1 Continuous Time Optimization

Transcription

1 University of British Columbia Department of Economics, International Finance (Econ 556) Prof. Amartya Lahiri Handout #6 1 1 Continuous Time Optimization Continuous time optimization is similar to dynamic problems in discrete time. Consider the following generic problem: subject to the flow constraint max e βt f (u t, x t ) dt t=0 ẋ t = g(u t, x t ), x 0 given. In this problem u is the control variable while x is the state variable. There are two basic ways to proceed. One method is pointwise optimization. This calls for maximizing the objective function by choosing u t for each t subject to the single lifetime constraint obtained by integrating the flow constraint. An alternative is the method of optimal control. Here we describe the optimal control methodology. Define the current value Hamiltonian to be H = f(u, x) + λg(u, x) where λ is the costate variable associated with the predetermined state variable x. Note that we have suppressed time subscripts for convenience. Define the present value Hamiltonian as H = e βt H 1 c Amartya Lahiri. Not to be copied, used, or revised without explicit permission from the copyright owner. 1

2 where H is the current value Hamiltonian. As long as f and g are both concave in their arguments, necessary and suffi cient conditions for an optimum are given by H u = 0 e βt λ t = H x While the first condition is a standard condition for any constrained optimization problem, the second condition is saying that at an optimum, the change in the present discounted value of the shadow value of the state variable should be equal to the marginal effect of the state variable on the discounted value of the objective. Effectively, this is saying that the marginal benefit of an additional unit of the state variable should, at an optimum, be equal to the cost of acquiring the asset net of the capital gain (here λ is the marginal value of the asset x). This is the counterpart of the Euler equation in discrete time. To see this note that the two first order conditions can be rewritten as f u = λg u λ = βλ f x λg x The second condition can be written as f x λ + g x + λ λ = β The left hand side is the additional benefit of an extra unit of x ( λ/λ is the capital gain on the asset) while the right hand side is the cost measured by the discount rate (since the benefit accrues tomorrow). 2 The basic monetary model Thus far we have considered real models with no discussion of money. We shall now relax this by introducing money into the basic model. The goal of introducing money is threefold. We want to understand the conditions under which money affects real variables in the 2

3 econom;, the implications for monetary policy and the optimal exchange rate regimes; and lastly, form a better understanding of events such as currency crises which are inherently monetary phenomena. Consider a small open economy inhabited by a representative agent who maximizes lifetime welfare given by V = t=0 e βt [u(c t ) + v(m t )] dt (1) where m = M denotes real money balances. The representative household s flow budget P constraint is ḃ = rb + y c where b are foreign bonds which pay r units of the good as interest every instant. Ṁ P r is assumed to be constant over time. The bonds are denominated in terms of the good. In the following we shall assume that y t = ȳ for all t. In the following we shall adopt the notational convention of π = P P µ = Ṁ M ε = Ė E where E is the nominal exchange rate price of foreign currency in terms of the domestic currency. In this one good model with free goods mobility and the normalization of the foreign currency price of the good to unity, we must have P = E Differentiating the expression m = M/P with respect to time gives ṁ = (µ π) m 3

4 Since Ṁ/P = µm, this implies that the flow constraint can be written as ḃ = rb + y + g c ṁ πm where g denotes lump-sum transfers received from the government. assets as a = b + m, one can rewrite the flow constraint as Defining total private ȧ = ra + y + g c im (2) where i = r + π denotes the nominal interest rate. In terms of the optimal control method highlighted above, the objective function is given by equation (1), the control variables are c and m, the state variable is a with its evolution over time being given by (2). Setting up the Hamiltonian as described previously and following the previously described steps gives the necessary and suffi cient conditions for optimality: u (c) = λ v (m) = iλ λ = (β r) λ Under the standard small open economy assumption of β = r, the above implies that λ is constant over time along any perfect foresight path. Hence, consumption c must be constant as well. The first two conditions can be combined to get 2.1 Government v (m) u (c) = i We assume that the government in this economy consists of a combined monetary and fiscal authority. The government prints money M, makes lump-sum transfers to the private sector g, and holds interest bearing foreign bonds (foreign reserves) R. flow constraint is given by Ṙ = rr + ṁ + πm g 4 The government s

5 Note that the government s revenue sources are interest earnings on foreign reserves, revenue from money creation (which is the sum of the expansion of real money balances and the inflation tax. The wing of the government that holds foreign reserves is the central bank. The central bank s balance sheet identity is where d = D P R + d = m is real domestic credit to the private sector. Time differentiating the central bank balance sheet identity gives Ṙ = ṁ d Substituting this into the government s flow constraint gives the transfer policy for g : d = g (rr + πm) This equation makes clear that for every domestic transfer policy (a path for g) there is a unique domestic credit policy (a path for d). Intuitively, the expression says that any government budget deficit, i.e., g > rr + πm, must be financed by domestic credit creation, i.e., d > Flexible exchange rates We shall start by analyzing a flexible exchange rate regime. When the central bank pursues a flexible exchange rate then the implicit operating rule is that it does not intervene in foreign exchange markets at all. Hence, the central bank operating rule becomes Ṙ = 0 i.e., it doesn t buy or sell international reserves. Note that this policy implies that ṁ = d at all times. Hence, the fiscal authority s transfer policy becomes rr + ṁ + πm = g In other words, the fiscal authority transfers as a lump sum to the households the revenues from money creation (ṁ + πm) along with the interest earnings on international reserves. 5

6 To make things simple also assume that R = 0. With this additional assumption we also have m = d from the central bank balance sheet. To complete the description of the government, we need to specify the nominal domestic credit rule. We assume that Ḋ t D t = µ D Hence, the growth rate of nominal domestic credit is assumed to be constant over time. 2.2 Equilibrium Combining the flow constraints of the private sector and the government yields the resource constraint for the economy: f = rf + y c where f = R + b denotes net foreign assets of the economy. Noting that c is constant over time, one can integrate this differential equation forward to get c = rf 0 + ȳ. The first order condition for optimal money holdings says that v (m) u ( c) = r + π. (3) But π = µ ṁ m equation above yields where µ is the rate of growth of nominal money. Since ṁ = d at all times, we must have Substituting this into the ṁ m = r + µ v (m) u ( c). (4) µ t = µ D for all t Recall that Ḋt/D t = µ D is a constant. Hence, µ must be constant at all times. In steady state we must have ṁ = 0. Hence, steady state real money balances m are defined by the condition v (m ) = u ( c) (r + µ D ), (5) 6

7 Note that the steady state condition ṁ = 0 implies that we must have π = µ = µ D in steady state. For future reference, note that equation (5) implies that an increase in the rate of domestic credit growth will raise µ which, in turn, will reduce the steady state m. to get To study the dynamics of this economy, differentiate equation (4) with respect to time ṁ m v m = (m) u ( c) > 0 Hence, this is an unstable differential equation. level of m then it stays there. If the system begins at the steady state Else, the economy drifts away secularly from steady state with either continuously increasing or continuously decreasing real money balances over time. Hence, the economy must jump to its steady state m at t = 0 itself. Thus, for any given M 0 (or equivalently, given initial R 0 and D 0 ), there is a unique P 0 which ensures that m 0 = m. Lastly, since the economy jumps to steady state right away, the preceding analysis also implies that π t = µ t = µ D for all t Unanticipated, permanent increase in µ D A permanent increase in the rate of domestic credit growth µ D will reduce the steady state m. Since the economy must jump to the new steady state right away, the price level jumps up to reduce real money balances to their new steady state level. Inflation rises instantaneously to its new higher steady state level. Money is super-neutral here: changes in the rate of growth of money supply leave the real side of the economy unchanged Unanticipated, one time, permanent increase in the level of D This is an increase in the level of money supply. However, this leaves the steady state m unchanged. Hence, M and P rise by the same proportion leaving real balances unchanged. 7

8 This shock also leaves the rate of inflation unchanged. model since there is no effect on the real side of the economy. Lastly, money is neutral in this 2.3 Devaluation or Inflation Rules The preceding analysis was carried out assuming that the central bank chose a rate of money growth and the exchange rate and the price levels were determined endogenously at each point. In open economies, central banks follow exchange rate rules (fixed exchange rate regimes or more generally, a predetermined path for the nominal exchange rate). Note that in this one good model, an exchange rate rule is equivalent to a price rule. Figure 1 shows the nominal exchange rates for a group of emerging economies till It makes clear the often used emerging economy practise of fixing or actively managing the exchange rate. In the context of the model developed here, the practise of fixing exchange rates is easy to formalize. Essentially, fixing the path of the nominal exchange rate implies that the central bank announces the price of foreign currency (the exchange rate) and delivers on its promise by buying and selling as much foreign currency for domestic currency as people want at the announced exchange rate. To make this operational we shall assume that the fiscal authority follows the transfer rule g = rr + πm (6) This transfer policy has two implications. First, real domestic credit must be constant over time. Second, we must Ṙ = ṁ Hence, international reserves and real money balances must move together. Intuitively, the transfer policy makes the exchange rate policy and the fiscal policy mutually consistent. If this were not the case then under some cases the fixed exchange rate peg may become unsustainable. To make matters general, we shall now assume that R > 0. The first-order conditions for the household are as before the household s problem is unchanged. Moreover, the real side of the economy is still independent of monetary 8

9 Figure 1: Nominal exchange rates in developing countries ARG_E BRAZIL_E INDONESIA_E KOR_E MEX_E THAI_E 9

10 conditions. Hence, c t = c = rf 0 + ȳ for all t. But now the central bank has exogenously fixed the path of the nominal exchange rate. Hence, the rate of devaluation of the nominal exchange rate is given exogenously: Ė t E t = ε t = π t, E 0 given. We assume that the central bank announces a constant rate of devaluation, i.e., ε t = ε for all t The first feature of this environment to note is that a fixed rate of devaluation ε implies that the nominal interest rate, i = r + π, is constant. Hence, from equation (3) real money demand is constant. Hence, µ = ε for all t. The transfer policy is still the same. Hence, d = 0. This implies that the rate of growth of domestic credit must equal the rate of inflation (equivalently, the rate of devaluation) at all times, i.e., µ D = ε for all t. Note that with an exogenously specified nominal exchange rate, monetary policy becomes endogenous; the central bank must supply as much nominal money as demanded by the private sector. As is obvious, real allocations in this economy are exactly the same as in the earlier economy where the central bank chose the rate of money growth. Moreover, it is easy to verify that the rates of growth of prices, nominal money balances, and the exchange rate in the two economies would be identical as long as the exogenously chosen µ D in the first economy is exactly equal to the exogenously chosen ε in the second economy. In other words, in this model, whatever the policymaker can do by choosing a money growth rule, she can also do by choosing the equivalent exchange rate rule. Lastly, since i = r + ε, interest rules generate isomorphic outcomes Unanticipated, permanent increase in ε This reduces the steady state m. Households try to reduce their real money balances by exchanging their nominal balances for foreign bonds. Note that this shock changes the rate 10

11 of currency depreciation but does not change the level of the nominal exchange rate at the date of the shock, i.e., E does not jump. The central bank sells foreign reserves in exchange for domestic money at the pre-announced exchange rate. The process stops once m falls to its new steady state level. In the new steady state, consumption is unchanged because net country foreign assets f remain unchanged. All that changes is the composition of country foreign assets. The private sector foreign bond holdings are greater while the central foreign bonds (foreign exchange reserves) are lower Unanticipated, discrete devaluation This shock implies that there is an unanticipated increase in E on the date of the shock. Since the rate of devaluation remains unchanged, the nominal interest rate doesn t change. Hence, real money balances do not change. To keep real money balances unchanged while P jumps up, the private sector demand more nominal money. The central bank accommodates this by buying foreign bonds from households and selling them domestic money at the new exchange rate. Once again, the real side of the economy is unaffected since net foreign assets of the economy do not change. 2.4 Welfare It is easy to derive the first-best monetary policy in this model. Effectively, the household derives utility from holding money. The cost of holding money is the nominal interest rate i. The first-best monetary policy in this model is to satiate the consumer with real money balances, i.e., set v (m) = 0 Hence, the cost of holding money should be set to zero. This is achieved by setting i = 0. Hence, the first best monetary policy in this model is ε = µ D = r. This is called the Friedman rule. 11