The collapse in oil prices has been truly remarkable and it get TH wondering how the oil price shock would affect a country’s income, so he cracked open his old textbooks and worked through what it should mean, at least in theory. The analysis here is for a country that imports oil. But the analysis goes in reverse for a country that exports oil (or another commodity, such as coal or iron ore, for that matter). This post is very technical and possibly boring, but TH thought it would be useful to write down and share what he figured out, just in case it is of use to anyone else.

Consider a simple small (island) open economy (call it Wallaby Island, or WI for short). WI produces one good (bread) and imports oil. Some of the bread is consumed and the rest is exported. It can borrow from or lend to the rest of the world if exports don’t match up with imports. There is no investment. We’ll keep prices of the locally produced consumption good fixed at $1 each (apparently the central bank is really good at maintaining price stability), but let the price of oil, which we’ll write as “p”, be something that can change (in response to developments in the global oil market). This last assumption is what we need to investigate the oil price shock.

Since all of WI’s income is derived from spending on the wheat it produces, we just add up that spending to get its national (gross domestic) income. Total expenditure on wheat equals WI’s total consumption spending (on wheat and oil), minus their spending on imported oil, plus their exports. Mathematically, where Y is income and M is the quantity of oil imports and the rest are what you think they are (C, consumption and X, exports), this is how we calculate income:

Y=C+X-p.M (1)

Next, we need to write down what determines spending on consumption and imports. We make the same assumptions as any first year textbook, and assume that the islanders spend a fixed amount (A) to cover basic needs plus a fraction of their income on consumption. They do the same for imports. Foreign spending on WI’s goods (i.e. WI’s exports) is assumed to be fixed for now.

C=A+cY (2)

M=B+mY (3)

Now we can start to figure out how income changes in response to a change in the oil price. Mathematically, when we want to write a change in a variable, we put a d in front of it, so, in the language of mathematics, we want to figure out dY due to dp.

When one thing in the economy changes, everything else changes too, so the total change in WI’s income, we need to add up all the changes in consumption and imports (you’ll see that we separate the change in spending on imports into the change in the quantity and the part due to a change in the price):

dY= dC + dX – dM – M.dp (4)

This is a good moment to highlight an important problem with how terms of trade changes are measured by statisticians in charge of the national accounts. GDP, which is the standard measure of national income reported in the news, is generally measured keeping prices constant (to avoid confusing inflation with economic growth). That means that the last term in the equation (the M.dp bit) gets left out of the GDP calculation. The statisticians are smart people and are fully aware of the short coming of the GDP measure, so they have an alternative measure, known as GDI, that includes the impact of a change in the terms of trade (i.e. the change on the price of imports). You will note that the change in GDI that we are calculating equals the change in GDP plus the change in import spending only due to the change in p. That is, dGDI=dGDP+M.dp .

Getting back to how everything changes, note that since A, B, and X are assumed to be fixed, the only changes in C and M that are triggered by the oil price are those that flow through changes in income.

dC=c.dY (5)

dM=m.dY (6)

We can now work out how a change in the oil price affects income by substituting (5) and (6) into (4) and collecting terms.

dY=c.dY-m.dY – M.dp (7)

dY =-M.dp /(1-c+m) (8)

dY = -(c-m)M.dp/(1-c+m) – M.dp (9)

What equation 9 tells us is that the change in national income equals an induced fall in GDP plus the direct reduction in income due to the higher price of oil. Of the total fall in income only a fraction, (c-m) of the total fall in income is accounted for in GDP statistics. For example, if c=0.7 and m =0.05, then only about two thirds of the total effect is captured by GDP.

We can multiply both sides of 8 by p/Y to work out the percentage change in GDP due to a one percent increase in the oil price as –pM/[Y{1-c+m)] =- m’/(1-c+m), where m’ is the share of import spending in income. Let’s throw in some hypothetical numbers. Let m’ bet 0.1, c = 0.7, and m be 0.05, then the percentage change in income from a percentage increase in the oil price is minus 0.25. So for our small open heavily oil dependent economy a 10 percent increase in oil prices reduces income by 2.5 percent. A 50 percent increase in the price of oil reduces income by about 12.5 percent (of which 8 percent shows up in GDP).

To do a fall in oil prices, we just reverse the sign, so a 50 percent fall in the price of oil causes a 12.5 percent rise in income (8 percent rise in GDP) for the residents of Wallaby Island. Most economies don’t import only oil and so the effect is not likely to be so large. On the other hand, the reverse of this analysis largely hold for exporters, and some economies like those in the Middle East or Venezuela depend heavily on oil exports. For these economies, large numbers like these seem reasonable. Either way, it gives you an idea of how important the oil price is.

This same simple technique can be used to figure out the impact on the world economy rather than the isolated effect on a small island economy. But that is left until a later post.