Friday, December 11, 2015

Scale invariance and constant returns to scale

As an addendum to this post, I thought I'd talk a bit more about 1) scale invariance and 2) constant returns to scale that may help with some understanding -- both yours and mine! There is not meant to be a point to this post. It's just general information and making some distinctions.

The information equilibrium Solow (Cobb-Douglas) production function model (with output in equilibrium with capital N ⇄ K and labor N ⇄ L) gives us the form

(N/N0) = (K/K0)ᵝ (L/L0)ᵞ

Actually this helps clear up one of the decent criticisms of the Cobb-Douglas production function that comes from Austrian economists. They say that the units of total factor productivity are meaningless unless β = γ = 1. It's true. If you take the function:

N = A Kᵝ Lᵞ

The units of total factor productivity are

[A] = n k⁻ᵝ l⁻ᵞ

Where [N] = n, [K] = k and [L] = l. If β and γ aren't 1, then total factor productivity has units of inverse capital to some fractional power times inverse labor to some fractional power times output. Of course, there are different places to put the A ... sometimes it goes with the L so that:

N = Kᵝ (A L)ᵞ

But that just requires a different kind of weirdness with units to make it work out. The information equilibrium version says that

A = N0/(K0ᵝ L0ᵞ)

and we have

N = N0 (K/K0)ᵝ (L/L0)ᵞ = A Kᵝ Lᵞ

... i.e. A represents a combination of the scale of labor, the scale of capital and the scale of the economy (or possibly A = L0 with some other bits in the other case). The main point is that it isn't its own scale and therefore it's fine if it has weird units.

Normally, economists use the transformation

K → α K
L → α L

to derive the necessary constraint on the exponents to maintain constant returns to scale. However, this transformation leaves the IT model equations invariant:

dN/dK = β (N/K) → (1/α) dN/dK = (1/α) β (N/K) ⇒ dN/dK = β (N/K)

dN/dL = γ (N/L) → (1/α) dN/dL = (1/α) γ (N/L) ⇒ dN/dK = γ (N/L)

As well as the solution:

(N/N0) = (K/K0)ᵝ (L/L0)ᵞ →
(N/N0) = (αK/αK0)ᵝ (αL/αL0)ᵞ = (K/K0)ᵝ (L/L0)ᵞ ⇒
(N/N0) = (K/K0)ᵝ (L/L0)ᵞ

Note that the equations are also invariant under:

N → α N
K → α K
L → α L

Essentially, we are scaling up the measuring rod with these transformations.

In contrast, constant returns to scale take

N → α N
K → α K
L → α L

but don't transform the scales

N0 → N0
K0 → K0
L0 → L0

Which leaves us with a non-trivial condition if we want constant returns to scale:

(N/N0) = (K/K0)ᵝ (L/L0)ᵞ →
(αN/N0) = (αK/K0)ᵝ (αL/L0)ᵞ = αᵝ αᵞ (K/K0)ᵝ (L/L0)ᵞ ⇒
α (N/N0) = αᵝ αᵞ (K/K0)ᵝ (L/L0)ᵞ

This implies that β + γ = 1 so that the α's on both sides cancel.

So what is the difference here? Constant returns to scale is an argument about extensivity of economic growth: double inputs means you double output. It's about what measurable changes in your system do to your system measured with scales in your system (N0, L0, K0, etc). Scale invariance is about whether the size of the measuring rod matters. The former is an experiment you can do. The latter is an experiment you can never do (if the theory is correct).

For example, with special relativity, you can never determine an absolute rest frame. No experiment will determine a privileged frame. However, the speed of light is a physical scale and so you can do experiments that test effects relative to that scale (for example, the self-rest-frame lifetime of muons is not long enough for them to make it through the atmosphere from cosmic rays, but they do make it through because of time dilation).

The extensivity  is a measurable property but doesn't generally apply when the constituents of the system being combined interact. A good example is batteries in a circuit. A bunch of batteries in series has voltage as an extensive property while a bunch of batteries in parallel has current as an extensive property. Some calculations speed up if done in parallel, while others do not.

That is to say constant returns to scale is a major assumption about how labor and capital interact to produce output. In the "double Earth" thought experiment, the two don't interact ... the new Earth is effectively separate from the old earth. So it makes sense that output just doubles and there are constant returns to scale. But that doesn't necessarily apply to doubling the capital and labor supply on a single Earth.

...

Side note: effective field theory

K → α K

dN/dK = β1 (N/K) + β2 (N/K)² →
(1/α) dN/dK = (1/α) β1 (N/K) + (1/α)² β2 (N/K)²

does not work, but

N → α N
K → α K

dN/dK = β1 (N/K) + β2 (N/K)² →
(α/α) dN/dK = (α/α) β1 (N/K) + (α/α)² β2 (N/K)² ⇒
dN/dK = β1 (N/K) + β2 (N/K)²

does work. Terms like

N d²N/dK², (N/K) dN/dK, and (N²/K) d³N/dK³

preserve the homogeneity of degree zero ... should represent higher order corrections to the IT model.