Int'l MacroEcon 笔记

本文是国际宏观经济学的笔记，教材是「International Macroeconomics: A Modern Approach」

L1 - Introduction

Positive vs. Normative

• Positive economics: How does the world work?
• Normative economics: How should the world work?

Balance of Payments Accounts

Current Account (CA)

• CA records exports and imports of G&S, int'l receipts and payments of income.
• (+) -> exports, income receipts
• (-) -> imports, income payments

Current Account Balance

• Trade Balance
  • Merchandise Trade Balance: NX of goods
  • Service Balances: NX of services
• Income Balance
  • Net Investment Income: interests, diidends
  • Net International Compensation to Employees: compensation to (domestic workers abroad - foreign workers)
• Net Unilateral Transfers
  • Personal Remittances
  • Government Transfers

Financial Account (FA)

• changes in a country's net foreign asset position.
• (+) -> Sell assets to foreigners
• (-) -> buy assets from abroad

Financial Account Balance

• Increase in Foreign-Owned Assets in Home Country - Increase in Home-Owned Assets Abroad
• Assets include: Securities, Currencies, Bank loans, Foreign direct investment (FDI)

Fundamental Balance-of-Payments Identity

• CA Balance = -(FA Balance)

Capital Account

• 3rd account besides CA and FA
• Records international transfers of financial capital
  • debt forgiveness
  • migrants' transfers
• Not discussed in this course

CA/FA Balance Practice

(a) An Australian resident buys a smartphone from South Korea for $500

Answer: CA = -500, FA = +500 (import goods)

(b) An Italian friend comes to Australia and stays at Plaza Hotel paying for $400 with his Italian VISA card.

Answer: CA = +400, FA = -400 (export serices)

(c) Australian government send $700 medicines to an African country affected by epidemic.

Answer: CA = 700 - 700 = 0, FA = 0 (unilateral transfer)

Reason: CA +700 (export goods); CA -700 (gives up receiving money)

(d) An Australian resident purchases shares from FIAT Italy with AUD.

Answer: CA = 0; FA = -400 + 400 = 0

Reason: FA -400 (buy shares from abroad); FA +400 (sell AUD to FIAT Italy)

Net International Investment Position (NIIP)

NIIP = A - L

• A -> foreign assets held by home residents
• L -> home assets held by foreigners

A CA deficit implies that the country has to

• Reduce A (int'l asset position)
• Increase L (int'l liability position)

Valuation changes: prices of financial instruments (currency, stock, bond) can change

ΔNIIP = CA + Valuation Changes

Valuation Changes Example

International assets:

• 25 shares of Italian carmaker Fiat
• 2 € per share
• 2 AUD per €

International liabilities:

• 80 Australian bonds held by foreigners
• 1 AUD per unit

NIIP = A - L = 25 · 2 · 2 - 80 · 1 = 20 AUD

【Event 1】Euro depreciates to 1 AUD per €

• NIIP = A - L = 25 · 2 · 1 - 80 = -30 AUD

【Event 2】Share price of Fiat increases to 7 € per share

• NIIP = A - L = 25 · 7 · 1 - 80 = 95 AUD

NIIP Practice

The question is about balance of payments of US. Exchange rate is EUR/USD = 1.5

Question 1

• US starts 2023 with 100 shares of Volkswagen (1 EUR/share)
• RW holds 200 units of US government bonds (2 USD/unit)
• Calculate $NIIP^{US}$

Answer 1

• $A^{US}$ = 100 · 1 · 1.5 = 150 USD
• $L^{US}$ = 200 · 2 = 400 USD
• $NIIP^{US}=A^{US}-L^{US}$ = -250 USD

Question 2

• US 2023 exports: Toys (7 USD)
• US 2023 imports: Shirts (9 EUR)
• Rate of return on Volkswagen shares: 5%
• Rate of return on Outlandian bonds: 1%
• US residents received 3 EUR from relatives living abroad
• US government donated 4 USD to a hospital in Africa
• Calculate TB, NII, Net Unilateral Transfers, CA, NIIP

Answer 2

• TB = 7 - 1.5 · 9 = -6.5 USD
• NII (Income Balance) = 0.05 · 100 · 1.5 - 0.01 · 400 = 3.5 USD
• Net Unilateral Transfers = 3 · 1.5 - 4 = 0.5 USD
• Current Account = -6.5 + 3.5 + 0.5 = -2.5 USD
• NIIP = -250 - 2.5 = -252.5 USD

L2.1 - NIIP-NII-Paradox

If no valuation changes: $NIIP_{t}=(CA_{t}+\dots+CA_{1})+NIIP_{0}$

NII and NIIP in the US

• NIIP < 0 (largest external debtor)
• NII > 0 (receives more investment income)

Dark Matter Hypothesis

NIIP > 0, but the BEA fails to account for all of it

• BEA -> Bureau of Economic Analysis (US)

Hypothesis:

• U.S. FDI contains intangible capital (e.g. brand capital)
• intangible capital not reflected in NIIP, but recorded in NII
• Therefore, NIIP < 0 and NII > 0

Validation:

• TNIIP (true NIIP) = NIIP + Dark Matter
  • NIIP = observed NIIP (-$11.1T in 2020)
  • NII = net investment income ($0.1909T in 2020)
  • r = interest rate on net foreign assets (0.05)
  • NII = r · TNIIP
• TNIIP = NII / r = $3.818T
• Dark Matter = TNIIP - NIIP = $14.9T
• It's unlikely for $14.9T to go unnoticed by BEA

Return Differentials

Hypothesis:

• A -> risky, high-return assets (foreign equity and FDI)
• L -> safer, low-return assets (US government bonds, T-bills)
• Let: $r^{A}$ (return on A), $r^{L}$ (return on L)
• If interest rate differential $r^{A}-r^{L}\gt 0$, then NIIP < 0 and NII > 0

Validation:

• $NII=r^{A}A-r^{L}L$
  • A = $32.2T
  • L = $46.3T
  • $r^{L}$ = 0.37%
  • NII = $0.1909T
• Therefore, $r^{A}$ = 1.12%, $r^{A}-r^{L}$ = 0.75%
• This is more plausible than Dark Matter Hypothesis

Flipped NIIP-NII-Paradox

$NIIP^{US}=A^{US}-L^{US}=L^{RW}-A^{RW}=-NIIP^{RW}$

$NII^{US}=r^{A}A^{US}-r^{L}L^{US}=r^{A}L^{RW}-r^{L}A^{RW}=-NII^{RW}$

The rest of the world (RW) must experience a flipped paradox: $NIIP^{RW}$ > 0 and $NII^{RW}$ < 0

• e.g. China

L2.2 - CA Sustainability

CA surplus and deficit

• CA surplus (pay < receive): CN, DE, JP
• CA deficit (pay > receive): US, UK

Perpetual Trade Balance (TB)

• NO: initial NIIP < 0 (debtor)
  • need TB surplus to service its debt
• YES: initial NIIP > 0
  • finance the deficit with interests from net investments abroad

Perpetual TB Deficit Analysis

Rules:

• The economy lasts for only two periods, periods 1 and 2.
• The interest rate r on investments held for one period is exogenously given
• Further assumptions:
  • no international compensation to employees
  • no unilateral transfers
  • no valuation changes of assets in this world

Notation:

• $TB_{1}$: Trade balance in period 1
• $CA_{1}$: Current account balance in period 1
• $B_{0}$: The country's NIIP at the beginning of period 1
• $B_{1}$: The country's NIIP at the end of period 1

From definition:

• $CA_{1}=TB_{1}+rB_{0}$ (CA balance definition, Assumptions 1 & 2)
• $CA_{1}=B_{1}-B_{0}$ (ΔNIIP definition, Assumption 3)

We can dirive:

• $B_{1}=(1+r)B_{0}+TB_{1}$
• $B_{2}=(1+r)B_{1}+TB_{2}$

We can get $B_{0}=\frac{B_{2}}{(1+r)^{2}}-\frac{TB_{1}}{(1+r)}-\frac{TB_{2}}{(1+r)^{2}}$

Transversality Condition

• Let T denote the terminal date of the economy, then $B_{T} = 0$
• If $B_{T} \lt 0$, RW can't collect debt from the country
• If $B_{T} \gt 0$, the country can't collect debt from RW
• In this analysis, $B_{2} = 0$

Given Transversality Condition:

• $B_{0}=-\frac{TB_{1}}{(1+r)}-\frac{TB_{2}}{(1+r)^{2}}$
• If $B_{0} \gt 0$, it's possible to have $TB_{1} \lt 0$ and $TB_{2} \lt 0$
• If $B_{0} \lt 0$, it must be $TB_{1} \gt 0$ or $TB_{2} \gt 0$

Since $NIIP^{US} \lt 0$ currently, it will have to run TB surplus in the future.

Perpetual CA Deficit Analysis

Same rules from Perpetual TB Deficit Analysis

We had already derived:

• $CA_{1}=B_{1}-B_{0}$ (ΔNIIP definition, Assumption 3)
• $CA_{2}=B_{2}-B_{1}$ (same reasoning)

We can get:

• $B_{0}=-CA_{1}-CA_{2}+B_{2}$

Transversality Condition ($B_{2}=0$):

• $B_{0}=-CA_{1}-CA_{2}$
• If $B_{0} \gt 0$, it's possible to have $CA_{1} \lt 0$ and $CA_{2} \lt 0$
• If $B_{0} \lt 0$, it must be $CA_{1} \gt 0$ or $CA_{2} \gt 0$

Since $NIIP^{US}$ < 0 currently, it will have to run CA surplus in the future.

Four ways of viewing CA

CA as Changes in NIIP

• $CA_{t}=B_{t}-B_{t-1}$
• Notations:
  • $CA_{t}$: The country's current account in period t
  • $B_{t}$: The country's NIIP at the end of period t
• If $CA_{t} \lt 0$, NIIP falls
• If $CA_{t} \gt 0$, NIIP rises

CA as Reflections of TB and NII

• $CA_{t}=TB_{t}+rB_{t-1}$
• Notation:
  • $TB_{t}$: The country's trade balance in period t
  • $r$: The interest rate on investments held for one period
• This is the original definition of CA

CA as the Gap Between Savings and Investment

• $CA_{t}=S_{t}-I_{t}$

CA as the Gap between National Income and Domestic Absorption

• $CA_{t}=Y_{t}-A_{t}$
• Note: domestic absorption $A_{t}\equiv C_{t}+I_{t}+G_{t}$

Perpetual TB Deficits (Infinite Economy)

Time periods $t = 1, 2, \ldots$

The interest rate $r (> 0)$ on investments held for one period is constant over time

As before $B_1 = (1 + r)B_0 + TB_1$, and thus $B_0 = \frac{B_1}{1 + r} - \frac{TB_1}{1 + r}$

Generalizing, for all $t \geq 1$, $B_{t-1} = \frac{B_t}{1 + r} - \frac{TB_t}{1 + r}$

Therefore, repeatedly using (13), for all $T \geq 1$,

$B_0 = \frac{B_1}{1 + r} - \frac{TB_1}{(1 + r)}$

$B_0 = \frac{B_2}{(1 + r)^2} - \frac{TB_1}{1 + r} - \frac{TB_2}{(1 + r)^2}$

$B_0 = \frac{B_3}{(1 + r)^3} - \frac{TB_1}{1 + r} - \frac{TB_2}{(1 + r)^2} - \frac{TB_3}{(1 + r)^3}$

$B_0 = \frac{B_T}{(1 + r)^T} - \frac{TB_1}{1 + r} - \ldots - \frac{TB_T}{(1 + r)^T}$ (14)

Assuming the transversality condition: $\lim_{T \to \infty} \frac{B_T}{(1 + r)^T} = 0$

Taking limits on both sides of (14) yields: $B_0 = - \sum_{t=1}^{\infty} \frac{TB_t}{(1 + r)^t}$ (15)

We obtain the same conclusion as in the 2-period case:

• If $B_0 > 0$, it is possible (but not necessary) to have $TB_t < 0$ for all $t \geq 1$
• If $B_0 < 0$, we must have $TB_t > 0$ for some $t \geq 1$

A country can run a perpetual trade deficit only if the country has positive initial NIIP

Perpetual CA Deficits (Infinite Economy)

We will show that in contrast to finite horizon economies, in infinite horizon economies perpetual CA deficits can be possible even if the country has a negative initial NIIP

We give the example of an economy that is growing and dedicates a growing amount of resources to pay interest on its external debt

Suppose that:

• $B_0 < 0$
• $r > 0$
• $\alpha \in (0, 1)$
• $TB_t = -\alpha rB_{t-1}$ (whenever $B_{t-1} < 0$ the country generates a TB surplus that suffices to cover a fraction $\alpha$ of its interest obligations)

Then $B_t = (1 + r)B_{t-1} + TB_{t-1} = (1 + r - \alpha r) B_{t-1} < 0, \text{ for all } t \geq 1$ (16)

• $(1 + r - \alpha r) > 0$
• NIIP is negative in all periods

Moreover, $CA_t = rB_{t-1} + TB_t = r(1 - \alpha)B_{t-1} < 0, \text{ for all } t \geq 1$

• $r(1 - \alpha) > 0$
• $B_{t-1} < 0$
• the country runs a perpetual CA deficit

(16) implies that for all $t \geq 1, B_t = (1 + r - \alpha r)^t B_0$

Thus, the transversality condition is satisfied:

$\frac{B_t}{(1 + r)^t} = \left(\frac{1 + r - \alpha r}{1 + r}\right)^t B_0 \to 0$ as $t \to \infty$

For all $t \geq 1, TB_t = -\alpha rB_{t-1} = -\alpha r(1 + r - \alpha r)^{t-1} B_0 > 0$, therefore, the trade balance grows unboundedly over time

Since $TB_t = X_t - IM_t = Q_t - C_t - I_t - G_t = Q_t - A_t$ and $A_t = C_t + I_t + G_t \geq 0$, the GDP $Q_t$ must grow unboundedly over time, as well.

L3 - CA Determination

Optimal Intertemporal Allocation

• makes intertemporal consumption and saving decisions
• smoothes consumption over time by borrowing and lending

Intertemporal Budget Constraint

Rules:

• Two-period small open economy: periods 1 and 2
• The single consumption good in the economy is perishable
  • cannot be stored across periods
• The single asset traded in the financial market is a bond
  • measured in units of the consumption good
• There is a single representative household (HH) in the economy endowed with
  • $B_{0}$ units of the bond at the beginning of period 1
  • $Q_{1}$ units of the good in period 1
  • $Q_{2}$ units of the good in period 2
• Interest Rates:
  • $r_{0}$ for the initial bond holdings
  • $r_{1}$ for the bonds held at the end of period 1
• The HH can reallocate resources between periods by purchasing or selling bonds (via international financial market)
  • $B_{t}=NIIP$

The HH's budget constraint in period 1 is:

• $C_{1}+B_{1}=(1+r_{0})B_{0}+Q_{1}$
• Notations:
  • $C_{1}$ is consumption in period 1
  • $B_{1}$ is the amount of bonds held at the end of period 1

The HH's budget constraint in period 2 is:

• $C_{2}+B_{2}=(1+r_{1})B_{1}+Q_{2}$
• Note: $B_{2}=0$ (transversality condition)

By combining the above, we obtain the HH's intertemporal budget constraint. The intertemporal budget constraint describes the consumption paths $(C_{1}, C_{2})$ that the HH can (just) afford.

$C_{1}+\frac{C_{2}}{1+r_{1}} =(1+r_{0})B_{0}+Q_{1}+\frac{Q_{2}}{1+r_{1}}$ (Consumption Values = Initial Assets + Total Income Values)

The slope of the budget constraint is $-(1+r_{1})$

Utility and Indifference Curves

The utility function $U(C_{1}, C_{2})$ can be represented by indifference curves (ICs).

Common Utility Functions

• Logarithmic: $U(C_{1},C_{2})=\ln C_{1}+\ln C_{2}$
• Square-root: $U(C_{1},C_{2})={\sqrt{C_{1}}}+{\sqrt{C_{2}}}$
• Cobb-Douglas: $U(C_{1},C_{2})=(C_{1})^{\alpha}(C_{2})^{1-\alpha}$ where $(0\lt \alpha\lt 1)$

Properties of Indifference Curves

• ICs do not intersect
• ICs are downward sloping
  • True if more is better
• The right-upper ICs indicate higher levels of utility
• ICs have a bowed-in shape towards the origin (convex toward the origin)

Slope of Indifference Curves (MRS)

${\mathrm{MRS}}={\frac{U_{1}(C_{1},C_{2})}{U_{2}(C_{1},C_{2})}}$, where:

• $U_{1}(C_{1},C_{2})=\frac{\partial U(C_{1},C_{2})}{\partial C_{1}}$
• $U_{2}(C_{1},C_{2})=\frac{\partial U(C_{1},C_{2})}{\partial C_{2}}$

Slope of IC containing $(C_{1},C_{2})$ at $(C_{1},C_{2})$ is $-{\frac{U_{1}(C_{1},C_{2})}{U_{2}(C_{1},C_{2})}}$

Optimal Intertemporal Allocation

The HH maximizes utility subject to the intertemporal budget constraint.

The optimal consumption path is point B.

At the optimal bundle $(C_{1},C_{2})$ the IC is tangent to the intertemporal budget constraint (IBC)

$-{\frac{U_{1}(C_{1},C_{2})}{U_{2}(C_{1},C_{2})}}=-(1+r_{1})$

or equivalently, $U_{1}(C_{1},C_{2})=(1+r_{1})U_{2}(C_{1},C_{2})$

TB and CA in Equilibrium

Exogenously given are $r_{0}$, $B_{0}$, $r^{*}$, $Q_{1}$ and $Q_{2}$. An equilibrium is a consumption path $(C_{1},C_{2})$ and an interest rate $r_{1}$ such that:

• Feasibility of the intertemporal allocation
  • $C_{1}+\frac{C_{2}}{1+r_{1}} =(1+r_{0})B_{0}+Q_{1}+\frac{Q_{2}}{1+r_{1}}$
• Optimality of the intertemporal allocation
  • $U_{1}(C_{1},C_{2})=(1+r_{1})U_{2}(C_{1},C_{2})$
• Interest rate parity condition
  • $r_{1}=r^{*}$ (free capital mobility)

In this economy, $TB_{1}=Q_{1}-C_{1}$, $TB_{2}=Q_{2}-C_{2}$

And $CA_{1}=r_{0}B_{0}+TB_{1}$, $B_{1}=B_{0}+CA_{1}$, $CA_{2}=r_{1}B_{1}+TB_{2}$

Also, since we don't have investments in this economy $(I_{1}=I_{2}=0)$, $S_{1}=CA_{1}$, $S_{2}=CA_{2}$

Therefore, the HH's willingness to save determines the TB and CA

Temporary Output Shock

Budget Constraint and Optimal Allocation

• $C_{1}$ and $C_{2}$ are normal goods (their consumption increases with income)
• Output in Period 1 = $Q_{1}-\Delta$
• Output in Period 2 = $Q_{2}$
• A is the old and A′ the new endowment point
• B is the old and B′ the new optimal consumption path
• $C_{1}$ declines by less than ∆
• $TB_{1}$ deteriorates

Summary

• Relatively small effect on the consumption path $(C_{1},C_{2})$
• Temporary negative income shocks are smoothed out by borrowing from the rest of the world
• Generally, one should expect the borrowing to move the country's trade balance and current account significantly

Permanent Output Shock

Budget Constraint and Optimal Allocation

• Output in Period 1 = $Q_{1}-\Delta$
• Output in Period 2 = $Q_{2}-\Delta$
• A is the old and A′ the new endowment point
• B is the old and B′ the new optimal consumption path
• $TB_{1}$ doesn't change much

Summary

• Relatively large effect on the consumption path $(C_{1},C_{2})$
• Generally, one should expect permanent negative income shocks to lead to similarly sized reductions in $C_{1}$ and $C_{2}$
• Generally, one should expect the country's trade balance and current account to not be much affected

Terms of Trade Shocks

Consider an economy which exports endowments of oil $(Q_{1},Q_{2})$ and imports food for consumption $(C_{1},C_{2})$

Then, the HH's budget constraints for period 1 and 2 are:

• $C_{1}+B_{1}=(1+r_{0})B_{0}+TT_{1}Q_{1}$
• $C_{2}=(1+r_{1})B_{1}+TT_{2}Q_{2}$

Terms of Trade (TT) are the relative price of exports in terms of imports:

• $TT_{1}=\frac{P_{1}^{X}}{P_{1}^{M}}$
• $TT_{2}=\frac{P_{2}^{X}}{P_{2}^{M}}$

With Terms of Trade Shocks, equlibrium would be:

• Feasibility of the intertemporal allocation (with $TT_1$ and $TT_2$)
  • $C_{1}+\frac{C_{2}}{1+r_{1}} =(1+r_{0})B_{0}+TT_1 Q_{1}+\frac{TT_2 Q_{2}}{1+r_{1}}$
• Optimality of the intertemporal allocation
  • $U_{1}(C_{1},C_{2})=(1+r_{1})U_{2}(C_{1},C_{2})$
• Interest rate parity condition
  • $r_{1}=r^{*}$ (free capital mobility)

Capital Controls

CA deficits are often viewed as bad for a country.

Assume the country's authority introduces a policy that prohibits borrowing, that is, requires $B_{1}$ ≥ 0

Result:

• The HH can't borrow
• The optimal allocation moves from B to A (IC is lower)
• Optimal consumption path changes to:
  • $C_{1}=Q_{1}$
  • $C_{2}=Q_{2}$
• $TB_{1}=TB_{2}=0$ $(TB_{t}=Q_{t}-C_{t})$

Logarithmic Utility Equilibrium

$U(C_{1},C_{2})=\ln C_{1}+\ln C_{2}$

Using derivatibe formula $\frac{d\ln(x)}{dx}=\frac{1}{x}$, we get:

$U_{1}(C_{1},C_{2})=\frac{\partial U(C_{1},C_{2})}{\partial C_{1}}=\frac{\partial(\ln C_{1}+\ln C_{2})}{\partial C_{1}}=\frac{1}{C_{1}}$

$U_{2}(C_{1},C_{2})=\frac{\partial U(C_{1},C_{2})}{\partial C_{2}}=\frac{\partial(\ln C_{1}+\ln C_{2})}{\partial C_{2}}=\frac{1}{C_{2}}$

Therefore, the equilibrium condition $U_{1}(C_{1},C_{2})=(1+r_{1})U_{2}(C_{1},C_{2})$ becomes ${\frac{1}{C_{1}}}=(1+r_{1}){\frac{1}{C_{2}}}$

From the above, we can derive $C_{2}=(1+r_{1})C_{1}$

By substituting into Feasibility of the intertemporal allocation, we get:

$C_{1}={\frac{1}{2}}[(1+r_{0})B_{0}+Q_{1}+{\frac{Q_{2}}{1+r_{1}}}]$

$C_{2}={\frac{1}{2}(1+r_{1})}[(1+r_{0})B_{0}+Q_{1}+{\frac{Q_{2}}{1+r_{1}}}]$

L4.1 - Int. R. Shocks & Tariffs in an Endowment Economy

World Interest Rate Shocks

World interest rate increase from $r^{*}$ to $r^{*}+\Delta$

HH as debtor

HH as creditor

Substitution effect (SE)

• An increase in the interest rate makes $C_{1}$ relatively more expensive and $C_{2}$ relatively cheaper
• Change in period 1 consumption $C_{1}$ due to change in relative "price"
  • $A=(C_{1}^{A},C_{2}^{A})$ = originally optimal path
  • $B=(C_{1}^{B},C_{2}^{B})$ = path that is optimal given budget line that passes through A and has slope ${(-1+r^{*}+\Delta)}$
  • $SE=C_{1}^{B}-C_{1}^{A}\lt 0$
• Saving in period 1 becomes more attractive

Income effect (IE)

• Change in C1 due to change in purchasing power
  • $B=(C_{1}^{B},C_{2}^{B})$ = previous path
  • $C=(C_{1}^{C},C_{2}^{C})$ = new optimal path
  • $IE=C_{1}^{C}-C_{1}^{B}$
• $r^{*}$ ↑ has two opposing effects on HH's purchasing power
  • The HH can purchase more on a fixed budget since the “price” $\frac{1}{1+r^{*}}$ of $C_{2}$ falls
  • The present value $\frac{Q_{2}}{1+r^{*}}$ of endowment $Q_{2}$ decreases
• Netting these effect, $r^{*}$ ↑ makes
  • debtors (for whom $Q_{1}\lt C_{1}$ and thus $Q_{2}\lt C_{2}$) poorer
  • creditors (for whom $C_{1}\lt Q_{1}$ and thus $C_{2}\lt Q_{2}$) richer
• Since $C_{1}$ is a normal good
  • IE < 0 if purchasing power decreases (debtor)
  • IE > 0 if purchasing power increases (creditor)

TE (total effect) = SE + IE

Int. R. Shocks Summary

Overall, if the interest rate increases:

• If the country is a debtor $(Q_{1}\lt C_{1})$:
  • IE < 0 as budget line shift that determines IE is to the left.
  • TE = SE + IE < 0 as SE < 0 and IE < 0.
  • Therefore, HH's savings increase after the shock.
• If the country is a creditor $(Q_{1}\gt C_{1})$:
  • It remains a creditor after the interest rate rise (by a revealed preference argument).
  • IE > 0 as budget line shift that determines IE is to the right.
  • TE = SE + IE (> or <) 0 depending on which of SE and IE dominates.

Import Tariffs

Given two-goods economy:

• exports endowments of oil $(Q_{1},Q_{2})$
• imports food for consumption $(C_{1},C_{2})$
• The HH treats $\tau_{t}$ and $L_{t}$ as exogeneously given

Assume that in each period t ∈ {1, 2}, the government

• imposes an import tariff $\tau_{t}\geq0$, and
• returns the revenue from $\tau_{t}$ via a lump-sum transfer $L_{t}$

Then, the HH's budget constraints for period 1 and 2 are:

$(1+\tau_{1})C_{1}+B_{1}=(1+r_{0})B_{0}+TT_{1}Q_{1}+L_{1}$

$(1+\tau_{2})C_{2}=(1+r_{1})B_{1}+TT_{2}Q_{2}+L_{2}$

HH's intertemporal budget constraint:

$(1+\tau_{1})C_{1}+\frac{(1+\tau_{2})C_{2}}{1+r_{1}}=(1+r_{0})B_{0}+TT_{1}Q_{1}+L_{1}+\frac{TT_{2}Q_{2}+L_{2}}{1+r_{1}}$

Slope: $[-\frac{1+\tau_{1}}{1+\tau_{2}}(1+r_{1})]$

The optimality condition (tangency of IC and IBC) becomes $U_{1}(C_{1},C_{2})=\frac{1+\tau_{1}}{1+\tau_{2}}(1+r_{1})U_{2}(C_{1},C_{2})$

• If τ1 = τ2, then there is no change from τ1 = τ2 = 0.
• If τ1 > τ2, then $C_{1}$ becomes relatively more expensive and, assuming diminishing marginal utilities, $C_{1}$ ↓ and $C_{2}$ ↑.
• If τ1 < τ2, then $C_{1}$ becomes relatively cheaper and, assuming diminishing marginal utilities, $C_{1}$ ↑ and $C_{2}$ ↓.

Import Tariffs Equilibrium

Exogenously given are $r_{0}$, $B_{0}$, $r^{*}$, $Q_{1}$, $Q_{2}$, $\tau_{1}$, and $\tau_{2}$.

An equilibrium is a consumption path $(C_{1},C_{2})$ and an interest rate $r_{1}$ such that:

• Feasibility of the intertemporal allocation
  • $C_{1}+\frac{C_{2}}{1+r_{1}} =(1+r_{0})B_{0}+TT_{1}Q_{1}+\frac{TT_{2}Q_{2}}{1+r_{1}}$
• Optimality of the intertemporal allocation
  • $U_{1}(C_{1},C_{2})=\frac{1+\tau_{1}}{1+\tau_{2}}(1+r_{1})U_{2}(C_{1},C_{2})$
• Interest rate parity condition
  • $r_{1}=r^{*}$ (free capital mobility)

Import Tariffs and TB

Question: Does an increase in import tariffs reduce imports and therefore improve TB?

$TB_{1}=TT_{1}Q_{1}-C_{1}$

An increase in import tariffs leads to

• TB ↑ if (present import tariff > expected future import tariff)
• TB - if (present import tariff = expected future import tariff)
• TB ↓ if (present import tariff < expected future import tariff)

Additional observation:

• In our model, $\tau_{1}\neq\tau_{2}$ leads to lower welfare than $\tau_{1}=\tau_{2}=0$ (no import tariffs)
• $\tau_{1}\neq\tau_{2}$ distorts the HH's optimal intertemporal allocation

L4.2 - CA Determination in a Production Economy

Now we introduce a representative firm in the economy, and firm's decision also significantly affects TB and CA.

Rules:

• Two-period small open economy: periods 1 and 2
• The single consumption good is perishable
• The single asset traded in the financial market is a bond (measured in units of the consumption good)
• There is a representative household (HH) endowed with $B_{0}^{h}$ units of the bond at the beginning of period 1
• There is a representative firm. The household owns the firm and obtains the firm's profits
  • $\Pi_{1}$ in period 1
  • $\Pi_{2}$ in period 2
• Interest Rates:
  • $r_{0}$ for the initial bond holdings
  • $r_{1}$ for the bonds held at the end of period 1

Firm

What does the firm do?

• makes investments in period $t$ that lead to output in $t + 1$
• finances its investments in period $t$ by issuing debt in $t$

Firm's Production Function

• ${Q}_{1}=A_{1}F(I_{0})$
• ${Q}_{2}=A_{2}F(I_{1})$
• Notations:
  • $F$ is a function
  • $A_{t}\gt 0\ (t=1,2)$ are technology parameters
  • $I_{0}$ - investment in period 0 and exogeneously given
  • $I_{1}$ - investment in period 1 and chosen by the firm

Firm's Debt

• The firm issues debt $D_{t}^{f} (t=1,2)$
• $D_{0}^{f}=I_{0}$ (exogenous; to be repaid in period 1)
• $D_{1}^{f}=I_{1}$ (to be repaid in period 2)

Production Function (MPK and MCK)

The firm chooses $I_{1}$ by maximizing profits

$\Pi_{2}=A_{2}F(I_{1})-(1+r_{1})D_{1}^{f} (\mathrm{where}\ D_{1}^{f}=I_{1})$

or $\Pi_{2}(l_{1})=A_{2}F(I_{1})-(1+r_{1})I_{1}$

The first-order condition for profit maximization is $\Pi_{2}^{\prime}(I_{1})=0$

Therefore, $0=A_{2}F^{\prime}(I_{1})-(1+r_{1})$

Thus, we have Optimal Investment Condition (MPK = MCK)

• $A_{2}F^{\prime}(I_{1})=(1+r_{1})$
• Marginal Product of Capital (MPK): ${\frac{d[A_{2}F(I_{1})]}{d I_{1}}}=A_{2}F^{\prime}(I_{1})$
• Marginal Cost of Capital (MCK): $(1+r_{1})$

We assume that F has the following properties:

• $F(0) = 0$
• Positive MPK: $F^{\prime}(l)=\frac{d F(I)}{dI}\gt 0$ for all $I\gt 0$
• Diminishing MPK: $F^{\prime\prime}(l)=\frac{d F^{\prime}(I)}{dI}\lt 0$ for all $I\gt 0$
• $\lim_{I \to 0}F^{\prime}(I)=\infty$ and $\lim_{I \to \infty}F^{\prime}(I)=0$
• The above properties guarantee:
  • the optimal investment condition has a unique solution
  • unique solution also maximizes profit
• Example: $F(I)={\sqrt{I}}$

Firm's Optimal Investment Decision

The firm chooses $I_{1}$ so that $A_{2}F^{\prime}(I_{1})=(1+r_{1})$

Household

Since the HH owns the firm, it receives the latter's profits:

• $\Pi_{1}=A_{1}F(I_{0})-(1+r_{0})D_{0}^{f}$
• $\Pi_{2}=A_{2}F(I_{1})-(1+r_{1})D_{1}^{f}$

The HH's budget constraint in period 1 is: $C_{1}+B_{1}^{h}=(1+r_{0})B_{0}^{h}+\Pi_{1}$

The HH's budget constraint in period 2 is: $C_{2}+B_{2}^{h}=(1+r_{1})B_{1}^{h}+\Pi_{2}$

Assume transversality condition $B_{2}^{h}=0$

From HH's budget constraint above, we obtain the HH's intertemporal budget constraint:

$C_{1}+\frac{C_{2}}{1+{r_{1}}}=(1+r_{0})B_{0}^{h}+\Pi_{1}+\frac{\Pi_{2}}{1+{r_{1}}}$

Economy's net foreign asset positions:

• $B_{0}=B_{0}^{h}-D_{0}^{f}$
• $B_{1}=B_{1}^{h}-D_{1}^{f}$
• Notations:
  • $B_{t}$: NIIP
  • $B_{t}^{h}$: Domestic Supply
  • $D_{t}^{f}$: Domestic Demand

By combining all the above, we obtain the economy's intertemporal resource constraint:

Steps

$C_{1}+\frac{C_{2}}{1+{r_{1}}}=(1+r_{0})(B_{0}+D_{0}^{f})+\Pi_{1}+\frac{\Pi_{2}}{1+{r_{1}}}$

$C_{1}+\frac{C_{2}}{1+{r_{1}}}=(1+r_{0})(B_{0}+D_{0}^{f})+[A_{1}F(I_{0})-(1+r_{0})D_{0}^{f}]+\frac{A_{2}F(I_{1})-(1+r_{1})D_{1}^{f}}{1+{r_{1}}}$

$C_{1}+\frac{C_{2}}{1+{r_{1}}}=(1+r_{0})(B_{0}+D_{0}^{f})+A_{1}F(I_{0})-(1+r_{0})D_{0}^{f}+\frac{A_{2}F(I_{1})}{1+{r_{1}}}-D_{1}^{f}$

$C_{1}+\frac{C_{2}}{1+{r_{1}}}+D_{1}^{f}=(1+r_{0})(B_{0}+D_{0}^{f}-D_{0}^{f})+A_{1}F(I_{0})+\frac{A_{2}F(I_{1})}{1+{r_{1}}}$

$C_{1}+\frac{C_{2}}{1+{r_{1}}}+D_{1}^{f}=(1+r_{0})B_{0}+A_{1}F(I_{0})+\frac{A_{2}F(I_{1})}{1+{r_{1}}}$

$C_{1}+\frac{C_{2}}{1+{r_{1}}}+I_{1}=(1+r_{0})B_{0}+A_{1}F(I_{0})+\frac{A_{2}F(I_{1})}{1+r_{1}}$

Equilibrium

Exogenously given are $r_{0}$, $B_{0}^{h}$, $r^{*}$, $I_{0}$, $D_{0}^{f}$, $A_{1}$ and $A_{2}$.

An equilibrium is $(C_{1},C_{2},I_{1},r_{1})$ such that:

• Feasibility of the intertemporal allocation
  • $C_{1}+\frac{C_{2}}{1+{r_{1}}}+I_{1}=(1+r_{0})B_{0}+A_{1}F(I_{0})+\frac{A_{2}F(I_{1})}{1+r_{1}}$
• Optimality of the intertemporal allocation
  • $U_{1}(C_{1},C_{2})=(1+r_{1})U_{2}(C_{1},C_{2})$
• Interest rate parity condition
  • $r_{1}=r^{*}$
• Optimal investment condition
  • $A_{2}F^{\prime}(I_{1})=(1+r_{1})$

Positive Productivity Shocks

Positive Anticipated Future Productivity Shock: Assume $\Delta\gt 0$ and that the period 2 technology parameter changes to $A_{2}^{\prime}=A_{2}+\Delta$, while $A_{1}$ remains unchanged

Firm's Decision

Productivity shocks affect the firm's optimal investment

• Positive shock: $I_{1}$ ↑, output ↑, $\Pi_{2}$ ↑
• Negative shock: $I_{1}$ ↓, output ↓, $\Pi_{2}$ ↓

Household's Reaction

Summary

Assuming that $C_{1}$ is a normal good:

• Δ optimal investment: $A_{2}\uparrow\to I_{1}\uparrow$
• HH's reaction: $A_{2}\uparrow\to\Pi_{2}\uparrow\to S_{1}\downarrow$
• CA deteriorates: $\Delta CA_{1}\lt 0\,(CA_{1}=S_{1}-I_{1})$

Other Productivity Shocks

Other productivity shocks includes:

• negative anticipated future productivity shocks
• temporary productivity shocks $(A_{1}^{\prime}\neq A_{1},\,A_{2}^{\prime}=A_{2})$
• permanent productivity shocks $(A_{1}^{\prime}\gt A_{1},\,A_{2}^{\prime}\gt A_{2})$ or $(A_{1}^{\prime}\lt A_{1},\,A_{2}^{\prime}\lt A_{2})$

The HH's reaction to such a shock depends on:

• shock is positive or negative
• shock is temporary or permanent
• shock is anticipated or not

L5.1 - Int. R. Shocks in a Production Economy

Firm's Reaction

A positive world interest rate shock:

• decreases the firm's investment level
• decreases output

Household's Reaction

If