Monday, January 11, 2016

thoughts on capital

While I can't quickly locate it, I'm sure I've written about GDP and related measures in the past; in particular, while the calculation of GDP subtracts out the costs of most inputs into final goods (so, for example, when cloth is produced, and a suit is produced from that cloth, the value of the cloth produced is not double-counted), it does not subtract out the capital that is used.  This isn't such a big deal when the capital being used is a sewing machine, which can produce suits worth many times the cost of the machine before it wears out, but is a bigger deal when much of the capital being used is computers and software that become obsolete within five years — and, for purposes of measuring growth, is an even bigger deal when the economy is switching to a large degree from the former to the latter.  From the beginning of 1980 to the beginning of 2010, the growth in GDP exceeded the growth in Net Domestic Product — i.e. GDP minus capital depreciation — by an average of .25 percentage points on an annualized basis for three decades, as the amount of depreciation as a fraction of GDP doubled.

There's an introductory microeconomics model in which the amount that a firm produces is a function of the quantity of labor used and the quantity of capital used, and a profit-maximizing firm will hire more workers as long as the "marginal product" — how much more can be produced with an additional worker — exceeds the cost of employing that extra worker.  Similarly, capital is employed at a rate such that the marginal product of capital is equal to its cost.  The cost of capital is typically denoted by the letter r, which in other contexts is used for the real interest rate, and sometimes sources will go so far as to call it the interest rate — but this is wrong.  It will include financing costs, indeed, but will also include depreciation; if I start the year with a $10,000 piece of equipment, and end the year with an $8,000 piece of equipment, using the capital has cost me both the financing cost (say $500) of the $10,000 [1] and the extra $2,000; in particular, if one imagined that there was a perfect market for buying and selling partially depreciated equipment, I could literally just buy it at the beginning of the year, sell it at the end, and it would be quite clear that I need to make an extra $2,500 for having used that equipment in order to justify its temporary ownership.[2]

Again, in these models, the production function takes capital and labor as its arguments; one occasionally sees land or natural resources added as an argument, but one rarely sees "intermediate goods" or the like — cloth, for example, for the tailor.  Obviously there is some sense in which that's a major oversight; a tailor couldn't double the production of suits with the same initial allocation of cloth.  "Production", here, is taken to mean something like "the value of production minus the cost of the materials" — a sort of value added by the labor and the capital to the other inputs to get the outputs.  Adding this up for all economic units, whether they are producing capital goods, final consumption goods, or intermediate goods, gets us to GDP; if you were either to exclude production of capital goods, or were to subtract out depreciation, you would get, at least in a long-run average, the Net Domestic Product.  As it stands, though, GDP is the better publicized figure, even though it essentially double-counts the capital used in production of goods.[3]

In these models one often sees the expression rK called the "capital share of income" or some such; similarly, wL is the "labor share of income", where w is for "wage" and L is for "labor".  It was noted in the early middle twentieth century that the income share of labor was uncannily stable[4] over time, and Cobb and Douglass wrote down an economy-wide aggregate production function now called the Cobb-Douglass production function that more or less explains that.  What that function would predict, as we move toward capital that depreciates more quickly, is that r would increase, the amount of capital per unit labor being used would decrease, and wL would continue to constitute the same fraction of GDP as before.  Insofar as can be discerned from the data, this is actually not what has happened; while it's still hard to definitively declare a break in the trend, wL seems to have started to decrease somewhat as a fraction of GDP — but maintained a fairly stable portion for NDP.  In the standard simplistic sense, workers are getting the same fraction of net production as they were in the twentieth century, but a smaller fraction of the "production" that the usual "production functions" are measuring.

I will also note here, though it's something of an undeveloped tangent, that rK is often viewed as though K should have units of dollars and r should have the units of an interest rate; I've adopted that above, mostly because it's standard.  Much of the model goes through unchanged, though, if we interpret K as the real units of capital (assumed, at least initially, to be implausibly homogeneous) and r as either units of output per unit time per unit capital, or even as dollars per unit time per unit capital if we incorporate the price of the output (but not the capital) into it.  This approach seems more useful to me as we attempt to improve the ability of our models to link financial developments with the real economy; in particular, the price of capital — not r, but the actual dollar cost of capital equipment — may change relative to the price of the firm's output, and it's going to be hard to treat the effect of that on the firm's behavior if you've bound it up implicitly with the quantity of capital.

[1]That is to say, the interest paid on a $10,000 loan to buy it, the return that would otherwise have been earned on $10,000 that was used to buy it, or some combination of these, perhaps with an adjustment for risk.

[2]I should perhaps let the complications go to a greater extent than I am, but will note that (1) accounting depreciation is an estimate of true economic depreciation, which, in the absence of perfect markets, may not be easy to make precise, especially over short periods of time, and (2) in the real world, where there aren't such markets, it will still work out; if and only if the use of the machine is worth its cost over the period of its ownership and use, there will be some way to attribute its depreciation over time such that its implied value started at its purchase price, ended at its disposal price (if any), and provided capital services equal to the capital costs along the way.  Depreciation will end up being a real cost, and often an important one, for any asset that decreases in value over time as it gets used.

[3]Surely part of the reason is that it is simpler.  Consider another wrinkle: your personal car.  It may, in fact, be a piece of equipment that is worth $10,000 at the beginning of one year and $8,000 at the beginning of the next; perhaps, in addition to gas, maintenance, etc., owning the car for that year in some sense costs $2,500.  We typically consider the car to be a "final good", but on some level the final good is your use of the car — a distinction that could in some sense be made of apples as well, but is less useful for goods that get purchased and used up quickly.  A careful accounting would perhaps include the $2,500 in "imputed rent" that you paid to use the car — and here I'll note briefly that some countries charge homeowners taxes on the "imputed rent" associated with the use of their own houses — but would then also subtract out the depreciation of your car.  It's easier and basically equivalent to count cars as being "consumed" when they are purchased rather than as they depreciate.  What we want to subtract out are the capital costs associated with the production of something else that is purchased later — the capital cost is part of its cost of production.  If you use your car for business, for example, it becomes capital used for the production of something else.  How much do you use it for personal purposes and how much for business?  If you have a car that costs $12,500 per year, and could be just as productive with one that costs $2,500, isn't $10,000 of that basically just your consumption?  Economics is simpler in theory than in practice sometimes; you can start to get a feel for why the corporate tax code is so complicated.

[4]It bounces up and down a bit, which becomes important later in the paragraph, but generally returns to its long-run average within several years.

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