Sudden Stops, Productivity and the Optimal Level of International Reserves for Small Open Economies

Transcription

1 Sudden Stops, Productivity and the Optimal Level of International Reserves for Small Open Economies Harun Nasir Department of Economics University of Reading Preliminary and incomplete Please do not cite or distribute without the author consent Abstract One of the most significant current discussions in international macroeconomics is rapid increase in international reserves for emerging economies. However, there has been little discussion about the optimal level of reserves as an insurance against sudden stop of capital inflows including the production structure of an economy. The aim of this research is to fill this gap in the literature providing new theoretical optimal reserves formulas for emerging economies in the framework of production economy. Our formulas imply a negative relationship with productive capital and a positive relationship with labour productivity for production economies, but both prove that investment is an increasing factor of international reserve to GDP ratio. JEL Classification: F31, F32, F33, F41 Keywords: International Reserves, Investment, Productivity, Sudden Stop I am grateful to my supervisors Dr Alexander Mihailov and Prof. Kerry Patterson for their invaluable guidance and support. I would also like to express great thanks to the participants of the Reading PhD seminar and workshop. All errors are mine. h.nasir@reading.ac.uk

2 1 Introduction One of the most significant current discussions in international macroeconomics is a rapid increase in international reserves for emerging economies. Traditionally, international reserves have subscribed to the belief that whether reserves are adequate or not until the 2000s. There are two main benefits of large reserve holdings. International reserves provide liquidity to smooth consumption and give a flexibility to manage large outflows during the crises. In addition, reserves can help with recovery after the crisis as well as reserve policies can prevent an economy from a crisis. More recently, literature has emerged that offers contradictory findings about international reserves with accumulating high level of reserves. The issue has grown in importance in light of the recent rapid increase of international reserves from adequate reserves to excessive reserves. International reserves have increased substantially in recent years, by almost 150 % from 2005 to A rapid increase of the international reserves can be seen in developing countries more clearly. Figure (1.1) shows, almost half of it holds by middle income countries. The issue has been discussed under the two main approaches that emerging economies hold a high level of reserves as a form of self-insurance against capital flows or what are the determinants of reserve holdings (Aizenman, Y. Lee et al., 2007; Aizenman and Marion, 2003; Aizenman and Marion, 2004; Chinn et al., 1999; Dominguez et al., 2012; Dooley et al., 2004; Eichengreen and Mathieson, 2000; Greenspan, 1999) and what is the optimal level of reserves for economies (Alfaro and Kanczuk, 2009; Caballero and Panageas, 2007; G. A. Calvo et al., 2012; Durdu et al., 2009; Jeanne and Ranciere, 2006; Jeanne and Ranciere, 2011). Recent developments in international reserves have heightened the need for a new formula on the optimal level of reserves. So far, there has been little discussion about the optimal level of reserves as an insurance against sudden stop of capital inflows, including the production structure of an economy, although, most studies in international reserves have only been carried out on the issue of active reserve management, new type of monetary mercantilism and optimal level of reserves. The model explored here draws heavily from Jeanne and Ranciere (2011),hereafter called the JR model. They provide an insurance model of optimal reserves for a small open economy and suggest that the consumer can smooth consumption during the sudden stops by holding a stock of reserves. The authors present a closed form expression for the optimal level of reserves and find consistent results with the literature. Their model predicts a reserve-to-gdp ratio of 9%. Jeanne and Ranciere (2011) suggest that their paper can be extended adding to productive capital and investment. Initial discussion is whether investment can provide a new margin to smooth consumption, which would tend to reduce the level of reserves or reserve offers a new benefit, which is to smooth investment and output. 1

3 Figure 1.1: Global International Reserves (in trillions US Dollars) Total International Reserves Years World Middle Income Countries Source: Authors computations based on World Development Indicators, International Reserves minus Gold. Following the JR model, this paper attempts to show that the effects of investment and production structure of the economy on the optimal level of reserves. Therefore, we develop two different models which are extended versions of the JR model. First, we explain an optimal level of reserve formula where capital is the only source of production (one factor of production model), and then, we develop two factor of production model of optimal level of reserves including labour for the production economies. In the first model, in order to show the effect of investment, we modelled the economy as in the framework of production economies. We employ an AK type production function in order to see the effect of investment plus productive capital. This model suggests a negative relationship between reserve level and capital productivity. Depending on productivity, the first model offers three different ratios, 12.4%, 11.3% and 11.1%, for low productivity, unit productivity and high productivity countries, respectively. Our second model benefits a more general production function introducing labour to the model. The constant returns to scale labour augmented Cobb-Douglas production function is employed to see the effect of labour productivity. Surprisingly, labour productivity was found to an increasing factor of reserve level. The second model suggests three different ratios as in the first model. For low productive countries 10.8%, unit productive economies 11% and 11.8% for high productive economies. Our two models indicate that addition of investment to the JR model implies a higher optimal level of the reserve ratio. However, because of productivity differences and production structure, the models suggest different reserve ratios. 2

4 This paper contributes to filling the gap in the literature through introducing the effect of investment and production structure on the foreign reserves for small open economies. Following the JR, our sample consists of 34 middle income countries over the period 1975 to Thus, we do not only present new formulas and also extended the range of the dataset. The rest of the paper is structured as follows. Section 2 presents an extended version of Jeanne and Ranciere (2011) theoretical model of optimal reserve holding by investment, using two benchmark extensions of the JR model (AK model and Cobb-Douglas model) for a production economy. The results of the calibration are provided in section 3. Section 4 concludes. 2 Theoretical Models: The Role of Productivity and Investment on the Optimal Level of Reserves Formula for Production Economies In this section, we describe the assumptions of the theoretical framework and present an optimal level of reserve formula considering a small open economy. We follow Jeanne and Ranciere (2011), to set up the environment and highlight how investment can play a critical role in the reserve formula. In this framework, our contribution to the JR model is introducing investment to the optimal reserve formula. In order to show differences and similarities between our model and the JR model, whenever possible we use the notation of Jeanne and Ranciere (2011). The original JR formula has been modelled as an endowment economy. The model does not explain the production structure of the economy. In our theoretical framework, we are going to analyse the structure of capital accumulation, hence; our representative economy will be modelled as a production economy. Moreover, our key parameters will be not only investment but also productivity which is going to play a critical role in terms of labour productivity and productive capital. In order to show the transition from the endowment economy to production economy, some of the original JR model assumptions have to be changed. When we introduce the production structure of the economy, we will employ two theoretical Solow type growth production function, AK model and Harrod neutral Cobb-Douglas, respectively. Using these growth models, original JR model is going to be extended by two ways. At the first subsection, we are going to use AK production function which is going to allow us to analyse the role of productive capital in the optimal level of reserve formula. We assume there is no technological growth and population growth at this stage since the AK production function is a special case of the Cobb-Douglas aggregate production function. In this framework, sustained economic growth is the main objective of our first extended model. 3

5 Second extension of the model is deterministic constant returns to scale labour-augmenting technology model. In this extension, the production function is going to be described as Harrod neutral Cobb-Douglas production function which allows us a possible balanced growth path in the long run. Using a more general form of Cobb-Douglas production function than the AK model, we will able to see the effect of productive labour as well as capital on international reserves. 2.1 The Optimal Level of Reserves with Deterministic AK Technology: Sustained Growth In this section, AK type growth model is employed to see the effect of productive capital and investment on international reserve holding. All the assumptions of the JR model holds but only production function is added to the model. Two main contributions of this AK technology model to the JR model are explanation of productivity in the reserve formula and allowing to capital accumulation throughout the period. Replacing total output, Y t by AK technology model, the effect of productive capital can be examined. AK model is a special case of constant returns to scale Cobb-Douglas production function (where θ = 1). When θ takes value of 1, then labour was normalized and the effect of productive capital can be seen easily. In this subsection, assumption of the model is going to be presented and then a formula of reserves with the AK model is going to be derived alongside of the sustained growth Assumption of the Optimal Level of Reserves with AK Model The model is largely based on the JR model. Hence, we model the optimal level of reserves for a small open economy looking for protection across the possibility of losing access to international credit market during the financial crisis. In this paper, our main contribution to the JR model is to introduce of investment in the optimal level of international reserve formula. As mentioned above, our representative small open economy consumes and produces one single good which is consumed domestically and abroad. The economy consist of discrete infinite times t = 0, 1, 2,.... There is no other source of uncertainty rather than a hazard of sudden stops in capital flows. In that sense, the country follows a deterministic trend and it has a risk of international liquidity problems (Jeanne and Ranciere, 2011). There are two domestic agents and one international agent in the model. A private sector and a government are two domestic actors in the model. International agent is the foreign insurers which provide reserves to the country. The representative private sector in the JR model consists of a continuum of atomistic and identical infinitely lived consumers. Their utility can be characterized as an intertemporal 4

6 utility function as follows, U t = E t [ Σ i=0,...,+ (1 + r) i u(c t+i ] (1) where the utility function of the private sector assumed to be of the constant relative risk aversion (CRRA) type, σ 0, and C is the aggregate consumption. here, u(c) = log(c) for σ = 1 u(c) = C1 σ 1 σ, σ 1 (2) Due to the introduction of investment to the model, JR s consumer s budget constraint has modified. In this framework, consumers maximise their current consumptions subject to budget constraint as, C t = Y t I t + L t (1 + r)l t 1 Z t (3) where Y t is domestic output, I t is investment, L t is external debt and Z t is a transfer from the government. The external debt has to be repaid with a constant interest rate, r. There is no default in paying back of external debt of the modelled economy. Once we describe budget constrain in that form, now we can analyse each element of the constraint. Using AK technology model the economy has a production function as, Y t = F (k t ) = AK t (4) Equation (4) shows that using the current technology level, A, and the current capital stock, K t, how much a country can produce at the period t. As mentioned above, our model differs from JR because of investment. According to our model, the representative economy not only consumes all of its resources, it also makes investment in order to increase tomorrow s output, hence, S t = I t = Y t C t + L t (1 + r)l t 1 Z t (5) Furthermore, investment is some proportion of total output as, I t Y t = s (6) where s is a constant saving rate, I t is investment, and Y t is domestic output. Equation (6) shows the investment during period t is a fixed proportion of total output at the beginning of the period using available technology. 5

7 Now we can describe investment in terms of capital accumulation. The most known definition of capital accumulation is: the increase in the capital stock (net investment) in current period, t, equals the difference between new investment and depreciated capital. K t+1 = I t δk t (7) which shows how the capital stock at the beginning of period t, accumulates over time where K t is the capital stock at the period t, δ is a constant proportion of the capital stock is assumed to depreciate each period. We assume that there is no population growth in this model. Then we could describe the saving rate (constant) in the model, We assume capital grows at a constant rate as, s = K t+1 + δk t AK t (8) K t+1 = (1 + g k )K t (9) where g k is the growth rate of capital. In terms of steady-state analysis of AK model, we assume that the growth rate of economy equals the growth rate of capital, g = g k. This is the condition for sustainable growth in the AK technology model. Adding to the investment our model, then our budget constraints takes the form of, C t = F (k t ) K t+1 + (1 δ)k t + L t (1 + r)l t 1 Z t (10) where F (k t ) is domestic output, K t+1 + (1 δ)k t is investment, L t is external debt and Z t shows government transfers, r is a constant interest rate. A critical assumption of the JR model is valid; the domestic private sector has to pay the external debt in the next period with certainty. In other words, there will be no failure for small open economy in terms of paying its external debt back. In this model, one of the critical assumptions is related to the external debt, L t. How much can a small open economy borrow from foreign lenders? There should be some limit on the amount of output can be guaranteed by the domestic private sector to foreign creditors. In the JR model, this restriction is given by the condition that the external debt must be completely paid back in the next period, which requires; (1 + r)l t α t F (k t+1 ) n (11) 6

8 where F (k t+1 ) n is trend output in period t + 1 and α t is a time varying parameter. The economy can only borrow according to this rule; hence α t, indicates the warranty of next period domestic output to external lender. Considering the agents know the value of α t and F (k t+1 ) n in current period t, equation (11) refers that external debt in period t is a default-free as long as it fulfilled. This model simply assumes the time varying parameter α t as an exogenous variable, because of the possibility of sudden stops, the rigidity of the consumer s external debt borrowing constraint can fluctuate over time and α t can be seen as a penalty for domestic agents if they default their debts (Jeanne and Ranciere, 2011) As in the JR model, there are two states in the economy: the normal state (denoted by n) or in a crisis state (denoted by s). In non-crisis state, output increases by a fixed rate g and the economy can guarantee a constant portion of the output, Y n t = (1 + g) t F (k t 1 ) (12) αt n = α (13) On the other hand, when the economy faces a sudden stop, domestic output decreases by a constant fraction, γ below its long run growth path and guaranteed output goes down to zero: Y s t = (1 γ)f (k t ) n (14) αt s = 0 (15) Due to normalization, the guaranteed output does not drop a positive level. Time varying parameter plus output loss is assumed less than one, α + γ < 1, in order to secure that the domestic private sector do not have difficulty to pay back of all the debt during the crisis and interest rate is assumed greater than the growth rate, r > g, to hold the private sectors intertemporal income limited as Jeanne and Ranciere (2011). The issue of how long do the capital inflows come back to the country when the crisis ends, is given in the JR model and it is valid in our model as well. Jeanne and Ranciere (2011) assume that after a sudden stop, the capital inflows converges to its pre-crisis pattern within a certain number of periods, υ, for the economy. Moreover, the country returns to normal state, n, at period t + υ + 1. In reality the country gain access to international liquidity as in its pre-crisis level in more than one year, if a sudden stop hits the economy at the current period t. Therefore, a sudden stop phase can be defined as a length of [t, t+] in the JR model. In other words, matching to the various times of a crisis stage s t = s 0, s 1,..., s υ, in a specific period t the country might be either in the non-crisis state, 7

9 s t = n, or in the crisis state of υ + 1, which are the sub-states of υ + 2 phases Jeanne and Ranciere (2011). In the JR model, dynamics of external debt depends on dynamics of output. Therefore, when the dynamics of domestic output during the sudden stop is described, dynamics of the external debt is defined, instinctively. Thus, dynamics of sudden stop output can be presented as, F (k t ) s t+τ = [1 γ(τ)]f (k t+τ ) n (16) αt+τ s = α(τ) (17) where τ = 0, 1,..., υ. In both equation (16) and (17), γ(τ) and α(τ) are exogenously determined since they depends on τ. Recall equation (14) and (15), we know γ(0) = γ and α(0) = 0 for τ = 0 in the JR model. Furthermore, it is assumed that the country converge to its pre-crisis pattern monotonously, because γ(τ), is non-negative, α(τ) 0 and decline in τ whereas α(τ) is also non-negative, α(τ) 0, and rising in τ. When the crisis ends, the private sector can be financed by international liquidity as in pre-crisis periods, so there will be no restriction to access foreign markets, hence, α(υ) = α (Jeanne and Ranciere, 2011). In the JR model and in our extended version of it, the risk of sudden stop is assumed as the only reason of uncertainty. Sudden stops have negative effects on consumption and investment decision of domestic consumers, and therefore it diminishes their welfare. It results in a decline the consumer s consumption. Economic crisis reduce the trend consumption because of consumers elasticity of intertemporal substitution of consumption is bounded. Moreover, it causes a reduction of domestic output which implies a decrease the consumer s intertemporal income (Jeanne and Ranciere, 2011). It is obvious that a reduction in domestic output and an abrupt fall in capital inflows lead a strong decrease in economic activity during the crisis state consumption goes down sharply during the crisis state (Jeanne and Ranciere, 2011). Eventually, consumption increases as foreign capital flows return in the economy after the sudden stop. However, it takes more than one year for investment to recover pre-crisis level. Investment continues to decrease after the sudden stop. Figure (3.2) shows the trend of economic activities in 5 year event periods. There might be many possible explanation of this impact of sudden stops on investment such as increasing cost of investment, and difficult to find foreign funds for investment, and multinational companies mostly select more stable economies rather than highly volatile economies. All these might be an explanation of why investment does not recovery immediately after the sudden stop as consumption. A probability of sudden stop hits the country is shown by π. As there is no other source of uncertainty rather than the probability of sudden stop, π, the country definitely turn to normal state, n, when the crisis ends. The effects of sudden stops on consumption is 8

10 analysed by Jeanne and Ranciere (2011). The effects of sudden stops on investment are going to be analysed in our model. Introducing investment to the JR model we are going to analyse how investment response to the sudden stops. The second domestic agent of the economy is the government-or monetary authority of the country, which plays a critical role in the JR model. The task of government is it to provide smooth domestic consumption between the normal states and crisis states. In order to provide smooth domestic welfare during the crisis, the government has a tool of reserve insurance contracts. A reserve insurance contract is a simple contract between government and foreign insurers. The aim of the government is to protect domestic agents from the case of a sudden reversal capital flows; therefore, the government forgoes some funds today in order to gain capital access during the crisis 1. In the JR model, reserve insurance contracts shows the trade-offs in reserve management since government has to foregone of some resources in normal times. The mechanism of reserve insurance contract follows the steps as in the JR model. Firstly, the government announces a settlement in the period 0 with external creditors. Then, the external fund providers receive a payment X t from the monetary authority at period t. This process continues till the crisis occurs. Once the crisis started at time t, the economy obtains a fund R t. After the sudden stop occurs, the monetary authority might sign a new reserve insurance deal when the sudden stop phase 2 ends. The government role can be seen in our budget constraint (5) since it shifts the funds coming from the agreement with foreign investors to the private sector as follows; if the country is at the stage of non-crisis stage, Z n t = X t (18) However, if a sudden stop arises, the government secure a payment in the form of, Z s t = R t X t (19) Equation (19) shows the government gain during the sudden stop of capital inflows. The economy earn R t from foreign insurers, but also it should pay the last payment of the reserve insurance contract, X t, in the duration of the sudden stop. Thus, the final transmission of the government access to international liquidity is the difference between R t and X t Jeanne and Ranciere (2011). 1 This could be seen as the cost of reserves and Jeanne and Ranciere (2011) shows that this kind of insurance should be financed by long-term liabilities. 2 The time of the crisis is unknown, an insurance contract signed in period 0 must be specified an infinite sequence of conditional payments (X t, R t) t=1,...,+ (Jeanne and Ranciere, 2011). 9

11 In terms of our contribution to the JR model, there is no change in the role of the government. Hence, we kept all the assumption related the government transfers which were made by Jeanne and Ranciere (2011). However, once we introduce the investment, the effect of investment could be seen on the level of reserves. The last agent in the JR model is foreign insurers. The role of external creditors is to supply international liquidity to the economy during the sudden stop via the reserve contracts. This definition requires a condition that foreign creditors should be agree on the price of the government contracts. This is a critical parameter in the JR model which shows the condition of foreign insurers participation. The marginal utility of funds for the investors at date t denoted by µ t. It is assumed that the marginal utility of funds is more expensive in the crisis state than the normal state (Jeanne and Ranciere, 2011). µ s t µ n t (20) The price of insurance is depending on the ratio between µ s t and µ n t for our representative economy. For simplicity, the JR model assumes that the price parity of funds in normal times to funds in the sudden stop episode is fixed and equal or less than one, as, p = µs t µ n t 1 (21) The JR model considers that the external investors are perfectly competitive and they have same discount rate for the future with domestic private sector. Under the these assumptions foreign insurers supply any reserve insurance contract (X t, R t ) t=1,...,+ whose present discounted value is non-negative (Jeanne and Ranciere, 2011), of the form, Σ + t=1 βt (1 π) t 1 [(1 π)x t µ n t π(r t X t )µ s t] 0 (22) There is no change in foreign insurers participation condition once we extended the JR model by adding investment. Therefore, all assumptions on foreign insurers participation condition are kept in our model A Formula for the Optimal Level of Reserves with AK Technology The economy is going to provide self-insurance across sudden stops, choosing the right amount of international reserves. The problem of optimality is going to be solved analytically in this section. Therefore a closed-form expression is going to be developed for optimal level of reserves. In order to solve this issue, we have to assume that the borrowing constraint (11) is 10

12 binding 3. Following, Jeanne and Ranciere (2011), we first describe country s short term debt to GDP ratio and then set up the Langrangian function for the optimality. Then, we present optimal level of reserve formula and, show the sufficient conditions for the equilibrium. The economy s short-term debt to output ratio is constant in non-crisis time since binding constraint (11) is always held. Short-term debt to GDP ratio is shown by λ. λ = Ln t AK n t = 1 + g 1 + r α (23) In order to provide smooth consumption government wants to maximise consumer s intertemporal utility (1) subject to constraints (10), (18), (19), and the binding credit constraints (11) and the external creditors participation condition (22) (Jeanne and Ranciere, 2011). L = Σ + t=1 βt (1 π) t {(1 π)u(c n t ) + πu(c s t ) + v[(1 π)x t µ n t π(r t X t )µ s t]} (24) Where v is the shadow cost of constraint (22), and the normal state consumption is given by, Ct n = AK t (1 + g k )K t + (1 δ)k t + α 1 + r F (k t+1) αf (k t ) X t ( ) Ct n (r g) (25) = K t A g k δ + Aλ( 1 + g ) X t Where v is the shadow cost of constraint (22), and the sudden stop episode consumption is given by, Ct s = (1 γ)ak t K t+1 + (1 δ)k t αak t + R t X t ( Ct s = K t (1 γ)a g k δ Aλ( 1 + r ) 1 + g ) (26) + R t X t The first order conditions indicates, u (C n t ) = pu (C s t ) (27) Equation (27) shows that the domestic consumption can be substituted at the same rate between the normal and crisis state by the private sector and external creditors as in the JR model. There is no difference at the definitions of government transfers in terms of reserves and probability of sudden stops between our model and the JR model. If we simply rewrite the 3 If the constraint is not binding, optimal level of reserves cannot solve under the closed form. Moreover, the condition (11) implies that there is no precautionary savings in the JR model and our model, since the reserve insurance contract plays a substitution role to the precautionary savings. 11

13 first order condition (27) and the external creditors binding condition (11), we can describe the government transfers X t, in form of, X t = π π + p(1 π) R t (28) Once we describe parameters, now we can solve the first order condition if borrowing constraint (11) is always binding. Assuming these conditions meet, we can present the optimal level of reserve formula under the AK type production economy as a ratio of the optimal level of reserves to domestic output, ρ ( ρ = R (γ + λ) 1 (r g) t = Y t 1 π π+p(1 π) Rt F (k, in form of, t n ) ( ) 1+g λ gk+δ A 1 p 1 σ ( ) (29) 1 p 1 σ where, γ is the output loss in the first period of capital outflows, λ is the short-term debt to GDP ratio, p is the price ratio of funds in different states (normal times and sudden stop episodes), r is the interest rate, g is the growth rate of the economy, g k is the growth rate of capital stock, δ is the depreciation rate of physical capital, A is the technology level of the economy, π is a crisis probability, and σ is the risk aversion. The optimal level of reserves is presented by equation (29) for a small open economy which has an AK type production function. The reserve ratio has the same relationship with some of the parameters of the original JR model. For example, it has a positive relationship with short term debt, λ; the output cost of sudden stops, γ; the probability of a sudden stop, π whereas it also decreases with growth rate of the economy, g. In addition to these parameters, we have three more parameters which are related to the production structure of the economy. First new parameter is the growth rate of capital, g k. It has a positive relationship with the optimal level of reserves. Second additional parameter is the depreciation rate of capital, δ. It has also a positive relationship with the reserve ratio. The last parameter is productivity level, A, which has an opposite relationship with the ratio. Adding a production function to the model, now we are able to analyse the components of saving rate in the production economy. If, we recall the equation (8) and after the some manipulation: saving rate is a ratio of growth rate of capital plus depreciation rate to technology level. The components of saving rate in the AK model allow us to interpret the role of investment in the reserve formula more explicitly. Our model indicates that a positive relationship between investment and reserve to GDP ratio. Furthermore, the reason of this positive relationship is because of the growth rate of the physical capital and depreciation rate since productivity has a negative relationship. In this sense, countries which require 12

14 more capital, they need to increase their reserves level. On the other hand, countries with higher productivity require less reserve level than less productive economies. In order to compare our model with the original JR model, we can rewrite our optimal level of reserve formula as follows; (γ + λ) ρ = ( ) 1 p 1 σ 1 π π+p(1 π) ( 1 p 1 σ ) [( ( )) p (1 π) 1 α γ + (λ + γ) (g k + δ) 1 ] π + p (1 π) A If compare our model with the JR model, there are lots of similarities but our model differs in terms of adding investment and productive capital by (g + δ)( 1 A ) To begin with similarities, first, if we compare our model with the JR model, there is no difference in the behaviour of left hand side of the equation. It is identically same and consistent with the JR model since ρ λ + γ. Furthermore, right hand side of the equation (30) is positive as well because p 1 and α = γ < 1 as in the JR model. Secondly, the case of p = 1 which implies that external insurers do not have preferences between the crisis and non-crisis state (Jeanne and Ranciere, 2011). In this special case of full insurance, the economy s reserve ratio equal to aggregate size of output loss and external debt, (γ + λ), which is the same result in both models as well. Thirdly, the behaviour of the risk aversion is the same in two models since reserves has positive relationship with the parameter of risk aversion, σ. Last similarity is the response to Greenspan-Guidotti rule. So, the unique case of p = 1 when there is no loss in output, λ = 0, which implies ρ = λ, the ratio of reserves to output equals to short term to output ratio as discussed in section three (Jeanne and Ranciere, 2011). The difference between the JR model and our model are the parameters of investment; the depreciation rate of physical capital, δ, the growth rate of capital, g k, and productivity level of the country, A. We explained the relationship between the optimal level of reserves and these parameters. A positive relationship indicates by the depreciation rate and the growth rate of capital and a negative relationship with productivity level. The main parameter is productivity since it spreads to country to country differently. Therefore in our calibration section, we are going to analyse three different scenarios when A = 1 and A < 1 and A > 1. These are the scenarios that, unit productivity, less productivity and high productivity. For a small open economy, higher growth may lead higher borrowing, which increases the external debt of the country, and then the economy may face a sudden stop risk, therefore holds more reserves. Depending on productivity, if A < 1, the economy needs, even higher reserves level than case of A > 1. (30) 13

15 2.2 The Optimal Level of Reserves with Deterministic Labour Augmenting CRS Technology: Balanced Growth Path At the previous framework, labour was normalised and productive capital was the only variable of interest. In this section we introduce labour to the model and show the effect of the more general production function on the decision of reserve holding. Therefore a constant return to scale (CRS) labour-augmenting 4 Cobb-Douglas production function has employed. Using more general production function, we will be able to analyse the effects components of output (productive labour and capital) on the level of reserves. As we are seeking a solution for the balanced growth path (BGP) in the long run, we employ this particular production function rather than the alternatives such as Hicks-neutral technology and Solow-neutral technology. Harrod-neutral technology is only one that gives a solution for BGP in the long-run (Acemoglu, 2008). It also gives us a chance to see particular effect of labour productivity on reserve holdings. At first extension, a negative relationship has proved between productive capital and reserve ratio whereas we are going to analyse productive labour in this extension. In this section, representative small open economy is going to be modelled as a production economy rather than endowment economy framework of Jeanne and Ranciere (2011). All the assumption of the previous section has holds, but only a different production function takes places instead of domestic output. We first introduce additional assumptions in order to set up model as a production economy with two factor of production. Then, our second optimal reserve formula is going to be presented Assumption of the Optimal Level of Reserves with Labour-Augmenting Technology This model differs from the first model in terms of introducing labour in the reserve formula. Therefore, all assumptions and restrictions related to SOE model in previous model are valid but production technology has changed. Our main contributions to the JR model are production function and investment. Adding these two key parameters to the reserve formula, we show the importance of production structure of the economy when determining reserve level in broader concept. For this aim, we used the assumption of the Solow type growth model for the production structure of the SOE. The original Solow model is a growth model for closed economy. However, relaxing the assumptions of external borrowing, we are able to analyse this model in an open economy framework. In addition this, the Solow model only valuable model in terms of explaining basic dynamics of the domestic economy. Adding external debt to Solow 4 Known as Harrod-neutral technology. 14

16 model, we are going to show how the simplest Solow model incorporates with a SOE which described in the previous sections. Our main interest point is sudden stops of capital flows; therefore external debt plays a critical role in our economy but before introducing external debt, Solow parameters are going to be presented in terms of steady state analysis. Our model economy carries all the assumption of the previous models in section three. The economy consists of discrete time with an infinite horizon. Hence, we are interested in the discrete time Solow model. We assume that all firms in this economy have an identical production function for the final goods. In other words, the economy has a representative firm with an aggregate production function (Acemoglu, 2008). Our aggregate production function in a form of CRS Cobb-Douglas function, in form of, Y t = F (k t ) = K θ t (AN) 1 θ t (31) Y t is the total amount of production function of the final good at time t, K is the capital stock, N is total employment, and A is technology at time t. θ shows capital share of income and economy assumes a state of constant returns to scale since the summation of capital share and labour share equals 1. Some simple features of this production function given by proposition (1); Proposition 1: Continuity, differentiability, positive and diminishing, and constant returns to scale are the fundamental properties of Cobb-Douglas production function. The production function F : R 3 + R + is twice differentiable in K and N, and satisfies F K (K, N, A) F (K,N,A) K > 0 F KK (K, N, A) 2 F (K,N,A) K 2 > 0 F N (K, N, A) F (K,N,A) N > 0 F NN (K, N, A) 2 F (K,N,A) N 2 > 0 We assume that, our production function shows constant returns to scale in K and N. This implies the condition that the production function is linearly homogenous of degree one in these two variables. Using this production function, we introduced labour, N, to the model and this contribution requires a description for population growth as follows, N t+1 N t = η; N t = (1 + η) t N 0 (32) where N 0 is the population level base period, η is the population growth rate. There is no change in the definition of budget constraint equation (3) but domestic output replaces by equation (31). Assuming economy holds a constant saving to output ratio 15

17 as equation (5) and (6), we could define the rate of growth of capital per-capita is given by; g k = k t+1 k t = s y t k t δ η (33) Then we can describe capital-output ratio of the economy as follows, k t y t = K t Y t = s g k + δ + η (34) Capital-output ratio is constant along the BGP and it equals the saving rate, s, over the sum of growth rate of capita, g k, depreciation rate, δ, and growth rate of labour, η. In this model capital accumulation has modified since we introduce labour to the model. It can be written as follows, K t+1 = sy t (δ + η)k t (35) After describing the capital accumulation, investment can be written as, I t = sy t = K t+1 + ( + η)k t (36) I t = (g k + δ + η)k t where investment as a proportion of output as in AK model. If we replace output by the production function in order to see the effect of the components of the domestic production structure on the optimal level of reserves, output can be written as, Kt θ (AN) t 1 θ = (g k + δ + η) K t (37) s where output is proportional to the capital stock. We assume the economy follows a balanced growth path (BGP), where all key variables grow at the same rate g, g = g k = g n = g A (38) After all these set up, budget constraint of the consumers can be written in the form of, C t = K θ t (AN) 1 θ t (g k + δ + η)k t + L t (1 + r)l t 1 + Z t (39) 16

18 Equation (39) is the modified version of the equation (10). The output has replaced by labour-augmented CRS Cobb-Douglas production function and investment takes form of equation (36). Following Jeanne and Ranciere (2011), in this subsection, the economy can be again two states, normal times and crisis state as the previous model. So, the equations (12) and (14) are valid. The economy still follows the restriction since there is no change on the pledgeable output equation (11). The role of monetary authority and external creditors participation condition are same with the original model as in equation (18), (19) and (22) A Formula for the Optimal Level of Reserves with CRS Labour-Augmenting Technology Model In line with Jeanne and Ranciere (2011), one of the vital assumptions for all previous models is binding constraint. We assume that the binding constraint (11) is always binding which result in a closed-form expression for this simple insurance problem of the economy. First of all the necessary conditions is going to be explained, and then our second reserve formula is going to be presented. Parameter lambda, λ, shows an external debt to output ratio as in the previous model, however it takes the form of; λ = L n t K θ t (AN)1 θ t = 1 + g 1 + r α (40) Since the country keeps a constant short term debt to output ratio when the external credit constraint (11) is always binding, λ is the same parameter as in previous model, but definition of output has changed. Using Cobb-Douglas production function, consumption in the non-crisis state can be written by, C n t = ( 1 s λ( r g ) 1 + g ) Kt θ (AN) t 1 θ X t (41) With same methodology, consumption in the sudden stop can be written, in the form of, C s t = ( (1 s) (γ + λ) λ( 1 + r ) 1 + g ) Kt θ (AN) t 1 θ + R t X t (42) Since there is no change the role of monetary authority, it enters a reserve insurance contract as described above in order to maximize the private sector s utility (1) subject to the constraints (39), (18), (19), the external borrowing constraint (11) and the external creditors participation (22). The Lagrangian can be written as in the previous section: L = Σ + t=1 βt (1 π) t {(1 π)u(c n t ) + πu(c s t ) + v[(1 π)x t µ n t π(r t X t )µ s t]} 17

19 where v is the shadow cost of constraint (22). We can solve the Lagrangian since there is no change on the first order condition, equation (27) and the form of government transfers, equation (28). The optimal level of reserves can be written as a constant reserves-to-output ratio and given by ρ, where k = K N. ( ( ) ) ρ = R (γ + λ) 1 (r g) 1 θ ( ) t 1+g λ (g k + δ + η) k A 1 p 1 σ = ( ) (43) Y t π 1 π+p(1 π) 1 p 1 σ The optimal level of reserves with deterministic constant returns to scale labour-augmenting Cobb-Douglas production function has many common parameters with the AK model such as; γ is the output loss for next period of crisis, λ is the short-term debt to output ratio, p is the price ratio of the funds in non-crisis and crisis state for external insurers, r is the interest rate, g is the growth rate of the economy, g k is the growth rate of capital stock, δ is the depreciation rate of physical capital, π is the probability of a sudden stop, and σ is the risk aversion. The differences between the two models are growth rate of labour, η, capital-labour ratio, k, and θ is the capital share of income. Our optimal level of reserve formula, equation (43) implies that a positive relationship with the output cost of a sudden stop, γ, the level of short term debt, λ, the probability of a sudden stop, π, the growth rate of capital, g k, depreciation rate, δ, interest rate, r, and risk aversion, σ whereas it implies a negative relationship with the growth rate of the economy, g. Those have similar relationships with previous models. In addition those parameters, changing production structure of the economy, we have three more parameters in this model. First parameter is population growth, η, which has a positive relationship with optimal level of reserves as expected 5. Second parameter is the capital share of income, θ, which shows a negative relationship between optimal level of reserves. Third parameter is capital-labour ratio with productive labour, k, which indicates a positive relationship with the optimal level of reserves. Moreover, if we analyse the each parameter of k, we found that labour productivity is an increasing factor of optimal reserve formula as well as capital stock. In order to compare two extensions of the JR model, AK model and Cobb-Douglas model, we need to analyse capital-labour ratio which is the capital over productive labour. It implies that a positive relationship between the optimal level of reserves and productive labour. This gives a different productivity result from the previous case. In the AK model 5 Earlier literature on optimal level of reserves assumes population is a scale variable and shows positive relationship between level of reserves (See: Heller (1966),Clark (1970),Kelly (1970),Hamada and Ueda (1977),Frenkel and Jovanovic (1981)). 18

20 productive capital implies less reserve level whereas our specific Cobb-Douglas model shows reserve to GDP ratio increases by labour productivity. If we compare our two specific production economy models (productive capital and productive labour) with investment, we found the main link between them is the saving rate. Using different production function, we defined constant savings rate in two different structures. Figure (3.3) shows this development. In order to compare the last model and the JR model, our optimal level of reserve can be written as follows, ( ) 1 p 1 (γ + λ) ρ σ = ( ) π 1 π+p(1 π) 1 p 1 σ [ ( ( )) ( ) ] (44) p (1 π) k 1 θ 1 α γ + (λ + γ) (g k + δ + η) π + p (1 π) A Equation (44) shows the comparison of our model and the JR model. If the last term of RHS ignored, the model would become the JR model. Therefore, it includes lots of similarity such as the behaviour of the LHS of the equation (44), the case of p = 1, the behaviour of the risk aversion parameter, and responding the Greenspan-Guidotti Rule as in previous models in this paper. However, introducing labour and production structure of the economy, our model shows an increasing effect on the optimal level of reserves by the growth rate of capital, depreciation rate, population growth, capital-labour ratio. 3 Calibration In this section we are going to analyse our models by the data of 34 middle income countries 6 from 1975 to A positive relationship between international reserves and investment can be seen at Figure (3.1). First of all, we are going to analyse domestic absorption to show the overall behaviour of the model parameters during the sudden stop year. To do this, we are able to see the distinction between our models and the JR model in terms of investment in sudden stop. Secondly, we are going to show the calibration of common parameters in the models. Finally, we will show the calibration of model specific parameters. 6 In order to make a comparison between our paper and Jeanne and Ranciere (2011), we used same sample of countries with Jeanne and Ranciere (2011) but they cover the data from 1975 to They classified countries as a middle income according to the World Bank s classification. This sample excludes major oil-producing countries 19

21 Figure 3.1: Reserves and Investment Reserves (% of GDP) Bolivia Malaysia Thailand Jordan Hungary Bulgaria Peru Uruguay Philippines Czech Rep. Korea Romania Botswana ParaguayPoland Morocco Brazil Costa Tunisia Honduras Chile Rica Guatemala Mexico South Turkey AfricaJamaica Colombia El Salvador Sri Lanka Dominican Rep. Egypt Argentina Ecuador China Gross capital formation (% of GDP) Source: Authors computations based on World Development Indicators, International Reserves minus Gold. In order to show the effects of the parameters, domestic consumption 7 can be defined in terms of domestic output, financial account, investment, income transfers and change in reserves. C t = Y t I t + F A t + IT t δr t (45) where F A t is the financial account, IT t is the income and transfers from abroad and R t shows the change in reserves. A sudden stop is defined as an unexpected decrease in financial account which leads a decrease in domestic consumption and it might cause a decrease on domestic output or it can be adjusted by international reserves (Jeanne and Ranciere, 2011). The above equation (45) shows the link with our budget constraint in the sudden stop as, C s t }{{} C t = (1 γ)y n t } {{ } Y t g k +δ A Y t ( 1 θ)yt (g k + δ + η) k + ( L t 1 ) + [ rl t 1 (π + ω)r t ] ( R t ) }{{}}{{}}{{} A) }{{} F A t IT t R t I t (46) 7 Equation (45) is a version of domestic absorption composition since domestic absorption equals sum of domestic consumption and investment, D t = C t + I t 20

22 Figure 3.2: Average Behaviour of Model Parameters in Sudden Stop Consumption Time Investment Time Domestic Output Time Change in Reserves Time Financial Account Time Mean One Standard Error Band Note: 0 indicates a year of sudden stop. All parameters showed in ratio of the GDP. As in the JR model a positive level of change in reserves shows reserve loss during the period. Source: Authors computations based on World Development Indicators, International Reserves minus Gold. where ω shows a pure risk premium 8. It might interpret as an opportunity of holding reserves. For our interest point, the equation (46) takes two different versions depending on our definition of savings. In the first model we described investment, using productive capital g as: k +δ A Y t whereas it is described as (g k + δ + η)( k A )1 θ Y t in the second model. Figure (3.2) shows the average behaviour of equation (46) in a five-year event window when the middle observation, 0 shows sudden stop year. Although, all components of equation (46) shows same pattern with JR model, our contribution to the JR model investment shows an interesting feature. Investment responds 8 Because of playing no role with the productivity and investment, the opportunity cost of holding reserves is not described in this model. However, in order to make a comparison between our models and the JR model, their methodology is followed as: X t = (π + ω)r t, (See: Jeanne and Ranciere (2011)). 21