501 Econ Fall 2008 Lecture 02 Part 1Page 1
The Real Economy Dynamic Open Economy Models
Readings: 1. Obstfeld and Rogoff, 1996, Foundations of International Macroeconomics, ch 1 and 2. Outline: 0. Small Open Endowment Economy (OR, 1.1 & 1.2) -- assigned reading 1. A General Equilibrium Model(s) of the International Economy (OR, 1.3) 2. Consumption and the Current Account (OR 2.12.3)
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1. A General Equilibrium Model(s) of the International Economy Objectives: 1. setup and characterize a simple GE model 2. explain the determination of the global real interest rate 3. explain the transmission of shocks between trading countries General Framework Two economies, domestic (home) and foreign. Two period model: period 1 (today) and 2 (tomorrow). We setup and characterize the choices in the domestic economy; the foreign is identical
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1.1 Domestic Economy A single representative agent (all agents identical and population is normalized to one); lives two periods; single tradable consumption good. Economy is small: faces a real interest rate r on international capital markets; pricetaker. Market discount factor: 1 R 1+r Consumer's problem Preferences: U (C1, C2) = u(C1) + u(C2) where is the subjective discount factor and u() satisfies standard regularity conditions. Endowments Y1 and Y2
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Net foreign assets: Bt+1 the stock of foreign assets at end of period t e.g Bt+1 returns (1 + r)Bt+1 in period t + 1. Flow constraints (period by period budget constraints): C1 + B2 = Y1 + (1 + r)B1 Period 1 C2 + B3 = Y2 + (1 + r)B2 Period 2 More generally, Ct + Bt+1 = Yt + (1 + r)Bt Intertemporal budget constraint: C1 + RC2 + RB3 = Y1 + RY2 + (1 + r)B1 Terminal condition: B3 = 0. B3 > 0 consumption could be higher in period 1 & 2 B3 < 0 foreign lenders' consumption could be higher period 1 & 2 Note B1 is exogenous (properly -- pre-determined).
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Consumer chooses {C1, C2} (and implicity B2) to maximize, u(C1) + u(C2) subject to, C1 + RC2 = Y1 + RY2 + (1 + r)B1 The first order conditions solve for {C1, C2} and the Lagrange multiplier as functions of the exogenous variables and parameters. The solutions for C1 and C2 are demand functions: C1 = C1(r, Y1, Y2) C2 = C2(r, Y1, Y2) Intertemporal Euler equation From the first order condition to the representative agent's optimization problem: u (C2) = R u (C1) MU future consumption Price of future consumption = MU current consumption Price of current consumption
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National Accounting All of our standard national accounting identities have counterparts in the simple model economy. The general period flow constraint is, Ct + Bt+1 = Yt + (1 + r)Bt (1)
We define the current account as the change in the value of a country's net claims on the rest of the world: CAt Bt+1 - Bt From (1), CAt = Yt + rBt - Ct (= St, National Saving) Bt+1 - Bt = (Yt - Ct) + rBt Current Account = Net Exports + Net Income from o/s
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Capital Account: the inflow of capital (domestic agents selling assets o/s): -(Bt+1 - Bt) Balance of Payments (Current account + Capital account = 0): (Yt - Ct) + rBt - (Bt+1 - Bt) = 0 This is just equation (1); the flow constraint and the balance of payments are the same. GDP and GNP: GNP: value of final product produced nationally plus income on overseas assets: GNPt = Yt + rBt GDP: value of final product produced nationally, Yt.
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Consumption Smoothing Suppose = R; then u (C2) =1 u (C1) which implies, C1 = C2 That is, whatever the pattern of {Y1, Y2}, C1 = C2. Consumption Tilting Suppose Y1 = Y2 and > R (relatively patient); then at endowment point, u (Y2) >R u (Y1) (since Y1 = Y2). Home will set C1 < Y1 (lend); delay consumption.
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Autarky real interest rate Without trade, C1 = Y1 and C2 = Y2. Then, 1 u (Y2) A R u (Y1) 1 + rA RA is the autarky terms of trade; it summarizes a country's (consumer's) preferences and endowments, useful for interpreting the open economy equilibrium. Suppose RA < R; then future consumption is relatively cheap domestically. Home has a comparative advantage in future consumption. Export future consumption, import current consumption. Borrow now, repay later. This occurs if (1) home has relatively more second period endowment and or (2) a low discount factor . If RA > R, then home has a comparative advantage in current consumption.
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Main Results: Result 1 Welfare gains from trade Intertemporal trade improves welfare (or at least doesn't worsen it). The welfare gains arise from the principle of comparative advantage. By themselves, CA deficits are not necessarily a policy concern Result 2 Consumption Smoothing If = R, then C1 = C2 irrespective of Y1 and Y2. Permanent changes in output, e.g. both Y1 and Y2 rise, have little or no affect on the current account (borrowing/lending patterns) of an economy Temporary changes in output are absorbed in the current account -- consumption smoothing. For example, a permanent increase in government spending has no effect on the current account; a temporary increase in government spending does affect the current account.
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1.2 Extension: Economy with Productive Capital Amend the above model to include a production sector. Production Production function: Yt = F (Kt), F > 0, F < 0 Capital accumulation: Kt+1 = Kt + It - It > 0 or It < 0 capital can be consumed after it has been used in production - Kt+1 is capital stock at end of t (analogous to Bt+1) - no depreciation
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National accounts (again) The flow constraint (balance of payments) is, Ct + It + Gt + Bt+1 = Yt + (1 + r)Bt The current account is, CAt = Bt+1 - Bt = Yt + rBt - Ct - Gt - (Kt+1 - Kt) CAt = St - It E.g. twin deficits hypothesis: large current account deficits are driven by government budget deficits. CAt = [(Yt + rBt - Tt) - Ct] + [Tt - Gt] - It = St + St - It
Private Public
But St
Private
and It will/may respond to changes in public saving.
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Consumer's Problem Consumer chooses {C1, C2, B2, I1, I2} to maximize u(C1) + u(C2) subject to, C1 + I1 + G1 + B2 = Y1 + (1 + r)B1 C2 + I2 + G2 = Y2 + (1 + r)B2 Y1 = F (K1) K2 = K1 + I1 Y2 = F (K2) K3 = K2 + I2
given G1, G2, B1, K1 and r. For simplicity, assume B1 = 0. K3 = B3 = 0 is optimal and implicit. Since K3 = 0, this implies I2 = -K2 which means, ... take the capital apart in period 2 (after using it for production in period 2) and consume it.
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As before, we can write an intertemporal budget constraint. The above constraints simplify to, C1 + I1 + G1 + B2 = F (K1) C2 + I2 + G2 = F (K2) + (1 + r)B2 C2 + G2 = F (K2) + K2 + (1 + r)B2 C2 + G2 = F (K1 + I1) + (K1 + I1) + (1 + r)B2 Combining: C1 + I1 + RC2 = (F (K1) - G1) + R(F (K1 + I1) + K1 + I1 - G2) Consumer's problem is, more simply: choose {C1, C2, I1} to maximize utility subject to the above constraint.
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First order conditions: u (C2) = R determines consumption path u (C1) F (K1 + I1) = r determines investment Separation of consumption and investment decisions -- no matter what desired consumption and saving, investment is determined by the world interest rate. Obtaining maximum wealth does not require domestic saving -- the resources can come from abroad. We can work with the model diagrammatically.
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Production possibility frontier Set G1 = G2 = 0 and B1 = 0 for simplicity. Recall model without production. The production possibility set can be constructed by considering the set of consumption allocations that are feasible without trade: C1 Y1 C 2 Y2 With investment the PPF is more complicated because we have some degree of autarkic intertemporal trade (through investment). Again, consider the consumption possibilities without trade: C1 + I1 = F (K1) C2 = F (K1 + I1) + K1 + I1 Which implies: C2 = F (K1 + F (K1) - C1) + K1 + F (K1) - C1 PPF
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Slope: d 2C2 dC2 = -(1 + F (K2)) < 0 2 = F (K2 )) < 0 concave dC1 dC1 Intercepts: Horizontal: Set I1 = -K1 so K2 = 0; C1 = F (K1) + K1 and C2 = 0. Vertical: Set I1 = F (K1); C1 = 0 and C2 = F (F (K1) + K1) + F (K1) + K1 Preferences, the IBC and the PPF determine the solution to consumption and investment. C1 + I1 + RC2 = (F (K1)) + R(F (K1 + I1) + K1 + I1) C2 = F (K1 + F (K1) - C1) + K1 + F (K1) - C1 IBC
PPF
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Diagram: - Slope of the intertemporal budget constraint is -(1 + r). - Slope PPF of is -(1 + F (K1 + I1)) - Tangency of PPF and IBC determines point of production. - Tangency of IBC and Indifference curve determines optimal consumption. - Consumption and investment decision are separate.
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Main Results: Result 1 Separation of consumption and investment Result 2 CA reflects Investment Opportunities investment opportunities If (1 + r) < (1 + rA) then opportunity cost of financing domestic investment is lower -- investment increases. Similarly, current consumption is relatively cheaper -- current consumption increases Current account deficit increases (borrowing to finance domestic investment and increased current consumption) Result 3 Real interest rate shocks An increase in the real interest rate: Lower investment Lower current consumption CA deficit reduced Future output falls -- contrast with simple Mundell-Fleming model
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1.3 Global Equilibrium -- Endowment Economy Home and Foreign Demand Functions For home and foreign, the consumer optimization problems solve (implicitly) for demand functions: C1 = C1(r, Y1, Y2) C2 = C2(r, Y1, Y2) Assume,
dC1/dr < 0 and dC1 /dr < 0 C1 = C1 (r, Y1, Y2) C2 = C2 (r, Y1, Y2)
Recall 1 + r is the relative price of current consumption so these behave like standard demand functions. Equivalently,
dS1/dr > 0 and dS1 /dr > 0
So, we are assuming that the saving functions are upward sloping.
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Two possible effects on consumption (saving) from a change in r: substitution effect and a terms of trade (income) effect. Substitution effect: a rise in r raises the relative price of current consumption, causing savings to increase. Terms of trade effect for a borrower: a rise in r worsens the country's terms of trade, lowering consumption (raising saving) -- like an income effect. SE and TTE both cause saving to rise. Terms of trade effect for a lender: a rise in r improves the country's terms of trade, raising consumption (lowering saving). SE and TTE work in opposite directions -- possible to have a backward bending savings function.
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Equilibrium Conditions:
C1 + C1 = Y1 + Y1 C2 + C2 = Y2 + Y2
Using the definition of savings and Walras' Law, we can focus on:
S1(r) + S1 (r) = 0
This equation can be solved for the equilibrium real interest rate r. Notice that the above implies, since investment is zero: CA1(r) + CA(r) = 0 1 For now, we assume that the savings functions are upward sloping (and monotonic). In this case, it is very easy to represent the world equilibrium in a simple diagram.
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Results: 1. The equilibrium interest rate will be between the autarky interest rates 2. Events in one economy affect the other through the real interest rate; including welfare effects Example: a decrease in saving for the Foreign economy, say due to changing demographics. Assume that home is initially a borrower. The effects make Home worse off because the interest rate rises. Its intertemporal terms of trade have been worsened (price of its exports, second period consumption good, has fallen). Analytical Solution for ln-utility model -- see assignment What's missing? 1. Is the savings function upward sloping? 2. Investment?
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1.4 Global Equilibrium with Investment -- Metzler Diagram Section 1.3.3 OR. Model Setup Two economies with production.
Assume no government and no initial assets (B1 = B1 = 0).
Representative agents make consumption and investment decisions. Production is Y1 = A1F (K1) Recall K2 = K1 + I1; ie. no depreciation. Production in period 1 is given, since K1 is given. Production in period 2 can be written in terms of I1: Y2 = A2F (K1 + I1). Agent can consume capital stock after production in period 2; that is, K2 = I1 + K1 is available for consumption. Y2 = A2F (K2) where A1 and A2 are productivity parameters.
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Domestic Representative Agent's Problem Choose {C1, C2, I1, } to maximize U (C1, C2) = u(C1) + u(C2) subject to, C1 + I1 + RC2 = A1F (K1) + R[A2F (K1 + I1) + K1 + I1] K1 given. IBC refers to the intertemporal budget constraint. The right hand side of the IBC is the present discounted value of agent's income. It might help to write it as, A1F (K1) + R[A2F (K2) + K2] IBC
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The Lagrangian for the above constrained optimization is,
C1 ,C2 ,I1 ,
max L = u(C1) + u(C2) + (A1F (K1) + R[A2F (K1 + I1) + K1 + I1] - C1 - I1 - RC2)
The first order conditions: L = C1 L = C2 L = I1 L =
u (C1) - = 0 u (C2) - R = 0 (R[A2F (K1 + I1) + 1] - 1) = 0 IBC = 0
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Summary of first order conditions for home (foreign identical): u (C2) = R u (C1) C1 + I1 + RC2 = A1F (K1) + R[A2F (K1 + I1) + K1 + I1] A2F (K1 + I1) = r Together, these determine consumption (savings) and investment functions; that is, they solve for C1 = C1(r, A1, A2) C2 = C2(r, A1, A2) I1 = I1(r, A1, A2)
C1 = C1 (r, A, A) 1 2 C2 = C2 (r, A, A) 1 2 I1 = I1 (r, A, A) 1 2
We can usefully define the first period savings function for home and foreign: S1(r, A1, A2) = A1F (K1) - C1(r, A1, A2) S1 (r, A, A) = AF (K1 ) - C1 (r...