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Economics/Why must net capital outflow = Trade Balance

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Question
Hi Eklimur,

Assume an open economy.

I am not convinced that Net Capital Outflow equal trade balance.

Specifically, I am referring to this equation:

 S-I = NX

where:

 S = Domestic Savings
 I = Domestic Investments

 NX = Net Exports

I think I don't understand this concept because I am not sure how certain transactions are reflected in this equation. For example:

If an American buys a PS4 from Japan. Then, from America's perspective:

 Net Exports goes down (because the PS4 is imported) -> Does consumption go up? If consumption goes up must it necessarily reduce saving?

Another example I have no idea at all on how to account for is if the American government issues bonds to China to say, build some infrastructure in America.

Please help me understand this identity.

Answer
Hi Sara,

Thank you for asking my help in connection with your understanding of a common international financial identity. As you didn’t give me any idea about your academic background, and as you seem to grapple with a rather simple set of macroeconomic variables in an open-economy setting, I assume you have an exposure to elementary algebra. Although full-dress treatment would call for advanced mathematics, I intentionally don’t wish to bring calculus and linear algebra into my answer because that may be too far-fetched and abstruse for you, with which you yourself could have easily sorted it out. So I keep to least possible technicality and use only minimal school-level algebra to get my explanation across to you.

So far I understand, you put forward one main question with two subsidiary implications.

First, as your main question on which devolves all my three-faceted answer, you want a clear answer to the relationship between the variables, which may be endogenous as well as exogenous, given essentially by the identity S – I = NX.

Second, you want to know if purchase of a commodity from a foreign country (such as from Japan) by another country (such as the U.S.) would lead to a rise in its consumption and subsequent simultaneous or in-tandem decrease in saving.

Third, you also put an inquiry into whether issuance of bonds by one country (such as the U.S.) to another country (such as China) will have any spinoff from issuing country’s building of some infrastructure.

Hence, before the other two answers would follow logically, I would like to treat of you first or the main question at the outset.   


1.   THE INTERNATIONAL-FINANCE IDENTITY
Take a look at any elementary macroeconomics textbook or, for details, at any book on economics of finance. You will see the familiar national-income decomposition:

Y = C + I + C + X−M      (1)

where Y is gross national product (please find the difference between gross national product and gross domestic product from any elementary textbook or by browsing the search engine), C is private consumption, I is private investment, G is government expenditure, and X−M stands for net exports (exports−imports).

Now transform equation (1) by subtracting T from both sides (the axioms of equation tell us that an equation is unchanged if equal amounts are added to or subtracted from, or multiplied/divided by, both sides). Then equation (1) can be written as equation (2) below:

Y –T = C +I + G – T + X−M   (2)

We subtract C from both sides of equation (2) and obtain equation (3).

Y –T – C = I + G – T + X−M   (3)

Next we bring two important concepts to our above equation. You are also aware that tax or T is a big source of revenue to the government, and T plays a double-edged role. One, T reduces people’s income which, in turn, reduces their savings, so Private Saving becomes Disposable Income (Y – T) minus Consumption C, or (Y –T – C ). Two, T forms government revenue, and the government makes spending G by an amount by which T is reduced, which leaves us with the Public Saving = T – G.  These two facets of T playing with Y and G yield the breakup of Savings or S into two parts –Private Saving (PS) and Public saving (GS):

S = PS + GS = (Y–T −C)+(T – G)   (4)

The identity we have developed in equation (4) above could be embedded in our equation (3). Applying the axioms of equation, we subtract (G –T) from both sides of (3) so that (G –T) in the right side vanishes and (T –G) in the left side pops up, and we obtain equation (5):

(Y –T – C) + (T –G) = I + X−M   (5)

Finally, we subtract I from both sides, and get

[(Y –T – C) + (T –G)] – I = X−M   (6)

Since, as given in our equation (4) above, we call Y –T – C = PS (Private Savings) and T – G = GS (Public Saving). Substituting these into equation (6) yields:

PS + GS – I = X – M       (7)

As PS+GS=S (Total Savings), PS+GS – I = S – I, and as X – M = NX, we at long last arrive at your identity:

S – I = NX         (8)

2.   THE IMPACT OF PURCHASE OF A FOREIGN COMMODITY ON DOMESTIC CONSUMPTION AND SAVING
The second part of your question can easily be deduced from our above analysis. Take it, first, that domestic investment can be greater or less than national saving, and the difference between saving and investment determines the current-account imbalance for an open economy. A current account surplus shows that some national saving is being invested abroad; and a current account deficit shows that net capital inflows from abroad are financing domestic investment. Let me give a simple numerical example.

The economy’s current account CA is defined as the difference between exports and imports of goods and services. Hence,  X – M = CA. This implies CA = S – I = NX from equation (8).

Suppose National Saving = National Investment or 100 = 100
The country could invest domestically 80 out of 100, i.e., DI=80
The country could invest abroad 20 out of 100, i.e., FI=20 so that DI+FI = 80+20 = 100, and National Saving is invested abroad which leads to CA surplus.
Again, suppose National Saving = National Investment or 100 = 100
The country could invest domestically 100 out of 100, and there is FI=20 i.e., DI=120, and there is CA deficit.

Now, if an American buys something from Japan, that purchase could either be consumption good or capital good. If that is a capital good, the above holds. Consumption does NOT go up, but Saving DOES go down –for the purchase is made out of savings, as seen from equation (6) either T or C or G has to go down, which means Saving goes down. On the other hand, if it is consumption good, then obviously American consumption goes up while Saving goes down.

3.   WHAT WHEN AMERICAN GOVERNMENT ISSUES BONDS TO CHINA FOR BUILDING AMERICAN INFRASTRUCTURE?
This indirectly means China has a stake in American infrastructure since their money is financing that, and they are holding the bonds as if making loan to the U.S. What does that tell us? It is the same scenario such as CA Surplus as explained under heading 2 above. Here, however, consumption does NOT fall, since national money remains the same, and there is no immediate effect on saving.

There is a catch, however. As multiplier can also be defined as the inverse of the marginal propensity to save, which implies it is the inverse of the marginal propensity to invest (since I=S in equilibrium), and since marginal propensity to save is one minus marginal propensity to consume, multiplier becomes 1/(1−MPC), so that as MPC is up (dynamically we assume because of the good effects of infrastructure development), there would be a multiplier effect on income propagation, which will also lead to a series of effects on consumption and income. [If you can get into any mathematics book on series, you will understand that better –but don’t worry, you don’t miss much.] Now, as an impact, investment may as well go up. This brings us to economic dynamics, but that (which may require advanced-level difference equations such as in cobweb theorem) shouldn’t concern you much at this moment.

CONCLUSION
Sara, this is a wide topic, and there could be many lectures devoted to your one single question. I have tried my best to give you as simple an answer as possible. I hope this satisfies your query. Best of luck.  

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Eklimur Raza

Expertise

It appears some students in this website are confused about elasticity of demand and the slope of the demand curve when they are trying to figure out why rectangular hyperbola comes up in case of unitary demand curve. First, they don't know that RH can be depicted in a positive quadrant of price,quantity plane. Secondly, they make the mistake that the slope of RH is constant at -1. Two points could help them: first, e=1 at each and every point of the RH, because the tangent at any point shows lower segment=upper segment (another geometric definition of e); yet slopes at different points,dQ/dP, are different; second, e is not slope but [(Slope)(P/Q)]in absolute terms. Caveat: only if we measure (log P) along the horizontal axis and (log Q) up the vertical axis, can we then say slope equals elasticity --in which case RH on P,Q plane is transformed into a straight-line demand curve [with slope= -tan 45 deg] on (log Q),(logP) plane, and e= -d(log Q)/d(log P). [By the way, logs are not used in college textbooks --although that is helpful in econometric estimation of elasticity viewed as an exponent of P, when demand equation is transformed into log-linear form.] I have not found the geometrical explanation I have given in any textbook followed in undergraduate and college classes in Canada (including the book followed in a university where I taught for a short time and in the book followed in George Brown College, Toronto, where I teach.

Experience

About 11 years' teaching economics and business studies, and also English, history and elementary French.Practical experience in a development bank, working with international donor agencies like the World Bank and the ADB. Experience in free-lance journalism, including Canada's "National Post."

Organizations
I teach micro- and macroeconomics at George Brown College (continuing education), Toronto, ON, Canada.

Publications
Many articles and editorials, on different subjects, in English newspapers. Recently an applied Major Research Paper, based on a synthesis of the Solow growth model and the Lewis two-sector model, has be accepted by Ryerson University, Toronto. Professors Thomas Barbiero and Eric Cam, Ryerson University, accepted the paper.

Education/Credentials
Master degree in Interantional Economics and Finance and diploma with honours in Business Administration from Canada.

Awards and Honors
Received First Prize in an inter-university Literary Contest.

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