I'm thinking about the best way to teach the investment demand curve in the "classical" (flexible price and wage) macroeconomic model. I don't like the way it's normally done. This post is an experiment, where I'm trying to come up with a better way. I'm not sure if I have succeeded yet, so if you read on you may be a slightly dissatisfied guinea pig.
Get a sheet of graph paper. Put "quantity of consumption goods produced per year (Consumption)" on the vertical axis. Put "quantity of capital goods produced per year (Investment)" on the horizontal axis. (You can have two different consumption goods and two different capital goods, if you like, provided you have 4-dimensional graph paper.)
Draw the Production Possibilities Frontier, taking the existing technology and employment of resources as given.
Is that PPF a curve, or a straight line? It matters.
Suppose the PPF is a straight line. That means it has the same slope everywhere along it. In competitive equilibrium, the slope of the PPF will equal (minus) the relative price of capital goods to consumption goods. That's because the price of capital goods will equal the marginal cost, in terms of consumption goods foregone. The price of capital goods (in terms of consumption goods) Pk will be determined by existing technology and resources, independently of preferences, saving, or anything else. (Provided there is an "interior solution" where strictly positive quantities of both goods get produced.)
(If you have 2 different consumption goods and 2 different capital goods then the straight line PPF becomes a 3-dimensional hyperplane, but everything else is the same.)
Assuming profit-maximising competitive firms, the real wage on labour will equal the marginal product of labour services MPL, and real rentals on capital goods will equal the marginal product of capital goods services MPK. Provided the price of capital goods is expected to stay the same over time (which it will if the slope of the PPF is expected to stay the same over time) the (one-period) rate of interest will equal the real rental rate on capital goods services divided by the real price of capital goods Pk. Which means that r=MPK/Pk (the rate of interest will equal the rate of return on owning capital goods).
Preferences will affect labour supply, which will affect employment, which will affect MPK, which will affect r. Other than that (or, for a given level of employment), the rate of interest will be determined by existing technology and resources, independently of preferences, saving, and anything else. An increase in desired saving will only affect the rate of interest slowly, over time, as the greater flow of investment slowly increases the stock of capital and reduces MPK.
Another way of saying the same thing is that desired investment will be perfectly interest-elastic. A small drop in the rate of interest will mean a small increase in the stock of capital goods K at which (MPK/Pk)=r, since MPK diminishes as K increases. But a small increase in the stock of capital goods requires an infinitely large increase in the flow of investment if it happens instantly.
If the PPF between capital goods and consumption goods is a straight line, then we can simply re-define the units for the capital good so that it becomes a straight line with a slope of minus one. The opportunity cost of producing one extra unit of the capital good is always one less unit of the consumption good. And the price of the capital good will always be the same as the price of the consumption good. Pk=1, and so r=MPK.
In which case, Y=C+I can be interpreted as a PPF, a statement about technology, an engineering relationship. Given existing technology and employment of resources, every extra unit of the capital good produced means one less unit of the consumption good produced.Now lets take the extreme opposite case, where the PPF is rectangular. Because some resources can produce consumption goods, but cannot produce any investment goods; while other resources can produce investment goods, but cannot produce any consumption goods. In this case, the slope of the PPF cannot have any influence on the price of capital goods Pk. Because the slope of the PPF is undefined at the rectangular point corresponding to full employment of all existing resources.
Suppose the rate of interest falls (because people become more patient and want to save more). The price of capital goods Pk increases, because the present value of the expected future stream of rentals increases when the rate of interest falls. But the increase in Pk has no effect on the quantity of capital goods produced. The desired investment curve is perfectly inelastic.
At one extreme, if the PPF is a straight line, the desired investment curve is perfectly interest-elastic. The horizontal investment curve determines the rate of interest, but you need to know preferences and the savings curve to determine the level of investment. At the other extreme, if the PPF is rectangular, the desired investment curve is perfectly interest-inelastic. The vertical investment curve determines the level of investment, but you need to know preferences and the savings curve to determine the rate of interest.
The real world is probably somewhere between those two extremes.
If some resources have a comparative advantage in producing capital goods, and other resources have a comparative advantage in producing consumption goods, the PPF will be curved and convex (bowed out). To say the same thing another way, if capital goods and consumption goods are best produced with different mixes (ratios) of existing resources, the PPF will be curved.
If the PPF is curved, its slope changes as we move along it, and so the marginal cost of producing an additional unit of the capital good (in terms of units of the consumption good foregone), and hence the price of capital goods in terms of consumption goods, will increase as we move along the PPF and produce more capital goods and less consumption goods. Pk will be an increasing function of investment.
With a curved PPF, the price of the capital good is no longer pinned down by existing technology and resources. And so the rate of interest is no longer determined by existing technology and resources. Preferences for saving matter too.
If people want to save more, the rate of interest will fall, the price of capital goods will rise, and there will be a movement along the PPF as existing resources move away from producing consumption goods towards producing investment goods.The desired investment curve slopes down. The interest-elasticity of desired investment depends on the curvature of the PPF between consumption and investment. The straighter the PPF, the greater the interest-elasticity.
We should not write "C+I=Y=F(K,L) and I=dK/dt" unless we believe the PPF between C and I is a straight line. We should instead write "H(C,I)=F(K,L) and I=dK/dt", where H(.) is some convex function. National Income Accountants can continue to write "C+Pk.I=Y" if they wish. There's nothing wrong with writing "C+Pk.I=Y", provided you remember that Pk is the slope of the PPF between C and I, and it will increase as you move along the PPF to produce more I and less C. Unless the PPF is a straight line.
Postscript: the normal way of teaching the investment demand curve starts out by implicitly assuming a straight line PPF with a slope of minus one. So the desired capital stock is a negative function of the rate of interest, and the desired flow of investment curve is perfectly interest-elastic. Then we fudge in some ad hoc story about adjustment costs to slow down the adjustment of the actual capital stock to the desired capital stock, so the investment curve slopes down.