Is "Y=C+I" a PPF?

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.

Forget about Government expenditure, Net eXports, and depreciation, for simplicity. So Y=C+I. How do we interpret that equation?

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.

Comments

[Lucasllach]

Isn't it easier to tell the guys: Y is where you get the goods from, C+I what u use them for. There's no other possible use than consuming them or investing the goods you produce, as long as we call unused goods "inventory accumulation" and define "I" so that it includes inventory accumulation? I'd say they'll get the message.

[Account Deleted]

Agree with lucasllach, teaching any accounting identity requires conceptualisation in stock-flow terms and a double entry balance sheet approach, where does it come from , where does it go to.

The point Nick misses though is the difference between ex-ante and ex-poste investment Y=C+I is an exante identity, but post production we have an addition to Y as output, here we can go to Y=C+Pk.I

But that would not quite be true as it neglects the distribution of the income from the increased production - as ever if you start with identities Kalecki has already likely to have been there.

[Account Deleted]

Also the remark about savings- why assume loanable funds?

[rsj]

Nick,

I like this post, and don't want to be more ornery than usual, but can you go thought the above analysis for a MC firm rather than perfect competition? I would do it myself, but I don't know how.

[Nick Rowe]

rsj: thanks! Does that mean you think this might actually work for teaching?

With Monopolistic Competition. Hmmm. Let me see.

Let "e" be the elasticity of an individual firm's demand curve. (e is infinite under PC, but finite and greater than 1 under MC). Assume e is constant everywhere along a demand curve, and constant even when demand shifts (because it gets very hard otherwise). Assume that e is the same across all firms in all sectors (it makes it easier). Let m=(1-1/e). (m is like a markup factor for prices over marginal costs, or a markdown factor of wages and capital rentals below marginal products.)

Suppose a perfectly competitive economy suddenly becomes monopolistically competitive. What happens?

Then real wages become W/P=m.MPL and real rentals on capital goods become R/P=m.MPK. (Note that as we approach Perfect Competition, e becomes infinite and m becomes 1, so it reverts to the original model.) If the labour supply curve slopes up, the lower real wages will mean a smaller quantity of labour supplied, so lower employment and lower output of all goods. But, for a given stock of capital K, the supply curve for the employment of that stock will be vertical (nobody would leave the machines he owns idle, if he could get any positive rentals), so employment of capital stays the same. So far, that's very standard. Just like New Keynesian models in their "Long Run" equilibrium.

With lower employment of labour, the PPF between C and I will be shifted inwards with MC compared to PC. (And its shape may change too, depending on whether I is more or less labour intensive than C.) But, *provided that e is the same for firms producing C as it is for firms producing I*, it will still be true that Pk will equal the slope of that PPF. (That's because both firms markup their prices by the same percentage over marginal costs, so the relative price of capital goods in terms of consumption goods Pk is unaffected.)

But since real rentals on capital goods are less than MPK, the relationship between Pk and the rate of interest now becomes r=m.MPK/Pk.

That is a rather interesting result, that I didn't see coming. What it means is that, even if the labour supply curve was vertical, so that employment of labour stayed the same, so that MPK stayed the same (because the K/L ratio was not affected by the economy suddenly becoming monopolistically competitive), desired investment will fall. Intuitively, because if you use part of your income to buy a machine rather than consumption goods, your future rentals from owning that machine will be less than the value marginal product of that machine.

The only case where equilibrium investment would not fall would be where the PPF between C and I is rectangular and the labour supply curve is vertical. (Or where desired saving is perfectly interest inelastic).

Hmmm. I think I've got that right, but I'm not 100% sure.

[Nick Rowe]

Lucasllach: but that's not the message I'm trying to get across here. Sure, all goods produced are either consumption goods or investment goods. But what determines the mix? In particular (I have ignored the savings part of the model, since that would just be a standard intertemporal consumption choice problem) what is the relation between the rate of interest, the price of capital goods, and the mix of investment and consumption goods that firms will choose to produce?

Andrew: this is a "classical" model. The "Keynesian" analysis comes later. There is no money in the model (or, if there is money, prices and wages are perfectly flexible and we are always at full employment). Say's Law is assumed to hold in this model. There are only two things you can do with your income from the production of goods: buy consumption goods or buy capital goods. What we are looking for is an equilibrium in which the mix of newly-produced capital and consumption goods demanded by households is the same as the mix of newly-produced capital goods and consumption goods supplied by firms producing them.

You can add "bonds" to the model and nothing changes. In equilibrium, people will be indifferent between holding bonds and holding real capital goods and renting them out to firms. (A "bond" is a promise to pay 1+r units of the consumption good next period in exchange for one unit of the consumption good this period.)

It's when you add monetary exchange (and sticky prices) to the model that everything changes. Then you need to start talking about liquidity preference as well as desired saving and desired investment.

[Nick Rowe]

Those comments by Lucasllach and Andrew were useful to me. They show me what I needed to make clear and didn't make clear. If I do teach this stuff, I need to make clear that equilibrium means where the mix of C and I supplied equals the mix of C and I demanded.

[Peter N]

Let's start at the beginning.

1) What is the rental for something you can't rent? Real companies depreciate capital goods that are unimpaired. If they are impaired, the company takes a loss, and a new value is established. The rules are quite tricky. What the company management cares about is having to show a loss in the company's operating results, which will affect its net earnings and managements' salaries and job security. In deciding to do with capital goods, the company compares their current use, proposed future uses and forms of disposal. It picks the scenario with the lowest total NPV cost to the company including all the associated costs such as opportunity costs. How do you get a rental cost out of this, and why do you care if the company doesn't care and doesn't use such analysis in making decisions.

2) As AndrewLaington points out Y=C+I is a quasi-static identity. It depends on GDP covering only "added value" and assumes consumption and production occur in some timeless universe. As Marx pointed out a long time ago, what we buy is what was previously produced (or even what will be produced) and we are paid for what we have produced or will produce. Y=C+I removes time from the model.

But in reality, a company obtains financing to produce, sells the product and repays the loan.

Workers are payed with delay, with the amount of delay depending on the nature of the job (pay periods, bonuses, expense reimbursement, options and other deferred compensation).

Credit is created and eventually shows up as earnings.

3)Interest rates can't merely reflect marginal productivity. Lenders require compensation for risk and uncertainty. This can in some case dominate all other consideration. IBM can borrow at 1.5%. Spain has to pay 7%. IIR Sweden was offering negative rates.

4) Doesn't your analysis presuppose intertemporal equilibrium and the Modigliani-Miller theorem? Is this one of the things that "all economists know", but the students don't.

I'm afraid I've been reading Leijonhufvud (Axel in Wonderland, The Wicksellian Heritage), and some of the testiness is rubbing off.

[Peter N]

I think I see part of the problem. You're trying for a simple model, but what is it a model of? When you say Y=C+I, are you talking about GDP as it is calculated in the real world, or some simplified model GDP with the warts removed.

If it's the latter, it would be good to say this up front and, state your preconditions (perfect competition, intertemporal equilibrium, rentable capital, restrictions on what is time dependent or whatever). Then students wouldn't assume the model applies more generally than you intend (and you'd have fewer confused comments to answer).

[Nick Rowe]

Peter: economics is a conversation, and like any conversation, what is actually said always leaves out a lot of stuff that the speaker and audience assume, unless the speaker says otherwise. I am thinking of this post as a way of teaching (maybe) upper year undergraduates in economics. If I don't explicitly talk about things like uncertainty, transactions costs, externalities, or theft, or whatever, my audience will automatically assume I mean these things are absent from the model.

In a world of certainty and zero transactions costs, there is no difference between debt and equity, so the Modigliani-Miller Theorem holds trivially.

"What is the rental for something you can't rent?"

You most certainly can rent capital goods. You can rent cars, trucks, office space, photocopiers, just like you can rent workers. Firms can either rent the capital goods directly from the households that own them, or sell debt or equity to finance their purchase of capital goods. And in a world of certainty and zero transactions costs it makes no difference which they do.

[rsj]

Nick, yes, it is a good teaching post -- diagrams would help a lot if you teach it.

But I am confused by how you make a distinction between the firm and investors. Equity investors own the firm. They *are* the firm. If a firm is earning excess profits, then equity owners are taking delivery of those profits, by definition.

I think there is something wrong with how you are treating capital goods and firms. Something that is only revealed in the MC case.

[Nick Rowe]

rsj: We can imagine the workers owning the firm, and renting all the equipment. Lawyers do something like this. Or we can imagine a restaurant renting all of its equipment, as well as all its labour. If someone owns that restaurant, what does he really own? The name? Even the name can be rented, with franchises, etc. Firms are pretty nebulous things, when you drill down a bit. But macro theory of investment normally ducks those questions.

MC assumes free entry and exit of firms (which really means anyone can set up a new firm and locate it wherever he wants to in "product space"). And that normally means zero profits in equilibrium anyway.

Yep. I think I need some diagrams.

[rsj]

Nick, I don't care *who* owns the firm, except that whoever that person is, they are the investor, and the profits of the firm are the returns on that investment.

That's what it means when you buy equity -- you own it. If you want a model with both bonds and equities, then you need to justify equity earning more than bonds via some form of risk premium.

MC is not a reason for equity returns to be greater than bond returns.

[Peter N]

Can you work options and insurance into the model. By combining options with sales, you can synthesize almost any form of investment transaction. With insurance and options, you might get finance to live inside your model.

And I'm assuming that there are no economic profits, and that ordinary profits appear as costs.

And where do services come in? What kind of goods does an accountant produce? Does it depend on whether the client is an individual/family or a business, or is a finished tax return a durable good (not that unreasonable. You have to keep it for 7 years)? I would guess that, a lawsuit result, divorce or acquittal would be pretty durable too. But is this the right way to deal with this stuff?

BTW Typepad says in their promotional material that your HTML capabilities depend on which Typepad package you are using, but there were no details. I can't tell what the HTML features available to commenters are, and whether they track those available to the site.

Maybe an experiment is in order.

[Frances Woolley]

Nick, I really like posts like this, ones I can get my head around, and get me thinking...

[Jacques René Giguère]

Concurs with Frances. This post takes a notion and tranfer it somewhere else, just to see where it leads. Gives you a whole new perspective. That's what makes this blog one of the most not only commented but discussed. And linked to by Mark Thoma ...

[Nick Rowe]

rsj: OK. You would need to introduce risk and moral hazard into the model. The owner of the firm becomes the residual claimant, because the bondholders can't monitor him to check he's making the right decisions, and he has to buy rather than rent capital goods because the owners of those goods can't monitor him to check he's not abusing the rental trucks.

Way beyond the scope of this model.

"Can you work options and insurance into the model."

Sure. (Or, someone could, not me.) But it's complicated enough already.

"And where do services come in?"

goods = goods + services. Some goods have mass, and other goods don't, and we call that second type of good "services". But it makes no economic difference whether they have mass.

"BTW Typepad says in their promotional material that your HTML capabilities depend on which Typepad package you are using, but there were no details."

Thanks. But I'm afraid the binding constraint might be *my* HTML capabilities, rather than Typepad's! (I'm not very good at techie stuff).

Frances, Jacques Rene: Thanks! I really appreciate that.

[rsj]

Nick,

rsj: OK. You would need to introduce risk and moral hazard into the model.

Woah, why make things so complicated?

Just assume that savers, when they purchase capital to rent out, are in fact purchasing shares of firms and all firm earnings go to the investors. Whether those earnings are the result of pricing goods above marginal cost or at marginal cost is irrelevant. Your model doesn't need to have any bonds at all, equity can be the sole saving instrument.

You can model it without risk, because the excess earnings of firms just mean that the share price is marked up above the tangible book value, so that the returns from purchasing a share of stock are the same (in the absence of risk) with the returns to purchasing a risk-free bond, even if the firm itself is earning excess profits. The more profitable firm is more costly to purchase so that the saver does not realize any excess profits when purchasing a share of the firm.

The discrepancy between the market value and the book value of the firm can be viewed as the cost of purchasing the brand/trade mark/patents or whatever non-tangible assets the firm has.

In a static equilibrium, we can assume that the funds spent on acquiring the market power of a firm by purchasing a share of the firm is equal to the funds required to create your own market power as a result of advertising, etc.

In principle, monopolistic competition has nothing to do with risk or moral hazard, so why go there?

The point being, that originally the revenues of the firm flowed to workers, savers, and "managers". Perfect competition eliminates the managers -- it just makes them workers. Monopolistic competition means you have to account for extra income again, but this extra income does not go workers, nor savers, nor managers. It goes towards expenditures on non-produced goods such as brand awareness or search costs that creates the MC position.

Now, do you buy what I am selling? :)

And yes, this is a great post! I hope I have not derailed it.

[Nick Rowe]

rsj: thanks! (Slightly off-topic, but not massively, and not a derailment.)

Suppose that every time you start up a new firm, you have to pay a once-and-for-all fixed cost. That is an investment that is quite apart from any capital equipment that is used to produce goods. You could motivate that fixed cost in a number of ways, as you suggest. You would have a downward-sloping ATC curve (if you assumed constant returns to scale in production), so you would need to assume Monopolistic competition, but that's OK. You could model that as a growing economy, with a growing number of firms, producing a growing variety of products. It wouldn't be terribly hard to do (provided you assumed a constant elasticity of demand, for simplicity). And it might be interesting in its own right, but I'm not sure it would add much to the theory of the investment demand curve.

[Peter N]

An image

some text bold italic

underlined

some text bold italic

bold underlined

center

some text

bold italic underlined

If this is all the HTML it will do, I wouldn't worry about the learning curve. Since almost no HTML seems to be supported, there's almost nothing to learn. OTOH, it's pretty useless. You may have more available to you than we do, or, you may not. There are HTML editors (like Amaya) which would layout formulas for you, but if the blog software doesn't support the necessary mathml features, you're out of luck.

[rsj]

Peter, you can embed images?

[rsj]

Nick,

Yes, except it adds another degree of freedom.

I.e. if the risk free return is 3%. Let's say that firms earn 2% -- 3% to for the risk-free rate, and 2% to buy a certain level of market power. Or they can earn 7%. Or 12%. Etc. In all cases, the market value goes up so that investors still earn only 3%, except more and more money is spent on non-produced goods.

Is this added degree of freedom important?

[Nick Rowe]

Coo! Look what I can do now! Thanks Peter!

[rsj]

damn. no tables. Pls, webmaster, delete my post above.

[Too Much Fed]

"In a world of certainty and zero transactions costs, there is no difference between debt and equity"

I'd like to see JKH comment on that one.

[Greg Ransom]

If you choose longer production processes producing superior output, some inputs will become economic goods which previously were NOT economic goods.

Similarly, if you shorten production process some goods will lose their economic goods status and will no longer have any use as an imput to production.

That changes the PPF curve ......

[Edeast]

I had to read it, along with the dutch capital theory, reswitching post, and chapter 3 of Financial Economics, by Fabozzi, and I think I got it. So your students will be fine.

[Edeast]

Nope, didn't get it. You have preferences affecting labour supply, wheras the picture I was looking at shows the indifference curve between a good t1 and t2 not meeting up exactly at the ppf pt, requiring finance to bridge the gap. Since I and C are offered in the same period i don't know what the equivalent to finance is, in your model. Labour?

[Edeast]

Your students will still be fine though.

[Edeast]

"What we are looking for is an equilibrium in which the mix of newly-produced capital and consumption goods demanded by households is the same as the mix of newly-produced capital goods and consumption goods supplied by firms producing them."

This is where I was hung up. The slope of the tangent from the ppf and the indifference curve may be equal, but the two points don't need to be exactly equal. What does it mean to borrow to buy more consumption goods today, when they aren't offered.

[Nick Rowe]

If everybody is the same, the two points must be equal. If they weren't, there would be an excess demand/supply of capital goods, and an excess supply/demand for consumption goods, and the Pk would rise/fall, and so r would fall/rise, and the budget line would swivel. If people are different, some people could be on one side, if other people are on the other side, of the tangency point with the PPF.

[Peter N]

This is a link to 19 Leijonhufvud papers, some of which I had heard of but had no access to. Have fun. BTW his take on DSGE models is devastating and amusing if you aren't one of the authors of the paper he is criticizing - Axel in Wonderland.

http://www-ceel.economia.unitn.it/staff/leijonhufvud/files/

[Ritwik]

More people getting into Leijonhufvud! Good good, very good.

His 'The Wicksell Connection :Variations on a Theme' from '78 is probably the most packed with insight macro paper I have ever read.