Obstfeld and Rogoff (1996), Chap1.
Topics in International Finance
From the previous session:
\[ CA=S-I. \]
This identity tells us how domestic expenditure is financed.
It does not explain:
We now replace accounting descriptions with optimizing decisions and market equilibrium.
I. The aggregate economy
Consumption, government, investment, and equilibrium
II. The two-region world economy
Global market clearing and the world interest rate
III. Foreign borrowing and lending
Shocks, capital flows, and intertemporal prices
IV. What comes next
Labor mobility and the Feldstein-Horioka puzzle
Choice \(\rightarrow\) equilibrium \(\rightarrow\) international adjustment \(\rightarrow\) empirical puzzle
Live controls: M opens the outline · C annotates the slide · B opens the writing board.
Begin with consumption across time, then add government and productive investment.
A representative consumer maximizes lifetime utility \(U\).With T=2
\[ U=u(C_1)+\beta u(C_2), \qquad 0<\beta<1. \]
Initially assume:
The economy is small: its choices do not change (r).
With endowments \((Y_1,Y_2)\) and no initial foreign assets:
\[ C_1+\frac{C_2}{1+r} = Y_1+\frac{Y_2}{1+r}. \]
The right-hand side is the present value of lifetime resources.
The relative price of future consumption in units of present consumption is:
\[ \frac{1}{1+r}. \]
The optimum satisfies:
\[ u'(C_1)=\beta(1+r)u'(C_2) \]
Giving up one unit of consumption today:
The Euler equation is an optimality condition, not an accounting identity.
An alternative interpretation:
\[ \frac{\beta u'(C_2)}{u'(C_1)}=\frac{1}{1+r} \] The MRS of present for future consumption equals the price of future consumption in terms od present consumption
| Condition | Optimal profile |
|---|---|
| \(\beta(1+r)=1\) | \(C_2=C_1\) |
| \(\beta(1+r)>1\) | \(C_2>C_1\) |
| \(\beta(1+r)<1\) | \(C_2<C_1\) |
When \(\beta \ne 1+r\), incentives to tilt the consumption path.
In the first case the rate of return more than compensates the postponement of consumption.
In an open economy there is no reason for the consumption to be tied to current output. If \(\beta(1+r)=1\), then \(C_1=C_2\), but \(Y_2\) can be different from \(Y_1\)
Let \(B_t\) denote net foreign assets at the end of period \(t\). Then, the CA balance over period \(t\):
\[ CA_t=B_{t+1}-B_t. \]
- This definition stresses the view of the CA as intertemporal trade
\(CA_1>0\): net foreign assets rise; the economy lends.
\(CA_1<0\): net foreign assets fall; the economy borrows.
With no capital acumulation or government spending:
\[ CA_t=Y_t+rB_t-C_t \]
Gross National Income \(= Y_t + rB_t\), thus
\[ CA_t = GNI_t - C_t \]
\(Y_t\): the output produced within a country’s borders, GDP
\(rB_t\): net international factor payments
Except for countries with either large stocks of foreign wealth or debt, \(GDP-GNI\) is typically close to zero
| Economy | GDP (US$ tn) | GNI (US$ tn) | GNI - GDP (US$ bn) | Difference (% GDP) |
|---|---|---|---|---|
| Japan | 4.44 | 4.72 | +280.0 | +6.3% |
| Germany | 5.05 | 5.23 | +182.8 | +3.6% |
| France | 3.37 | 3.43 | +59.6 | +1.8% |
| Korea, Rep. | 1.87 | 1.90 | +32.2 | +1.7% |
| Saudi Arabia | 1.28 | 1.29 | +10.7 | +0.8% |
| Italy | 2.55 | 2.55 | +2.9 | +0.1% |
| Canada | 2.32 | 2.32 | +2.5 | +0.1% |
| China | 19.50 | 19.38 | -113.8 | -0.6% |
| United States | 30.77 | 30.59 | -183.9 | -0.6% |
| United Kingdom | 4.00 | 3.98 | -24.7 | -0.6% |
| Russian Federation | 2.56 | 2.54 | -17.9 | -0.7% |
| India | 3.96 | 3.91 | -49.4 | -1.2% |
| Türkiye | 1.60 | 1.57 | -23.3 | -1.5% |
| South Africa | 0.43 | 0.42 | -8.2 | -1.9% |
| Argentina | 0.68 | 0.67 | -13.9 | -2.0% |
| Indonesia | 1.45 | 1.41 | -38.4 | -2.7% |
| Australia | 1.80 | 1.75 | -50.1 | -2.8% |
| Mexico | 1.83 | 1.78 | -56.3 | -3.1% |
| Brazil | 2.28 | 2.21 | -74.2 | -3.3% |
The difference reflects income earned on cross-border labor and investment positions, not the current account itself.
Source: World Bank, World Development Indicators, 2025. Indicators NY.GDP.MKTP.CD and NY.GNP.MKTP.CD; API updated July 13, 2026. EU and African Union excluded to avoid overlapping aggregates.
\(B_1 = B_3 =0\). Thus,
\(CA_1 = B_{2} -B_1 = Y_1 -C_1\)
\(CA_2 = B_{3} -B_2 = Y_1 -C_1 = -CA_1\)
In a two period model with neither initial nor terminal assets \(CA_1+CA_2=0\)
From the budget constraint:
\[
C_2 = Y_2 - (1+r)(C_1+Y_1)
\]
The benefits of trading are clear in a simple two-period consumption diagram .
Suppose current income is temporarily low:
\[ Y_1<Y_2. \]
Autarky requires \((C_1=Y_1)\) and \((C_2=Y_2)\)
A current-account deficit can be the optimal response to uneven income—not evidence of economic failure.
Under capital-market access:
Suppose Government consumption, \(G\), enters the utility function in the form \(u(C) + \nu(G)\).
The Government appropriates \(G_1,G_2\) over the two periods in taxes from the private sector. As they reduce resources available for private consumption, the lifetime budget constraint becomes:
\[ C_1+\frac{C_2}{1+r} = Y_1+\frac{Y_2}{1+r} - G_1-\frac{G_2}{1+r}. \]
And the current account:
\[
CA_t=Y_t+rB_t-C_t-G_t.
\]
Government spending affects the current account through national saving.
A temporary government-spending increase is partly smoothed
Assume:
\(\beta(1+r)=1\) \(\Big[\Rightarrow C_1=C_2= \bar{C})\big]\),
\(Y_1=Y_2=\bar Y\), and
\(\boxed{G_1>0;G_2=0}\)
Incentives to borrow against a relatively larger second period after-tax income. Then:
\[ \bar{C}=\bar{Y}-\Big[\frac{1+r}{2+r}\Big]G_1. \] Therefore:
\(CA_1 = \bar{Y}-\bar{C} - G_1 = -\frac{G}{2+r}<0\)
Note: Private consumption falls by less than current government spending. With no \(G\) \(\bar{C}=\bar{Y}\).
If government spending rises equally in both periods:
\[
G_1=G_2=\bar G,
\]
then under the same smoothing assumptions:
\[ \bar C=\bar Y-\bar G. \]
The current account need not change because private consumption reflects the persistent loss of resources.
\(CA_1 = \bar{Y}-\bar{C} - \bar G\)
Persistence—not merely the sign of a fiscal shock—determines the response.
The result depends on:
The identity \(CA=(S_p-I)+(T-G)\) does not establish a one-for-one causal effect.
Now let:
\[ Y_t=F(k_t),\qquad F'(k)>0,\quad F''(k)<0, \]
and:
\[ k_{t+1}=k_t+I_t, \]
ignoring depreciation.
The current account is:
\[ CA_t=Y_t+rB_t-C_t-G_t-I_t. \]
With \(k_1\) given and no terminal investment:
\[ C_1+I_1+\frac{C_2}{1+r} = F(k_1)-G_1+\frac{F(k_1+I_1)-G_2+k_1+I_1}{1+r}. \]
Investment uses one unit today and raises future resources through production and the remaining capital stock.
The first-order condition for \(I_1\) is:
\[ \boxed{F'(k_2)=r},\qquad k_2=k_1+I_1. \]
Invest until:
\[ \text{marginal product of capital} = \text{world cost of funds}. \]
A profitable project should not be rejected merely because domestic saving is low.
Under the benchmark assumptions:
Therefore the desired capital stock does not depend directly on \(\beta\).
This separation fails with credit limits, default risk, country spreads, taxes, or incomplete markets.
<div class="control-row"><label for="inv-a">Future productivity, A₂</label><input id="inv-a" type="number" value="2" step="0.1"></div>
<div class="control-row"><label for="inv-alpha">Capital elasticity, α</label><input id="inv-alpha" type="number" value="0.35" min="0.05" max="0.90" step="0.05"></div>
<div class="control-row"><label for="inv-r">World rate, r (%)</label><input id="inv-r" type="number" value="10" step="1"></div>
<div class="control-row"><label for="inv-k1">Initial capital, k₁</label><input id="inv-k1" type="number" value="10" step="1"></div><div class="metric-row">
<div class="metric"><span>Desired k₂</span><strong id="inv-k2">—</strong></div>
<div class="metric"><span>Investment I₁</span><strong id="inv-i1">—</strong></div>
<div class="metric"><span>F′(k₂)</span><strong id="inv-mpk">—</strong></div>
</div>
<div class="identity-status" id="inv-reading">—</div>
<p class="small">Illustration: \(F(k)=A_2k^\alpha\). Negative \(I_1\) means desired capital is below \(k_1\).</p>In autarky, period-1 consumption and investment jointly determine period-2 resources.
The intertemporal production possibility frontier satisfies:
\[ \frac{dC_2}{dC_1} = -\left[1+F'(k_2)\right]. \]
Its curvature reflects diminishing marginal productivity.
At the closed-economy allocation:
The common tangent implies:
\[ \frac{u'(C_1)}{\beta u'(C_2)} = 1+r^A = 1+F'(k_2). \]
If:
\[ r^A>r, \]
the world cost of borrowing is below the domestic autarky return.
The economy can:
The gain from trade now comes from consumption smoothing and production reallocation.
With production:
\[ S_t=Y_t+rB_t-C_t-G_t, \]
so:
\[ CA_t=S_t-I_t. \]
Saving and investment respond jointly to common shocks and prices.
Their movements cannot be interpreted causally from the identity alone.
The world interest rate becomes an equilibrium price rather than an exogenous parameter.
Consider Home and Foreign, with starred variables for Foreign.
In the endowment economy:
\[ Y_t+Y_t^*=C_t+C_t^*. \]
Equivalently:
\[ CA_t+CA_t^*=0. \]
One region can lend only if the other borrows.
Suppose:
\[ r_H^A<r_F^A. \]
When markets open:
\[ r_H^A<r^w<r_F^A. \]
International capital flows equalize one intertemporal price across regions.
With productive capital:
\[ S_1+S_1^*=I_1+I_1^*. \]
Therefore:
\[ CA_1+CA_1^* = (S_1-I_1)+(S_1^*-I_1^*)=0. \]
The world interest rate clears the global saving and investment market.
For each region:
World equilibrium satisfies:
\[ [S(r)-I(r)]+[S^*(r)-I^*(r)]=0. \]
Efficiency requires:
[ MRS_H=MRS_F=1+r^w, ]
and:
[ F’(k_2)=F{’}(k_2^)=rw. ]
Goods markets also clear.
This conclusion depends on complete markets and the absence of distortions.
Shocks alter desired saving, investment, the world rate, and capital flows.
If (Y_1) rises temporarily in Home:
In two-region equilibrium, the world interest rate tends to fall.
Foreign benefits from cheaper borrowing even though its own output did not change.
Let:
[ Y_2=A_2F(k_2). ]
Optimal investment satisfies:
[ A_2F’(k_2)=r^w. ]
If (A_2) rises:
A higher (r^w):
Large-economy shocks are transmitted abroad through a price, not only through trade quantities.
An interest-rate increase combines:
The elasticity of intertemporal substitution helps determine which force dominates.
An upward-sloping saving schedule is a benchmark result, not a universal law.
For a period-1 lender:
For a period-1 borrower:
If Home becomes less patient:
[ , ]
desired saving shifts left.
In equilibrium:
If future productivity rises in Home:
An investment-driven deficit differs fundamentally from a consumption boom.
With present consumption as numeraire:
[ p_2=. ]
An increase in (r) lowers the present price of future goods.
The welfare effect also depends on the economy’s net foreign asset position.
A deficit may optimally finance:
Risk rises when borrowing finances persistent absorption without sufficient future resources or creates currency and rollover exposure.
A country reports a current-account deficit of (4%) of GDP.
Predict movements in (C_1), (S_1), (I_1), and (r^w) under:
The same deficit can emerge from different structural mechanisms.
Capital mobility raises questions about the movement of factors and finance.
If workers can move as well as capital, cross-country differences can adjust through:
The next extension asks how labor mobility interacts with capital flows and welfare.
If capital is highly mobile, domestic investment should not need to track domestic saving closely.
Yet Feldstein and Horioka found a surprisingly strong cross-country association:
\[ \frac{I_i}{Y_i} = \alpha+\beta_{FH}\frac{S_i}{Y_i}+\varepsilon_i. \]
Why is (_{FH}) high when capital is internationally mobile?
Tópicos Selectos de Finanzas Internacionales | Isai Guizar