A suggestion for complicating the simple aggregate production function (bleg)

I wrote this post. Then I realised it was wrong. I really wish my math were better. So I'm turning it into a sort of bleg. I should have written the technology in implicit form as F(C,I,K,L)=0 rather than H(C,I)=F(K,L). Because the way I wrote it makes Pk depend only on I/C, when it should also depend on K/L as well. I can't think of any plausible underlying story that would make H(C,I)=F(K,L) legitimate and reasonably general. But F(C,I,K,L)=0 is ugly and unintuitive and unteachable, even though it works fine theoretically, and is just a little bit more complicated.

Maybe someone has some ideas?

Here's what I originally wrote:

Macroeconomists like to aggregate things. To keep it simple. Especially for teaching. But we don't want it too simple, so we have to wave our hands when we want to talk about things that can't happen in the model.

Here is the simple aggregate technology macroeconomists often assume:

C + I = F(K,L)  where I = dK/dt  (I have ignored depreciation for simplicity).

Some economists object to the right hand side of that equation. They complain that it aggregates all labour into one type of labour L. And they complain that it aggregates all capital goods into one type of capital good K.

But I object more to the left hand side of that equation.

It aggregates newly-produced consumption goods C with newly-produced capital goods I. It assumes they are perfect substitutes in production. It assumes the Production Possibilities Frontier between C and I is a straight line with a slope of minus one. It assumes the opportunity cost of producing one more capital good is always and everywhere one less consumption good. It means that the price of the capital good will be always one consumption good. And that means that the (real) rate of interest will always equal the marginal product of capital.

We don't assume a straight line PPF between two different consumption goods. Why should we assume a straight line PPF between consumption goods and capital goods?

Let's relax that left hand side assumption. Let's instead assume:

H(C,I) = F(K,L)

Where H( ) is some convex function, so the PPF between C and I is bowed out. That means that the marginal cost of investment (in terms of foregone consumption) will be an increasing function of investment. So the price of the capital good Pk (in terms of the consumption good) will also be an increasing function of investment.

Let's continue to assume, as macroeconomists usually do, constant returns to scale. We assume that for both H( ) and F( ). So if we double both K and L we can also double both C and I. So the derivatives of F( ) with respect to K and L depend only on the K/L ratio. And the derivatives of H( ) with respect to C and I depend only on the I/C ratio.

The price of the capital good Pk (in terms of the consumption good) will equal the marginal cost of producing one more capital good (in terms of consumption goods foregone):

Pk = -dC/dI = H_c/H_i which is an increasing function of I/C.

The real wage W (in terms of the consumption good) will equal the marginal product of labour (the extra consumption goods produced):

W = dC/dL = F_L/H_c which is an increasing function of I/C and an increasing function of K/L.

The real capital rental R (in terms of the consumption good) will equal the marginal product of capital (the extra consumption goods produced):

R = dC/dK = F_K/H_c which is an increasing function of I/C and an decreasing function of K/L.

In equilibrium, the real rate of interest r (in terms of the consumption good) must equal the rate of return from owning one unit of the capital good. That rate of return will equal R/Pk, plus the annual percentage rate at which Pk is rising. (If you pay $100 to buy the machine, rent it out for $5 per year, and the price of machines rises by 2% per year, your rate of return will be 5%+2%=7%, and if the rate of interest is also 7% you will be just indifferent between buying and not buying that machine.)

r = R/Pk + (dPk/dt)/Pk

Substituting for R and Pk we get:

r = F_K/H_i + [d(H_c/H_i)/dt]/(H_c/H_i)

So that r will be a decreasing function of K/L, a decreasing function of I/C, and an increasing function of the rate at which I/C is rising over time. (In steady state the C/I ratio will be constant over time, so that second term will be zero.)

In the standard model, r is a decreasing function of K/L only.

In the standard model we get a perfectly elastic investment demand curve. An increase in desired saving and hence investment has no immediate effect on the rate of interest; it reduces the rate of interest slowly over time as the capital stock grows over time. K cannot jump, so r cannot jump (unless L jumps).

In the revised model we get a downward-sloping investment demand curve. An increase in desired saving and hence investment causes Pk to increase immediately and r to fall immediately.

I think that's a lot cleaner than the "adjustment costs" approach to getting a downward-sloping investment demand curve.

And it lets us talk about how changes in desired savings and the rate of interest will affect the price of capital goods.

It also shows what's wrong with "r = MPK", in a simple model.

You could add in a second capital good if you like. Just add K2 to F( ), and I2 to H( ), then you get a second equation for Pk2, for R2, and for r as a function of Pk2 and R2. But I don't think it makes as much difference. The problem is not aggregating capital goods. The problem is aggregating the capital good with the consumption good.

To complete the model we need to add a labour supply function and a savings function. One simple savings function would be a consumption-Euler equation where r is an increasing function of the growth rate in consumption, and so is an increasing function of I/C.

But is it simple enough to teach? I need to think up some diagrams, and a good name for the H( ) function, so students can understand it.

I don't know if anyone else has done it like this before. They may have.

I don't know if I got any of the math wrong. I may have. By the way, what am I implicitly assuming when I write H(C,I)=F(K,L) instead of G(C,I,K,L)=0? I originally planned to write the technology that second way, but thought the first way was a bit more intuitive.

(I thank Bob Murphy for sending me a copy of one of his papers, that inspired me to do this. (Got a link, Bob?). I think Bob and I are saying at least roughly the same thing. I'm just leaving out all the "what Samuelson said wrong" and "what Bohm-Bawerk said right" stuff that Bob goes into. I'm trying to keep it simple.)

Comments

[Simon Van Norden]

"Macroeconomists like to aggregate things."

Any standard micro textbook will work you through production functions with "netput" vectors (Varian is the classic reference for our generation.)
However, you should note that those who work seriously with data and production functions tend to use a KLEM or KLEMS framework ([K]apital, [L]abour, [E]nergy, [M]aterials and [S]ervices) because they think this better captures important features of the real economy.

[Nick Rowe]

Simon: so the upper year undergrad and grad students would be fine with F(C,I,K,L)=0 ?

I could just write it as C=F(I,K,L), which is equivalent, but a little more inutuitive.

(It's the investment demand curve you get from C+I=F(K,L) that really bugs me, and bolting on that adjustment costs story as an afterthought to make Id slope down is such a fudge.)

[Contraryneal.blogspot.com]

Before I jump into the math - have you ever considered using MathJax? http://www.mathjax.org/ Basically, it lets you typeset LaTeX in your webpage by adding a line to your site's header.

[Nick Rowe]

Contraryneal: I learned LaTeX once, but forgot it. I'm as scared of learning new computer programs as I am of math!

[Nick Rowe]

Neal: I just skimmed your post. Good to see you have the same curved PPF between C and I.

Here's what's at the back of my mind:
There's a technology for producing consumption goods C=c(Kc,Lc), and a technology for producing investment goods I=i(Ki,Li), and Kc+Ki=K and Lc+Li=L, so F(C,I,K,L)=0 is the envelope thingy.

If c() and i() are identical functions, with constant returns to scale we can write is as C+I=F(K,L). I'm OK with constant returns to scale, but I'm not OK with assuming c() and i() are identical functions, and so I'm not OK with assuming the PPF between I and C is a straight line.

[ivansml]

I understood your model like this: first, capital and labor are used to produce intermediate good, with standard production function Y = F(K,L); then, intermediate good is used to produce consumption and investment goods, with the tradeoff captured in H(C,I)=Y constraint. This seems somewhat restricting (e.g. it would be impossible for such model to capture investment adjustment costs which depend both on I and K).

Mas-Colell et al. textbook discusses this in section 20.C. For a two-sector growth model, they describe technology with "transformation" function C <= G(K,L,K'), where K' is capital next-period and G() is a maximum amount of consumption given K,L and K' (this is in discrete time). Similar formulation is often used when the model is solved in recursive form (so K' becomes control variable in the Bellman equation). Also, K and K' could be in principle vectors of disaggregated capital goods.

[Ken Houghton]

If you were getting CRS for Consumption Goods, economic shifts from Manufacturing to Services would make more sense--but the shift from Ag to Mfg wouldn't.

Conceptually, the model should be F(I, K, L) with a constraint that wL>=C. But that could be even harder to teach.

[Bob Murphy]

Nick I will try to look at this over the weekend...

The paper I sent you was published here, but I don't think it's online for free.

However, what people can see right away is my

[Bob Murphy]

Hmm I must have screwed up the formatting in my hyperlink, above. What I was trying to say was:

However, what people can see right away is my dissertation. In the mathematical appendix I came up with a formal result where r=MPK is the special case, of a more general equilibrium relationship. So that's a good summary of the way I was approaching this issue.

[Simon Van Norden]

Glancing through my (1977...okay, I'm old) copy of Varian, his notation is something along the lines of
define vector y = [C,I,K,L]
define set Y as the set of all feasable points in R4 for a given value of y.
This is the first thing Varian does (pages 1-5.) Have a look.

[Nick Rowe]

Ivansml: "I understood your model like this: first, capital and labor are used to produce intermediate good, with standard production function Y = F(K,L); then, intermediate good is used to produce consumption and investment goods, with the tradeoff captured in H(C,I)=Y constraint. This seems somewhat restricting (e.g. it would be impossible for such model to capture investment adjustment costs which depend both on I and K)."

Interesting interpretation. But I think that would give a straight line PPF between C and I, unless you assumed decreasing returns to scale.

Ken: "Conceptually, the model should be F(I, K, L) with a constraint that wL>=C."

Why would you want to impose wL >= C? (Or maybe you meant C>=wL? But even then, why assume all wage income gets consumed? Some of us workers save and invest and become capitalists too?)

Thanks Bob.

Simon: But what would Varian mean by "a given value of y"? What is y? Income? The way I'm looking at it is that income (measured in units of the consumption good) is (Pk.I + C). But you don't need to define income.

All: I'm starting to think that perhaps C = F(I,K,L) isn't so bad after all. [Or, if you include depreciation at rate d, where I is net investment, C= F(I+dK,K,L).] And if you impose CRS that becomes C/L=F(I/L, K/L). Which should give a simple enough steady state for things like the Solow Growth Model.

[Frank Restly]

Nick,

What is the contraint that holds F(C,I,K,L) = 0 or C = F(I,K,L)?
If labor then why not just say:

%L = Labor force dedicated to production of capital goods

I = F1 (K, %L)
C = F2 (K, 1 - %L)

[Alex Godofsky]

Frank: in your model the division of output between I and C is solely a function of the allocation of labor, but we can also vary the composition of output by changing the allocation of capital.

[Frank Restly]

Alex,

You are correct.

%K = Capital goods dedicated to production of capital goods

I = F1 (%K, %L)
C = F2 (1 - %K, 1 - %L)

[Neal]

Thanks - I just stole the picture from Garrison's book. (My equation was linear: Y = C + I.)

LaTeX for casual use is unsophisticated: _ for subscripts, \frac{}{} for fractions, and the occasional \partial for partial derivatives. I always use http://en.wikipedia.org/wiki/Help:Displaying_a_formula

I need to think more about the math in the model.

[Alex Godofsky]

Frank:

Frank: but now your model doesn't give us double the investment and consumption if we double the labor and capital inputs, and it doesn't understand what happens if we increase the total capital stock without adding more labor.

It's also possible that we can't disentangle what fraction of labor is "allocated to investment" and what fraction is "allocated to consumption". Some activities may be allocated to both more-or-less simultaneously. Thus Nick's most general possible formulation of F(C,I,K,L)=0.

[ivansml]

Nick: "But I think that would give a straight line PPF between C and I, unless you assumed decreasing returns to scale."

It doesn't have to, e.g. have H(C,I) = (C^a + I^a)^(1/a), for a>=1, which has constant returns (and PPF is line for a=1, circle for a=2, square for a=infinity, etc.). Anyway, I think that once you write your technology as H(C,I)=F(K,L), you're implicitly already assuming this kind of intermediate good story.

[Simon Van Norden]

Nick: in Varian's notation, y is just the netput *vector* (call it q if you find that less confusing.) Of course, it will contain all the components of GDP (ie. output) that you want as well as all the factors of production.

[Frank Restly]

Alex,

One last try:

I = F1 (%K, %L, K, L)
C = F2 (1 - %K, 1 - %L, K, L)

I just don't think you can get there with one equation to describe both C & I unless Nick is assuming some constraint or resource limitation not obvious in the equation.

[JF]

Prof. Rowe,

You could try looking into Frank Portier and Paul Beaudry's paper When Can Changes in Expectations Cause Business Cycle Fluctuations in Neo-Classical Settings, they have specific examples that could rationalize
H(C,I) = F(K,L)

[Peter N]

I think calculating everything in the consumption good hides certain aspects of the problem and, in a way, assumes what you seek to prove. An investment is made based on the expected price (or marginal utility, if you prefer) of the consumption good (assuming this is what is being directly produced). The money/utilities involved can't be assumed to track the consumption good.

I don't borrow 100,000 bolts, I borrow money equal to the current price of bolts. Whether I can repay the loan by producing 100,000 bolts depends on the price of bolts when I produce and sell them. I can't just hand my creditor 100,000 bolts and be quit.

This colors the result of your robot models, where the value of human labor in widgets is found to be stable. But if widgets lose 99% of their value, human employment in widget production will cease.

At one time, screw making was a skilled profession. Screws were hand filed from bronze or brass casting or rods. The first automated screw factory produced 30,000 screws a day with 30 employees. That was it for hand filed screws. Nobody with mechanical skill would work for what a day's hand production (maybe 50 screws) would sell for.

[Nick Rowe]

Peter N. Agreed. You need to add preferences. It's the utility of future consumption that matters. That is understood. This is the part where preferences get brought in: "To complete the model we need to add a labour supply function and a savings function. One simple savings function would be a consumption-Euler equation where r is an increasing function of the growth rate in consumption, and so is an increasing function of I/C."

JF: good find. it's here, for those interested

The technology they assume is (in my notation) C=F(I,K,L). They show how it is a general case of a two sector model that includes the C+I=F(K,L) and adjustment cost model as special cases.

Simon: thanks. I must borrow a copy of Varian.

[Alex Godofsky]

Frank:

Your equations are still vulnerable to identifying what %L and %K are, and implicitly assume resources are only ever allocated towards one thing at a time (what if a given production process produces C but also I as a byproduct?). But it's also unnecessary. If you start with your first equation:

I = F1(%K, %L, K, L)

Imagine we know I, K, and L, but not %K or %L. This gives us a curve in %K and %L of feasible points. Apply that curve to the equation:

C = F2 (1 - %K, 1 - %L, K, L)

Where we already know K and L, and that gives us a surface of points in C, %K, and %L. You can maximize C on this surface because %K and %L are constrained between 0 and 1.

But what we've just done with this process is develop a (possibly non-algebraic) equation of the form:

C = G(I, K, L)

So we didn't gain anything from introducing %K and %L as variables, and possibly lost something because we aren't sure they correspond to real things.

[Kathleen]

Do you really mean for the PPF to be convex? Normally they are concave to the origin. I understood that to mean the PPF would bow outwards.

[Nick Rowe]

Kathleen: I always get concave and convex muddled. Yep. PPF should bow out.

[Frank Restly]

Alex,

Even if you ignore allocation of resources between consumption and production goods

I = F1 (K,L)
C = F2 (K,L)

If you just know K & L, you should be able to solve for both I & C, whereas:

C = G(I, K, L)

Presumably, I is an independent variable from both K and L which means that to find C you must know I, K, and L.

[Alex Godofsky]

Frank:

If you just know K & L, you should be able to solve for both I & C, whereas:

(equation elided)

Presumably, I is an independent variable from both K and L which means that to find C you must know I, K, and L.

Yes, that's the point. Just knowing K and L doesn't tell you both I and C; as you noted, there is a question of allocation between investment and consumption. The problem is that you're trying to express that allocation as "% of each factor allocated to consumption", but that doesn't actually capture all of the possible systems. A given production process may produce both consumption goods and investment goods, and it may be impossible to decide which fraction of that process's inputs are allocated towards the consumption goods.

Example: you have a grain farm. Inputs are land (K) and labor (L). It produces two qualities of grain. High-quality grain is sold to humans as food (C). Low-quality grain is fed to cows (I). How do you decide what fraction of L is allocated to C?

Note that I'm not claiming you can't come up with some halfway-reasonable algorithm for this specific example; the point is that the question is extremely unclear and plausibly unanswerable for some example, so we shouldn't even attempt it. The behavior you are trying to capture is already sufficiently explained by making I an independent variable.

[2slugbaits]

I'm not a professional economist (not even close), but doesn't the function F(C,I,K,L)=0 confuse stock and flow variables? C & I are income and are therefore flow variables. K & L are stock variables and should be regarded as wealth or an endowment position. So doesn't the equation H(C,I) = F(K,L) equate flows and stocks? For example, when you say:

We don't assume a straight line PPF between two different consumption goods. Why should we assume a straight line PPF between consumption goods and capital goods?

you have me confused. Isn't the PPF derived from F(K,L) and doesn't H(C,I) represent the budget line? The language you're using here (i.e., referring to consumption "goods" and capital "goods") slides into thinking of C &I as physical stocks of things. Shouldn't C & I be interpreted as utility flows from C & I rather than as physical goods?

To be honest, I'm not sure what lesson you're trying to teach your students. Is the goal here to show that C & I are constrained by the PPF F(K,L)? Or is the goal to derive the locus of C & I equilibrium points subject to the prices of K & L? Are you trying to show something like the old four-quadrant Keynesian curves relating wages, the production function and IS-LM? I guess I'm not sure where this is headed. It's interesting, but I just don't see the takeaway.

Thanks.