rate at t , e ft = forward market exchange rate, P ft = price of theforward contract at t . G t = funds available to each bank at t (accumulated from past profits andnew deposit raised) and is given by G t = η t − (1 + R Ut − ) G t − − K Ut − [(1 − λ t − )(1 + r Ut − ) + λ t − (1 + R ∗ t − ) E t ( e t e t − )] + K Ut . The profit of each bank in the developing country is thus given by π Ut = η t (1 + R Ut ) G t − K Ut [(1 − λ t )(1 + r Ut ) + λ t (1 + R ∗ t )( e t +1 e t )]+(1 + R Ut ) G t ( e t +1 e ft − − P ft , (2.6)When foreign exchange markets are either informationally efficient or whenever risk premium is anon decreasing function of the level of transaction, forward contracts are priced in such a way thatmarginal benefit from the contract equals marginal cost of the contract. The last two terms in (2.6)vanish. Hence, the instantaneous mean variance utility function is given byΩ Ut = E t ( π Ut ) − βV t ( π Ut )2= E t ( η t )(1 + R Ut ) G t − K Ut [(1 − λ t )(1 + r Ut ) + λ t (1 + R ∗ t ) E t ( e t +1 e t )] − β R Ut ) G t V t ( η t ) + ( K Ut ) λ t (1 + R ∗ t ) V t ( e t +1 e t )] . The problem of the bank in the Bellman framework is given by V U ( G t ) = max λ t { Ω Dt + B U E t ( V U ( G t +1 ) } . (2.7)subject to fund constraint (2.2) where V U ( G t )is the value function assumed to be stationary and B U is the discount factor. As in the case of developed country the Bellman principle can be appliedhere on the quadratic value function to solve the dynamic programming problem. Thus, V U ( G t ) = x + yG t + zG t , (2.8)with x, y, z to be determined by equating coefficients. Again, the parameter z has to be negative for the same reason as explained earlier. heorem Under the linear quadratic framework for the existence of unique positive solution λ t for all t it is necessary that R Ut ) V t ( η t )+(1+ R Ut ) ( E t ( η t )) < B U < R Ut ) [ V t ( η t )+( E t ( η t )) ] , whenever A t = (1 + r Ut ) − (1 + R ∗ t ) E t ( e t +1 e t ) > . Further, there exists β s neither very large nor very small(in an interval, ( β , β ) , for some < β < β ) such the solution λ t ∈ (0 , exists for all t and isgiven by λ ∗ t = A t [(1 + B U y ) + 2 B U z (1 + R Ut ) E t ( η t ) G t − B U zr Utk ] K Ut [(1 + R ∗ t ) V t ( e t +1 e t )( β − B U z ) − B U zA t ] (2.9) with r Utk = K Ut (1 + r Ut ) − K Ut +1 where y and z are given in (6.17) and (6.5) . The proof of this theorem is relegated to Appendix B.It may be noted that for a small value of β (implying low perception of risk) allows the developingcountry banks to borrow larger amount, possibly even λ >
1, as long as R ∗ t remains lower than thelevel given by A t >
0. On the other hand, if β is too high then λ can become zero or negative.Thus for a solution of λ ∈ (0 ,
1) one must choose a β ∈ ( β , β ), for some 0 < β < β . There are two markets in this model, viz. international loan market and the foreign exchangemarket. Total supply of loans in the international market is the sum total of loans by the developedcountry banks and total demand for international loans is the sum total of demand by individualbanks in the developing country. In equilibrium aggregate demand equals aggregate supply at each t so that m D µ ∗ t F t = m U λ ∗ t K Ut e t . (2.10)The R.H.S is aggregate demand and the L.H.S. is aggregate supply of loans. It may be noted thatdeveloping country banks cannot raise loans from the international market at their home currency,it has to be in hard currency, such as US Dollar or EURO. The equilibrium in the foreign exchangemarket is obtained by the equality of the demand for foreign exchange comprising of repayment ofloans of previous period and the supply comprising of net exports plus current period loan. Thecurrent net exports is assumed to be a linear increasing function of current period exchange rate, N t = − N + N e t (2.11)where N equals relative price of foreign goods (imports) vis-`a-vis home goods (exports), with N > N , N following uniform distribution in the intervals ( N , N ) and ( N , N )respectively and N >
0. Stochastic coefficients of the net export function adds dynamics to themodel. It is clear from (2.11) that net exports is actually rising function of price of foreign goodsexpressed in home currency relative to price of home goods. It may be noted that at zero exchange6ate or a very low price of foreign goods relative to home goods there is a negative net exports, i.e.there is positive imports. The equilibrium in the foreign exchange market is given by − N + N e t + m U λ ∗ t K Ut e t = m U (1 + R ∗ t − ) λ ∗ t − K Ut − e ∗ t − . (2.12)The LHS gives the supply of foreign exchange while the RHS the demand for foreign exchange.Equations (2.10), (2.12) (with substitutions from (2.5), (2.9)) simultaneously determine, R ∗ t and e ∗ t at each t corresponding to realizations of random shocks in the respective periods. Accordinglywe get the dynamics of capital inflow, interest rate and exchange rate. Existence of equilibriumsolutions are given by the following theorem. Theorem Under the assumptions of Theorems 1 and 2 there exist unique positive solutions to (2.10) and (2.12) provided N > N m U K Ut (2.13)For the proof of this theorem please see Appendix C.The value function in each case is assumed to be stationary. If c < z < c < B D (1 + R Dt ) > B D (1 + R Dt ) < ( Z t +(1+ R ∗ t ) V t ( (cid:15) t )) [(3 / Z t (1+ R ∗ t ) V t ( (cid:15) t )+((1+ R ∗ t ) V t ( (cid:15) t )) and 0 < B D < B D is small enough to make the product less thanunity. Similarly z < < B U (1 + R Ut ) E t ( η ) < R ∗ t ) E t ( (cid:15) t ) > (1 + R Dt ) is also a necessary condition for µ ∗ t > . The developing country bank borrows only when the international interest rate adjusted for theexchange rate is less than the domestic interest rate i.e.,(1 + R ∗ t ) E t ( e t +1 e t ) < (1 + r Ut ) is also a necessary condition for λ ∗ t > . Combining these two conditions we have(1 + R Dt ) E t ( (cid:15) t ) < (1 + R ∗ t ) < (1 + r Ut ) E t ( e t +1 e t ) . (2.14)In the next section, the endogenous variables, viz. R ∗ t , e t and capital inflow are numerically solvedfor given initial values and parameters of the model and explain how the equilibria behave overtime. 7 .4 Simulation The two equations, (2.10) and (2.12)solve for equilibrium values of R ∗ t and µ t for a given parametricconfiguration and exogenous variables. We run the simulation for 30 time periods. It may be notedthat the algorithm used for finding the equilibrium values of the endogenous variables employs aversion of E-M Algorithm (Estimation Maximization employed in Maximum Likelihood Estimationin econometrics) in a dynamic setup. Given below the initial values chosen for two countries andvalues for parameters of the model equilibrium values are calculated using the algorithm providedin Appendix D. We have set B D = 0 . , B U = 0 . G = 10 , K U = 20 and F = 10 , K D = 10. Inthe entire analysis we have assumed that the new deposits raised K Ut and K Dt are constant overtime. Table 1: Initial values of stochastic parameters E ( e t +1 e t ) E ( (cid:15) t ) V ( (cid:15) t ) V ( e t +1 e t ) E ( η t ) V ( η t ) N N N N γ β m D m U K D K U R D r D R U r U R ∗ e G F R ∗ t , e ∗ t and total capital inflow (international borrowing/ lending) for 30 timeperiod obtained via simulation are provided in Figs. 1 through 3 below.Figure 1: Capital Inflow8igure 2: e t Figure 3: R ∗ t It is evident from the above figures that capital inflow and exchange rate follow stationary patternwith a shift in the trajectory of the former between the period between 15 and 20. The internationalinterest rate shows a rising trend in a small band. This is because of the fact that a static expectationof next period exchange rate with a value less than unity implies banks in the developing countryfinds it optimal to borrow from the international market. The ever increasing demand for loansraises the interest rate over time. In the process the role of dynamic terms of trade operates via(2.12). If there is an adverse supply shock in the foreign exchange market (due to, say a deficitin the balance of trade) then the second component on the L.H.S. of (2.12) has to fall to matchthe given value on the R.H.S. This interacts with the R.H.S. of (2.10) to achieve equilibrium inthe foreign exchange market. The effect of current period exchange rate becomes more pronouncedbecause the expectation of next period exchange rate is static - invariant to current information set9bout the future. Eventually in the final equilibrium interest rate effect in each period dominatesso that aggregate capital inflow to the developing country hovers around a stationary path. Thefluctuation around the stationary path is governed by the stochastic shocks to net exports. Inthis model balance of trade has a dominant role in the determination of the intertemporal path offoreign capital inflow. As a matter of fact the trajectory of capital inflow is almost a mirror imageof the trajectory of the exchange rate.In Table 3 below we provide the mean and variance of the endogenous variables.Table 3: Mean and variance of the endogenous variablesVariable Mean SD Coefficient of Variation R ∗ t e t . . We consider changes in some of the parameter values of interest for a better understanding of theeconomics of foreign capital inflow with emphasis on the primary focus of the model. It may benoted that for the comparative dynamic analysis we have retained the same realised values of shockto the net export function to make comparisons meaningful. The set of baseline trajectory of theendogenous variables are plotted in black while the new set are plotted in red throughout thispaper. γ An increase in γ implies that banks in the developed country perceive a higher risk for a givenlevel of expected profit (see equation (2.2)). In this particular exercise γ is raised from 4 to 15. If the expectation formation is in tune with rational expectations hypothesis then current exchange rate alsoreflects expected exchange rate provided the foreign exchange market is efficient. In such a situation dynamic termsof trade effect may not be as strong as observed here. R ∗ t . In the final equilibrium capital inflow to the developingcountry decreases and international rate increases for each t .Figure 4: Effect of increased γ on capital inflow (Bottom plot for higher γ )Figure 5: Effect of increased γ on R ∗ t (Top plot for higher γ )11igure 6: Effect of increased γ on e t (difference with baseline)Figs. 4 and 5 depict the new intertemporal trajectory (in red) of the capital inflow and theinternational interest rate along with the baseline (in black). The trajectory of capital inflowshows a downward shift from the original trajectory while the international interest rate shifts up.However, there is very little change in the exchange rate. To get a clear picture we plotted thedifference of the new exchange rate from its baseline for each t and plotted them in Fig. 6. Thetrajectory follows a stationary path around zero within a band from -0.027 to 0.02. There is noappreciable change in the exchange rate because the shock does not affect the net exports and sothe balance of trade which as we argued earlier has a dominating role in the determination of theequilibrium trajectories.Table 4: Mean and variance of variables-increased γ Variable Mean SD Coefficient of Variation R ∗ t e t . . R ∗ t . However, the percentage fall in foreign capital inflow 16.12% is more thanthe rise in R ∗ t γ is 275% from its initial value. The rate ofchange in the average capital inflow with respect to γ is -0.0447 while the rate of change of R ∗ t is0.00014. The elasticity of capital inflow and interest rate with respect to γ are -0.0379 and 0.0014(evaluated at sample averages). Clearly the sensitivity of capital inflow to risk perception is muchhigher than the sensitivity of interest rate. 12 .2 Effect of a rise in E t ( e t +1 e t ) : Expected depreciation of exchange rate When banks in the developing country (borrowers in the developing country in general) expect adepreciation of exchange rate in the future, meaning an increase in the expected exchange rate vis-`a-vis current rate, E t ( e t +1 e t ), cost of borrowing (in terms of foreign currency) increases. This changein expectation leads to a decrease in the demand for foreign borrowing which in turn reduces R ∗ t in the final equilibrium in each period. E t ( e t +1 e t ) is raised from 0.92 to 0.98 (6.52% rise), withno change in the other parameter values. Comparing the new intertemporal trajectory with thebaseline trajectory we find that the capital inflow attains a low level trajectory though the natureof the time path remains similar to the baseline trajectory.Figure 7: Effect of depreciation of E ( e t +1 e t ) on capital inflow (Bottom plot for higher E ( e t +1 e t ))Figure 8: Effect of depreciation of E ( e t +1 e t ) on e t (difference from baseline)13igure 9: Effect of depreciation of E ( e t +1 e t ) on R ∗ t (Bottom plot for higher E ( e t +1 e t ))Table 5: Mean and variance of variables - Increased E ( e t +1 e t )Variable Mean SD Coefficient of Variation R ∗ t e t R ∗ t decreases by 0.76% on an average. The rate ofchange of capital inflow with respect to expected exchange rate is -33.53 which is much higher thanthe rate of change of interest rate (-0.0183). The elasticities of capital inflow and interest rate withrespect to expected exchange rate are found to be -15.57 and -0.121. The current exchange ratedoes not change to any appreciable level. Difference of the current exchange rate from the baselineplotted in Fig. 8 shows that the range of variation is not significant (from − .
14 to 0 . E t ( η ) When there is a decrease in the productivity of the domestic sector of the developing country, E t ( η t ), profit in the production sector falls which in turn leads to decreased demand for foreignloans by the developing countries. We conduct this exercise by reducing E t ( η t ) reduced from 0.85to 0.70 (roughly 17.64% decrease). Given below the plots of endogenous variables with the new E t ( η t ). 14igure 10: Effect of a lower E t ( η ) on Capital Inflow (Bottom plot for lower E t ( η ))Figure 11: Effect of a lower E t ( η ) on R ∗ t (Bottom plot for lower E t ( η ))15igure 12: Effect of a lower E t ( η ) on e t (in difference)From the plot it is seen that the whole trajectory of R ∗ t as well as the trajectory of the capitalinflow shift below. Comparing the averages with that of the baseline (see Table 6) it is evident thatthere is no appreciable change in e t while there is fall in both the capital inflow and R ∗ t by 6.73%and 0.004% respectively. Thus the fall capital inflow is much higher and the fall in R ∗ t is a slightlylower compared to the decrease in productivity, E ( η t ). The rate of change of capital inflow withrespect to E ( η t ) is 6.7 while that for R ∗ t is 0.004 and the corresponding elasticities are 2.05 and0.022 respectively. Table 6: Mean and variance of variables-shock to E t ( η )Variable Mean SD Coefficient of Variation R ∗ t e t The study addresses the economics of foreign capital inflow based on trade theoretic explanationoperating via terms of trade effect in a dynamic context. The model is set up and solved in a twocountry group framework - developing or borrowing group and developed or lending group. Eachindividual agent in both the country groups chooses her respective loan portfolio for each period byoptimization of an intertemporal objective function of mean-variance variety. The equilibrium in theloan market together with the foreign exchange market solves the endogenous variables of the modelusing numerical methods. Net exports which constitutes the supply side of the foreign exchangemarket dominates intertemporal trajectory of endogenous variables. Simulation exercises are thenconducted to derive comparative dynamic results to assess the impact of shocks to parameters ofthe model on the intertemporal trajectory of the endogenous variables, viz. international interestrate, foreign capital inflow and exchange rate.Given the structure of the model a change in the risk perception in the developed country reducescapital inflow to the developing country group as a whole. An increase in the expected exchangerate vis-`a-vis the current exchange rate raises the effective cost of foreign loans leading to a decreasein demand which in the final equilibrium also reduces the international interest rate. A fall in the17roductivity in the domestic sector of the developing country leads to a fall in foreign capital inflowand a fall in the international interest rate. However, the responsiveness with respect to expectedexchange rate vis-`a-vis current exchange rate is much more compared to the other comparativedynamic analyses considered here.The paper can be extended in several directions. A restrictive assumption of the model is staticexpectation of exchange rate. An extension with endogenous determination of expected exchangerate is capable to add richer dynamics to the model. Another interesting extension could be theendogeneity of risk of default that is dependent on the outstanding borrowing of the developingcountries. Finally, a multi country extension within the borrowing country group can be undertakento show the contagion effect when a shock originating in one of the borrowing countries spreads tothe other members of the group.
Proof of Theorem 1:
From (2.3) a + bF t + cF t = max µ t { Ω Dt + B D E t ( V D ( F t +1 ) } , where Ω Dt = F t (1 − µ t )(1 + R Dt ) + F t µ t (1 + R ∗ t ) E t ( (cid:15) t ) − ( γ/ F t µ t (1 + R ∗ ) t V t ( (cid:15) t )and E t ( F t +1 ) = F t [(1 − µ t )(1 + R Dt ) + µ t (1 + R ∗ t ) E t ( (cid:15) t )] − (1 + r Dt ) K Dt + K Dt +1 . To solve this, denote M Dt = Ω Dt + B D [ a + bE t ( F t +1 ) + cE t ( F t +1 )], then the value function takes theform, V D ( F t ) = ma x µ t { M Dt } . First order condition to the above equation yields ∂M Dt ∂µ ∗ t | µ ∗ t = 0 . (5.1)Now, ∂ Ω Dt ∂µ ∗ t = − F t (1 + R Dt ) + F t (1 + R ∗ t ) E t ( (cid:15) t ) − γF t µ t (1 + R ∗ t ) V t ( (cid:15) t ) . ⇒ ∂E t ( F t +1 ) ∂µ ∗ t = − F t (1 + R Dt ) + F t (1 + R ∗ t ) E t ( (cid:15) t ) . ⇒ E t ( F t +1 ) = V t ( F t +1 ) + ( E t ( F t +1 )) . Now, ∂ ( E t ( F t +1 )) ∂µ ∗ t = 2 E t ( F t +1 ) ∂E t ( F t +1 ) ∂µ ∗ t and, V t ( F t +1 ) = µ t (1 + R ∗ t ) F t V t ( (cid:15) t ) . ∂V t ( F t +1 ) ∂µ ∗ t = 2 µ t (1 + R ∗ t ) F t V t ( (cid:15) t ) ⇒ ∂M Dt ∂µ ∗ t | µ ∗ t = ∂ Ω Dt ∂µ ∗ t + B D b ∂E t ( F t +1 ) ∂µ ∗ t + B D c ∂V t ( F t +1 ) ∂µ ∗ t + B D c ∂ ( E t ( F t +1 )) ∂µ ∗ t . This implies,0 = − F t (1 + R Dt ) + F t (1 + R ∗ t ) E t ( (cid:15) t ) − γF t µ t (1 + R ∗ t ) V t ( (cid:15) t )+ B D b [ − F t (1 + R Dt ) + F t (1 + R ∗ t ) E t ( (cid:15) t )] + B D c [2 µ t (1 + R ∗ t ) F t V t ( (cid:15) t )]+2 B D c [ F t [(1 − µ t )(1 + R Dt ) + µ t (1 + R ∗ t ) E t ( (cid:15) t )] − r Dtk ] × [ − F t (1 + R Dt ) + F t (1 + R ∗ t ) E t ( (cid:15) t )] . (5.2)Rearranging, µ ∗ t = F t Z t [(1 + B D b ) + 2 B D cF t (1 + R Dt ) − B D cr Dtk ] F t [ γ (1 + R ∗ t ) V t ( (cid:15) t ) − B D c ( Z t + (1 + R ∗ t ) V t ( (cid:15) t ))]= Z t [(1 + B D b ) + 2 B D cF t (1 + R Dt ) − B D cr Dtk ] F t [ γ (1 + R ∗ t ) V t ( (cid:15) t ) − B D c ( Z t + (1 + R ∗ t ) V t ( (cid:15) t ))] (5.3)with Z t = [(1 + R ∗ t ) E t ( (cid:15) t ) − (1 + R Dt )] (5.4)and r Dtk = (1 + r t ) K Dt − K Dt +1 , (5.5)where b and c are determined as given below. Let µ ∗ t be the optimum portfolio, then the optimumportfolio satisfies a + bF t + cF t = Ω Dt ( F t , µ ∗ t )+ B D E t ( V ( F t +1 )( µ ∗ t )) from equation (2.3) This equationholds for every F t , so a , b , c are determined by comparing coefficients of F t , F t and constant onboth the sides of the equation. This implies a + bF t + cF t = F t [(1 − µ ∗ t )(1 + R Dt ) + µ ∗ t (1 + R ∗ t ) E t ( (cid:15) t )] − γ F t µ ∗ t (1 + R ∗ t ) V t ( (cid:15) ) + B D a + B D b [ F t [(1 − µ ∗ t )(1 + R Dt ) + µ ∗ t (1 + R ∗ t ) E t ( (cid:15) t )] − (1 + r Dt ) K Dt + K Dt +1 ]+ B D c [( E t ( F t +1 )) + V t ( F t +1 )] . (5.6)This is so because E t ( F t +1 ) = ( E t ( F t +1 )) + V t ( F t +1 ) Comparing the coefficients of F t on both thesides of the equation, the following equality is obtained: c = (1 + R ∗ t ) V t ( (cid:15) t ) (cid:18) B D c (1 + R Dt ) Z t µ denom (cid:19) ( − ( γ/
2) + B D c ) + (1 + R D ) B D c + (2 B D c (1 + R Dt ) Z t ) µ denom , (5.7)19here µ denom = ( γ − B D c )(1 + R ∗ t ) V t ( (cid:15) t ) − B D cZ t . This yields, µ denom c (1 − B D (1 + R Dt ) ) = (2 B D c (1 + R Dt ) Z t ) [ µ denom − (1 / γ − B D c )(1 + R ∗ t ) V t ( (cid:15) t )] ⇒ ( A − cE ) (1 − T ) = 2 cZ t B D T ( A − cE ) ⇒ ( as c (cid:54) = 0 , otherwise the value function would be linear( A + c E − cAE )(1 − T ) = 2 cZ t B D T ( A − cE ) ⇒ c ( E (1 − T ) + 2 Z t B D T E ) − c ( AE (1 − T ) + Z t B D T A ) + A (1 − T )(5.8)where A = γ (1 + R ∗ t ) V t ( (cid:15) t ), E = 2 B D ( Z t + (1 + R ∗ t ) V t ( (cid:15) t )) = 2 B D ( Z t + Aγ ), T = B D (1 + R Dt ) and E = 2 B D ( Z t + A γ ). Hence, Note, since A , E and T all are positive, the coefficient of c in (5.8)is negative. Therefore, if 0 < T ≤
1, then all solution of c are positive real, as the discriminantis positive, since for 0 < T ≤ A E (1 − T ) = A E (1 − T ) , 2 B D Z t A ET (1 − T ) ≥ B D Z t A E T (1 − T ) and ( B D ) Z t A T >
0. Hence, the necessary condition is
T >
1. But thenthe discriminant may not always be positive as the 2nd condition may not always hold and thecoefficient of c may also be negative. So, the main idea would be to choose B D , such that T > c is negative, but the coefficient of c is positive. For T >
1, when coefficient of c is positive then there exists exactly one solution of c which is negative. Therefore c < B D (1 + R Dt ) > R Dt ) ( Z t + (1 + R ∗ t ) V t ( (cid:15) t )) > T [( Z t + (1 + R ∗ t ) V t ( (cid:15) t )) − Z t ( Z t + (1 / R ∗ t ) V t ( (cid:15) t )) ]= T [(3 / Z t (1 + R ∗ t ) V t ( (cid:15) t ) + ((1 + R ∗ t ) V t ( (cid:15) t )) ⇔ B D (1 + R Dt ) < ( Z t + (1 + R ∗ t ) V t ( (cid:15) t )) (3 / Z t (1 + R ∗ t ) V t ( (cid:15) t ) + ((1 + R ∗ t ) V t ( (cid:15) t )) . (5.9)20or the coefficient of F t , b = 2 B D c Z t (1 + R Dt ) µ denom + (1 + R Dt ) − γ R ∗ t ) V t ( (cid:15) t )4 B D c (1 + R Dt ) Z t µ denom [(1 + B D b ) − B D cr Dtk ]+ B D b [(1 + R D ) + Z t µ denom B D c (1 + R Dt )] + B D c [ 2 Z t (1 + R Dt ) µ denom [(1 + B D b ) − B D cr Dtk ]+ 2 Z t (1 + R Dt ) µ denom B D c [(1 + B D b ) − B D cr Dtk ] − r Dtk [(1 + R Dt ) + 2 B D c Z t (1 + R Dt ) µ denom ]]+ B D c (1 + R ∗ t ) V t ( (cid:15) t )4 B D c (1 + R Dt ) Z t µ denom [(1 + B D b ) − B D cr Dtk ]= (1 + R Dt )(1 + 2 B D c Z t µ denom )[(1 + B D b ) − B D cr Dtk ]+2 B D c (1 + R Dt ) Z t µ denom (1 + 2 B D c Z t µ denom )[(1 + B D b ) − B D cr Dtk ]+(1 + R ∗ t ) V t ( (cid:15) t )4 B D c Z t µ denom )[(1 + B D b ) − B D cr Dtk ][ B D c − γ B D b ) − B D cr Dtk ](1 + R Dt )(1 + 2 B D c Z t µ denom )(1 + 2 B D c Z t µ denom )+[(1 + B D b ) − B D cr Dtk ](1 + R ∗ t ) V t ( (cid:15) t )4 B D c (1 + R Dt ) Z t µ denom )[ B D c − γ B D b ) − B D cr Dtk ][(1 + R Dt )(1 + 2 B D c Z t µ denom ) − ( γ − B D c )(1 + R ∗ t ) V t ( (cid:15) t )4 B D c (1 + R Dt ) Z t µ denom ] . (5.10)This implies, b (1 − B D L ) = (1 − B D cr Dtk ) L ⇒ b = (1 − B D cr Dtk ) L − B D L , (5.11)where L = (1 + R Dt ) (cid:18) B D cZ t µ denom (cid:19) − ( γ − B D c )(1 + R ∗ t ) V t ( (cid:15) t )(1 + R Dt ) 2 B D cZ t µ denom = (1 + R Dt ) (cid:18) − − B D cZ t µ denom (cid:19) + (1 + R Dt ) ( γ − B D c )(1 + R ∗ t ) V t ( (cid:15) t ) µ denom − B D cZ t µ denom = (1 + R Dt ) (cid:18) − − B D cZ t µ denom (cid:19) + (1 + R Dt ) (cid:18) − − B D cZ t µ denom (cid:19) − B D cZ t µ denom , (5.12)since µ denom − ( γ − B D c )(1 + R ∗ t ) V t ( (cid:15) t ) = − B D cZ t . Hence, L = (1 + R Dt ) (cid:18) − − B D cZ t µ denom (cid:19) (cid:20) − − B D cZ t µ denom + − B D cZ t µ denom (cid:21) = (1 + R Dt ) (cid:18) − − B D cZ t µ denom (cid:19) . (5.13)This immediately implies that B D L <
1, as z is negative and − B D cZ t µ denom <
1. Also, note1 + B D b = (1 − B D cr Dtk ( B D L ))1 − B D L . µ ∗ t may be found from the condition as given below. µ ∗ t > ⇔ − B D cr Dtk ( B D L )1 − B D L + 2 B D c [ F t (1 + R Dt ) − r Dtk ] > ⇔ − B D cr Dtk ( B D L ) + 2 B D c (1 − B D L )[ F t (1 + R Dt ) − r Dtk ] > ⇔ − B D cr Dtk + 2 B D c (1 − B D L )[ F t (1 + R Dt )] > ⇔ B D c [ r Dtk − (1 − B D L )[ F t (1 + R Dt )]] < . (5.14)If r Dtk > (1 − B D L )[ F t (1 + R Dt )], then it always holds as c <
0. This should be the case, as it means F t , is tool small compared to r Dtk . On the other hand, if r Dtk < (1 − B D L )[ F t (1 + R Dt )], which isoften the case, then there exists γ > γ < γ would imply the last condition, as c islinear in γ . This means if the loan do not have too high a risk premium then µ ∗ t > µ ∗ t < µ ∗ t < ⇔ Z t (cid:18) − B D cr Dtk ( B D L )1 − B D L + 2 B D c [ F t (1 + R Dt ) − r Dtk ] (cid:19) < µ denom F t ⇔ Z t (cid:0) (1 − B D cr Dtk ( B D L )) + 2 B D c (1 − B D L )[ F t (1 + R Dt ) − r Dtk ] (cid:1) < µ denom F t (1 − B D L ) ⇔ Z t (cid:0) (1 − B D cr Dtk ) + 2 B D c (1 − B D L )[ F t (1 + R Dt )] (cid:1) < µ denom F t (1 − B D L ) ⇔ Z t < F t (1 − B D L ) µ denom + 2 B D cZ t [ r Dtk − (1 − B D L ) F t (1 + R Dt )] . (5.15)Since µ denom is linear in γ , as c is, from (5.14) it is clear that there exists 0 < γ < γ , such thatfor γ > γ the last condtion holds. This means if the riskpremium is to low then there is always apossibility of over-subscription of lending.Hence the proof of existence of µ ∗ ∈ (0 ,
1) for γ ∈ ( γ , γ ). Upper bound γ may be infinity if F t remains too small. Proof of Theorem 2:
To solve the maximization problem, denote M Ut = Ω Ut + B U E t ( V U ( G t +1 )) . Then V U ( G t ) = max λ t M Ut . First order condition for maximization implies, ∂M Ut ∂λ t | λ t = λ ∗ t = 0 (6.1)Now, ∂ Ω Ut ∂λ t = K Ut A t + βλ t K U t (1 + R ∗ t ) V t ( e t +1 e t )where, A t = (1 + r Ut ) − (1 + R ∗ t ) E t ( e t +1 e t ) . E t ( G t +1 ) = E t ( η t )(1 + R Ut ) G t − K Ut [(1 − λ t )(1 + r Ut ) + λ t (1 + R ∗ t ) E t ( e t +1 e t )] + K Ut +1 . So, ∂E t ( G t +1 ) ∂λ t = K Ut A t , and, E t ( G t +1 ) = ( E t ( G t +1 )) + V t ( G t +1 ).Now, ∂ ( E t ( G t +1 ) ∂λ t = 2 E t ( G t +1 ) ∂E t ( G t +1 ∂λ t ⇒ V t ( G t +1 ) = K U t λ t (1 + R ∗ t ) V t ( e t +1 e t ) + V t ( e (cid:82) t θ u du ta t ) G t (1 + R Ut ) . Thus ∂V t ( G t +1 ) ∂λ t = 2 K U t λ t (1 + R ∗ t ) V t ( e t +1 e t ) . ⇒ ∂M Ut ∂λ t = ∂ Ω Ut ∂λ t + B U y ∂E t ( G t +1 ) ∂λ t + B U z ∂V t ( G t +1 ) ∂λ t + B U z ∂ ( E t ( G t +1 )) ∂λ t . This implies,0 = K Ut A t − βλ t K U t (1 + R ∗ t ) V t ( e t +1 e t ) + B U y ( A t K Ut ) + B U z [2 K U t λ t (1 + R ∗ t ) V t ( e t +1 e t )]+2 B U z ( A t K Ut )[( E t ( η t )(1 + R Ut ) G t − K Ut [(1 − λ t )(1 + r Ut ) + λ t (1 + R ∗ t ) E t ( e t +1 e t )] + K Ut +1 ] . Rearranging, λ ∗ t = K Ut A t [(1 + B U y ) + 2 B U z (1 + R Ut ) E t ( η t ) G t − B U zr Utk ] K U t [(1 + R ∗ t ) V t ( e t +1 e t )( β − B U z ) − B U zA t ]= A t [(1 + B U y ) + 2 B U z (1 + R Ut ) E t ( η t ) G t − B U zr Utk ] K Ut [(1 + R ∗ t ) V t ( e t +1 e t )( β − B U z ) − B U zA t ] (6.2)where r Utk = K Ut (1 + r Ut ) − K Ut +1 x, y, z are determined as in below. If λ ∗ t is optimal then x + yG t + zG t = Ω Ut ( G t , λ ∗ t ) + B U E t ( V U ( G t +1 )) . (6.3) x, y, z are obtained by comparing the coefficients of G t , G t and constants on both the sides of theequation. Substituting for value function E t ( V U ( G t +1 )) we get z = − β (1 + R Ut ) V t ( η t ) + B U z (coeffof G t in ( E t ( G t +1 )) + B U z (coeff of G t in V t ( G t +1 )), i.e.,for the coefficient of G t , z = − β R Ut ) V t ( η t ) + (1 + R ∗ t ) V t ( e t +1 e t ) A t λ denom B U ) z (1 + R Ut ) ( E t ( η t )) ]+ B U z [(1 + R Ut ) ( E t ( η t )) + A t λ denom B U ) z (1 + R Ut ) ( E t ( η t )) + A t λ denom B U z (1 + R Ut ) ( E t ( η t )) ]+ B U z [(1 + R Ut ) V t ( η t ) + (1 + R ∗ t ) V t ( e t +1 e t ) A t λ denom B U ) z (1 + R Ut ) ( E t ( η t )) ] , (6.4)23here λ denom = [(1 + R ∗ t ) V t ( e t +1 e t )( β − B U z ) − B U zA t ] . This implies, zλ denom = λ denom (1 + R Ut ) V t ( η t )[ B U z − β R ∗ t ) V t ( e t +1 e t ) A t B U ) z (1 + R Ut ) ( E t ( η t )) [ B U z − β B U z (1 + R Ut ) ( E t ( η t )) [ λ denom + A t B U ) z + λ denom A t B U z ] . (6.5)Note that, if we bring everything to the left and equate it to zero, we obtain, coefficient of z as,(2 B U ) H − (2 B U ) H B U (1 + R Ut ) V t ( η t ) − (2 B U ) (1 + R ∗ t ) V t ( e t +1 e t ) A t (1 + R Ut ) ( E t ( η t )) B U − (2 B U ) (1 + R Ut ) ( E t ( η t )) H B U − (2 B U ) A t (1 + R Ut ) ( E t ( η t )) B U − (2 B U ) A t (1 + R Ut ) ( E t ( η t )) ( − B U ) H = (2 B U ) H [1 − B U (1 + R Ut ) V t ( η t ) − B U (1 + R Ut ) ( E t ( η t )) ] + (2 B U ) A t (1 + R Ut ) ( E t ( η t )) B U H, (6.6)where H = ( A t +(1+ R ∗ t ) V t ( e t +1 e t )). This coefficient is positive if B U (1+ R Ut ) [ V t ( η t )+( E t ( η t )) ] < B U (1 + R Ut ) E t ( η t ) < z , in the similar expression as above is, β R Ut ) V t ( η t )[(1 + R ∗ t ) V t ( e t +1 e t ) β ] , which is positive. Since the coefficient of z and z are both positive it guarantees a negativesolution of z .If we can get a condition which gives the coefficient of z to be negative then the Descartes’ rule ofsign says there is at most one negative root.Now the coefficient of z is,[(1 + R ∗ t ) V t ( e t +1 e t ) β ] − B U (1 + R Ut ) V t ( η t )[(1 + R ∗ t ) V t ( e t +1 e t ) β ] +( β/ R Ut ) V t ( η t )( − B U ) H [(1 + R ∗ t ) V t ( e t +1 e t ) β ] − B U (1 + R Ut ) ( E t ( η t )) [(1 + R ∗ t ) V t ( e t +1 e t ) β ] = [(1 + R ∗ t ) V t ( e t +1 e t ) β ] [1 − B U (1 + R Ut ) V t ( η t ) − B U (1 + R Ut ) ( E t ( η t )) ] − β B U (1 + R Ut ) V t ( η t ) A t (1 + R ∗ t ) V t ( e t +1 e t ) . (6.7)This expression is negative if,3 B U (1 + R Ut ) V t ( η t ) + B U (1 + R Ut ) ( E t ( η t )) > . (6.8)Thus there exists exactly one negative solution z if and only if13(1 + R Ut ) V t ( η t ) + (1 + R Ut ) ( E t ( η t )) < B U < R Ut ) [ V t ( η t ) + ( E t ( η t )) ] . (6.9)24or the coefficient of z , in a + a z + a z + a z = 0 is − B U H ( β (1 + R ∗ t ) V t ( e t +1 e t ) − B U (1 + R Ut ) [( E t ( η t )) + V t ( η t )][( − B U ) H ( β (1 + R ∗ t ) V t ( e t +1 e t )]+( β/ R Ut ) V t ( η t )( − B U ) H + ( β/ R Ut ) ( E t ( η t )) (1 + R ∗ t ) V t ( e t +1 e t ) A t B U ) − B U (1 + R Ut ) ( E t ( η t )) A t B U (1 + R ∗ t ) V t ( e t +1 e t ) β = − B U H ( β (1 + R ∗ t ) V t ( e t +1 e t ) + 4( B U ) (1 + R Ut ) E t ( η t ) H ( β (1 + R ∗ t ) V t ( e t +1 e t )+( β/ B U ) (1 + R Ut ) V t ( η t ) H − ( β/ B U ) (1 + R Ut ) ( E t ( η t )) (1 + R ∗ t ) V t ( e t +1 e t ) A t . (6.10)For the coefficient of G t , y = (1 + R Ut ) E t ( η t ) + A t λ denom B U z (1 + R Ut ) E t ( η t ) − β R ∗ t ) V t ( e t +1 e t )[ 2 A t λ denom B U z (1 + R Ut ) E t ( η t )[(1 + B U y ) − B U zr Utk ]]+ B U y [(1 + R Ut ) E t ( η t ) + A t λ denom B U z (1 + R Ut ) E t ( η t )]+ B U z [2(1 + R Ut ) E t ( η t )[ A t λ denom [(1 + B U y ) − B U zr Utk ] − r Utk ]+ A t λ denom B U z (1 + R Ut ) E t ( η t )[ A t λ denom [(1 + B U y ) − B U zr Utk ] − r Utk ]]+ B U z (1 + R ∗ t ) V t ( e t +1 e t )[ A t λ denom B U z (1 + R Ut ) E t ( η t )[(1 + B U y ) − B U zr Utk ]] . (6.11)This implies y = (1 + R Ut ) E t ( η t )[1 + 2 B U zA t λ denom ]+ B U y [(1 + R Ut ) E t ( η t )][1 + 2 B U zA t λ denom ]+ B U z [2(1 + R Ut ) E t ( η t )[ A t λ denom [(1 + B U y ) − B U zr Utk ] − r Utk ]][1 + 2 B U zA t λ denom ]+(1 + R ∗ t ) V t ( e t +1 e t )[ 2 B U zA t λ denom (1 + R Ut ) E t ( η t )[(1 + B U y ) − B U zr Utk ]][2 B U z − β ] . (6.12)25hus, y = (1 + R Ut ) E t ( η t )[1 + 2 B U zA t λ denom ] + B U y [(1 + R Ut ) E t ( η t )][1 + 2 B U zA t λ denom ]+ B U y [(1 + R Ut ) E t ( η t )[ 2 B U zA t λ denom ][1 + 2 B U zA t λ denom ]]+ B U z [2(1 + R Ut ) E t ( η t )[ A t λ denom [1 − B U zr Utk ] − r Utk ]][1 + 2 B U zA t λ denom ]+ B U y (1 + R ∗ t ) V t ( e t +1 e t )[ 2 B U zA t λ denom (1 + R Ut ) E t ( η t )][2 B U z − β ] . +(1 + R ∗ t ) V t ( e t +1 e t )[ 2 B U zA t λ denom (1 + R Ut ) E t ( η t )[1 − B U zr Utk ]][2 B U z − β ] . (6.13)Therefore, y = (1 + R Ut ) E t ( η t )[1 + 2 B U zA t λ denom ] + B U y [(1 + R Ut ) E t ( η t )][1 + 2 B U zA t λ denom ] + B U z [2(1 + R Ut ) E t ( η t )[ A t λ denom [1 − B U zr Utk ] − r Utk ]][1 + 2 B U zA t λ denom ]+ B U y (1 + R ∗ t ) V t ( e t +1 e t )[ 2 B U zA t λ denom (1 + R Ut ) E t ( η t )][2 B U z − β ] . +(1 + R ∗ t ) V t ( e t +1 e t )[ 2 B U zA t λ denom (1 + R Ut ) E t ( η t )[1 − B U zr Utk ]][2 B U z − β ] . (6.14)26ence, y [1 − J + J ] = (1 + R Ut ) E t ( η t )[1 + 2 B U zA t λ denom ]+ B U z [2(1 + R Ut ) E t ( η t )[ A t λ denom [1 − B U zr Utk ] − r Utk ]][1 + 2 B U zA t λ denom ]+(1 + R ∗ t ) V t ( e t +1 e t )[ 2 B U zA t λ denom (1 + R Ut ) E t ( η t )[1 − B U zr Utk ]][2 B U z − β ]= (1 + R Ut ) E t ( η t )[1 − − B U zA t λ denom ]+[(1 + R Ut ) E t ( η t )] 2 B U zA t λ denom [1 − B U zr Utk ][1 − − B U zA t λ denom ]+(1 + R Ut ) E t ( η t )( − B U zr Utk )[1 − − B U zA t λ denom ] − (1 + R Ut ) E t ( η t )[1 − B U zr Utk ][ ( β − B U z )(1 + R ∗ t ) V t ( e t +1 e t ) λ denom ][ 2 B U zA t λ denom ]= (1 + R Ut ) E t ( η t )[1 − − B U zA t λ denom ][1 − B U zr Utk ]+[(1 + R Ut ) E t ( η t )] 2 B U zA t λ denom [1 − B U zr Utk ][1 − − B U zA t λ denom ] − (1 + R Ut ) E t ( η t )[1 − B U zr Utk ][1 − − B U zA t λ denom ][ 2 B U zA t λ denom ]= (1 + R Ut ) E t ( η t )[1 − − B U zA t λ denom ][1 − B U zr Utk ] , (6.15)since λ denom − ( β − B U z )(1 + R ∗ t ) V t ( e t +1 e t ) = − B U zA t , where J = B U [(1 + R Ut ) E t ( η t )][1 + B U zA t λ denom ] , and J = B U (1 + R ∗ t ) V t ( e t +1 e t )[ B U zA t λ denom (1 + R Ut ) E t ( η t )][ β − B U z ].As in L (5.12) and (5.13) in the computation of y , the expression for J − J can be simplifiedfurther. Note, J − J = B U [(1 + R Ut ) E t ( η t )] (cid:34)(cid:18) − − B U zA t λ denom (cid:19) + ( β − B U z )(1 + R ∗ t ) V t ( e t +1 e t ) λ denom (cid:18) − B U zA t λ denom (cid:19)(cid:35) = B U [(1 + R Ut ) E t ( η t )] (cid:34)(cid:18) − − B U zA t λ denom (cid:19) + (cid:18) − − B U zA t λ denom (cid:19) (cid:18) − B U zA t λ denom (cid:19)(cid:35) = B U [(1 + R Ut ) E t ( η t )] (cid:20)(cid:18) − − B U zA t λ denom (cid:19) (cid:20)(cid:18) − − B U zA t λ denom (cid:19) + (cid:18) − B U zA t λ denom (cid:19)(cid:21)(cid:21) = B U [(1 + R Ut ) E t ( η t )] (cid:18) − − B U zA t λ denom (cid:19) = B U J, (6.16)27here J = [(1 + R Ut ) E t ( η t )] (cid:16) − − B U zA t λ denom (cid:17) . Thus, y = (1 + R Ut ) E t ( η t )[1 − − B U zA t λ denom ][1 − B U zr Utk ]1 − B U J = J (1 − B U zr Utk )(1 − B U J ) . (6.17)From this, y can be found immediately once the z is known.Now (6.17) further implies 1 + B U y = (1 − B U zr Utk ( B U J ))1 − B U J .
Positivity of the λ ∗ t may be found from the condition as given below: λ ∗ t > ⇔ − B U zr Utk ( B U J )1 − B D J + 2 B U z [ G t (1 + R Ut ) E ( η t ) − r Utk ] > ⇔ − B U zr Utk ( B U J ) + 2 B U z (1 − B D J )[ G t (1 + R Ut ) E ( η t ) − r Utk ] > ⇔ − B U zr Utk + 2 B U z (1 − B D J )[ G t (1 + R Ut ) E ( η t )] > ⇔ B U z [ r Utk − (1 − B D J )[ G t (1 + R Ut ) E ( η t )]] < . (6.18)If r Utk > (1 − B U J )[ G t (1 + R Ut ) E ( η t )], then it always holds as z <
0. This should be the case,as it means G t , is tool small compared to r Dtk (or perhaps negative). On the other hand, if r Utk < (1 − B U J )[ G t (1 + R Ut ) E ( η t )], then there exists β > β < β would imply the lastcondition, as z is linear in β . This means if the loan do not have too high a risk premium then λ ∗ t > λ denom is linear in β , as z is, from (6.18) it is clear that there exists 0 < β < β , such thatfor β > β the last condtion holds. This means if the risk premium is too low then there is alwaysa possibility of over-subscription of borrowing.Hence the proof of existence of λ ∗ t ∈ (0 ,
1) for β ∈ ( β , β ). Upper bound β may be infinity if G t remains too small or negative.Similarly, the condition for λ ∗ t < λ ∗ t < ⇔ A t (cid:18) − B U zr Utk ( B U J )1 − B U J + 2 B U z [ G t (1 + R Ut ) E ( η t ) − r Utk ] (cid:19) < λ denom K Ut ⇔ A t (cid:0) (1 − B U zr Utk ( B U J )) + 2 B U z (1 − B U J )[ G t (1 + R Ut ) E ( η t ) − r Utk ] (cid:1) < λ denom K Ut (1 − B U J ) ⇔ A t (cid:0) (1 − B U zr Utk ) + 2 B U z (1 − B U J )[ G t (1 + R Ut ) E ( η t )] (cid:1) < λ denom K Ut (1 − B U J ) ⇔ A t < (1 − B U J ) λ denom K Ut + 2 B U zA t (cid:0) r Utk − (1 − B U J ) G t (1 + R Ut ) E ( η t ) (cid:1) . (6.19)Since λ denom is linear in β , as z is, from (6.18) it is clear that there exists 0 < β < β , such thatfor β > β the last condtion holds. This means if the risk premium is too low then there is alwaysa possibility of over-subscription of borrowing. 28ence the proof of existence of λ ∗ t ∈ (0 ,
1) for β ∈ ( β , β ). Upper bound β may be infinity if G t remains too small or negative.Since it has multiple roots, we consider only those roots which belong to [0,1]. Remark:
Since Z t > A t >
0, are the economic requirement for the flow of Capital, we stillhave the same restrictions on R ∗ t , as(1 + R Dt ) E t ( (cid:15) t ) < (1 + R ∗ t ) < (1 + r Ut ) E t ( e t +1 e t ) . Besides, hypothesis of Theorem 3 is also required, i.e., N > N m U K Ut , to have a solution for ( R ∗ t , e t ). Proof of Theorem 3:
Define L ( R ∗ t , e t ) = m D µ ∗ t F t − m U λ ∗ t K Ut e t and L ( R ∗ t , e t ) = N e t − ( N + C t − ) e t + m U λ ∗ t K Ut (7.1)where C t − = m U (1+ R ∗ t − ) λ ∗ t − K Ut − e ∗ t − . Note, L is a quadratic function of e t for every R ∗ t . Thus,equating L with zero to get e t = ( N + C t − ) ± (cid:112) ( N + C t − ) − N m U λ ∗ t K Ut N . (7.2)Since N > N + C t − >
0, (7.2) has a unique positive solution if and only if ( N + C t − ) > N m U λ ∗ t K Ut . But the later holds if N > N m U K Ut as λ ∗ t ∈ (0 ,
1) by Theorem 1.Again, from the assumption of Theorems 1 and 2,(1 + R Dt ) E t ( (cid:15) t ) < (1 + R ∗ t ) < (1 + r Ut ) E t ( e t +1 e t ) . (7.3)Thus, for every e t , L is positive if (1 + R ∗ t ) = (1+ r Ut ) E t ( et +1 et ) (i.e., R ∗ t = (1+ r Ut ) E t ( et +1 et ) −
1) as A t = 0 implies λ ∗ t = 0. Similarly, for every e t , L is negative if (1 + R ∗ t ) = (1+ R Dt ) E t ( (cid:15) t ) , (i.e., R ∗ t = (1+ R Dt ) E t ( (cid:15) t ) − > Z t = 0 implies µ ∗ t = 0. Also, for every e t , L is clearly an increasing function of R ∗ t . Hencethere exist an unique positive solution of R ∗ t for the equation L = 0 for every e t . However, uniquepositive solution e t is guaranteed by the above assumption of the Theorem 3. Hence the proof. uniroot command in rootSolve package in R is used for the root approximation in a specified interval. Here too, for some parametric configurations, existence of root in [0,1] is not guaranteed. 1 − λ ∗ t < ⇒ λ ∗ t > G t is less than the total foreign loan taken by the bank. Appendix D
Algorithm of simulation • Start with initial values of endogenous variables R ∗ = 0 .
14 and e ∗ = 40 and other parametricvalues. • Calculate µ ∗ and λ ∗ from R ∗ using equations (5.3) and (6.2) respectively. • To calculate e t , R ∗ t , λ ∗ t and µ ∗ t at each t, fixed point iteration method is used. This is a versionof majorize-minimization or an E-M algorithm in a dynamic context. • Step 1: Start with an initial value λ tj , calculate e tj given λ ∗ tj using (2.12) • Step 2: λ ∗ tj = constant µ ∗ tj ⇒ λ ∗ tj = f ( R ∗ tj ) from (5.3) • Step 3: As λ ∗ tj is given(initial value), R ∗ tj is determined as the root of above polynomial. • Step 4: Obtaining R ∗ t , update λ ∗ t ( j +1) using (6.2),go back to step 1 and repeat the processuntil each variable converges. • Step 5: Having obtained all the exogenous variables at time period t , repeat the process toget those values at time period t + 1. Distribution of (cid:15) t , η t (cid:15) , η t are taken as two point random variables. Given two points and expectation, their varianceis fixed. Similarly given two points and variance, their expectation is fixed. So, for a two pointdistribution expectation and variance cannot be arbitrarily chosen given two points. Here η t takevalues { η } and η is chosen in such a way to match expectation and variance and so do for E t ( (cid:15) t ). η t is simulated randomly from a distribution which satisfies the given expectation varianceproperty. Since V ( η t ) is very low, E t ( η t ) can be used as a random number simulator from theunknown distribution. Compliance with Ethical Standards
Disclosure of potential conflicts of interestResearch involving Human Participants and/or AnimalsInformed consent 30 eferences [1] Allen, F., Gale, D.: Understanding Financial Crises, OUP: Oxford, UK, Clarendon Lecture inEconomics (2007)[2] Banerjee, A. V.: A simple model of herd behaviour, Quarterly Journal of Economics 107, 797-817 (1992).[3] Aghion, P., Bacchetta, P., Banerjee, A.V.: A Simple Model of Monetary Policy and CurrencyCrises, European Economic Review, 44, 728-38 (2000)[4] Basak, G.K., Das, P.K., Marjit, S.: Foreign capital inflow, exchange rate dynamics and potentialfinancial crisis, Vol 103, page:673-684, Current science, India (2012)[5] Bellman, R. and Dreyfus, S.: Applied Dynamic Programming, Princeton Univer-sity Press, New Jersy