Well, hi, guys. Interesting to discover that I am a subject of your discussion. Wow. I'm famous...

So, you seem to take issue with what I said. And have decided that since Ramsey was trained by Keynes that Somehow that means Ramsey was wrong and Keynes was right. Well,

KEYNES WAS A JACKASS.

Sorry, Ramsey was right, and Keynes didn't know his ass from a hole in the ground. Keynes' model is crap.

Do you really want the long vversion? OK.

BTW, how's your calculus?

For the record, it's time for the Ramsey model. Pay attention, you might learn something. The results are summarized at the end if you can't understand the process.

Individual labor supply will be assumed to be constant for simplicity.

Our economy will consist of one good (you should like that) that can be either consumed for utility or invested for production of more goods.

the capital stock at time t will be designated Kt for aggregate, or kt for per capita.

Individual consumption at time t will be designated ct, Ct for aggregate.

There is an infinite time horizon.

The capital stock depreciates at a constant rate, d.

Individuals discount the future at a constant rate, p.

Individuals desire to maximize the discounted present value of their lifetime utility

All individuals are considered to be identical, for simplicity.

There is no uncertainty, likewise.

Consumption produces utility through some utility function u(c) . Total population is N.

Capital K, Y, and C indicate aggregate values, lower case k, y, c represent per capita values. k = K/N, y = Y/N, c = C/N.

Capital continually produces income at time t, yt, through some production function yt = f(kt) such that f(0) = 0, f(infinity) = infinity, f'(k)

0, f''(k) < 0. For society as a whole, the production function is Yt = F(Kt,Nt). No, f() and F() are not the same function, though they share the same properties.

The instantaneous change in any variable, say X, is dX/dt which I will call Xdot (the usual notation would be an 'X' with a dot over it).

The rate of change in any variable is Xdot/X = dX/dt / X = d lnX / dt.

The population, N, grows at the rate n, or Ndot / N = n.

Society's total capital stock is K. Each individual has K/N = k capital to work with.

The present value of an individual's lifetime utility is: (integral) u(ct) * e^-pt dt.

The instantaneous change in the total capital stock (Kdot) is total output, (Y), minus total consumption, (C), minus depreciation on existing capital, (dK), or Kdot = Y - C - dK.

k = K/N lnk = lnK - lnN d(lnk)/dt = d(lnK)/dt - d(lnN)/dt kdot/k = Kdot/K - Ndot/N kdot/k = (Y-C-dK)/K - n kdot/k = (1/N / 1/N) * (Y-C-dK)/K - n. kdot/k = (f(k) - c - dk)/k - n kdot = f(k) - c - dk - n*k

kdot = f(k) - c - (n+d)*k

In the steady-state kdot = 0, and cdot = 0 (or the state wouldn't be steady, natch, there's a page of equations available to prove this if you don't believe it).

The problem is to maximize (integral) u(ct) * e^-pt dt subject to the constraint kdot = f(k) - c - (n+d)*k, with the decision variables being c and k.

For continuous time we form the Hamiltonian:

H = u(c) e^-pt + m[f(k) - c - (n+d)k], m being some (unknown) constant.

define m = l * e^-pt. So the problem then becomes to maximize:

H = [u(c) + l * [f(k) - c - (n+d)*k] * e^-pt

The first-order conditions are:

dH/dc = 0, dH/dk = 0, dH/dm = 0 (implying dH/dl = 0).

For the steady state we require:

dk/dt (kdot) = 0, dc/dt (cdot) = 0, and dm/dt = 0 (also implying dl/dt = 0). (we don't actually, need all of the above, they're included for completeness)

dH/dc = [u'(c) - l] e^-pt = 0 u'(c) - l = 0 u'(c) = l

dm/dt = ldot * e^-pt - l * p * e^-pt = 0

dH/dk = l * [f'(k) - (n + d)] * e^-pt = 0

since dm/dt = 0 = dH/dk, we can arrange things thusly:

ldot * e^-pt - l * p * e^-pt = - l * [f'(k) - (n + d)] * e^-pt

to get:

ldot - l * p = - l * [f'(k) - (n + d)]

rearranging:

ldot/l = (n + p + d) - f'(k)

From above:

l = u'(c)

so

ldot = u''(c) * cdot

Substituting:

u''(c)*cdot / u'(c) = (n + p + d) - f'(k)

cdot = u'(c)/u''(c) * [(n + p + d) - f'(k)]

The following are the heart of the matter:

cdot = 0 implies

---

n+p+d = f'(k)

---

kdot = 0 implies

---

c = f(k) - (n+d)k

---

Per capita output/income equals

f(k)

Per capita consumption equals

f(k) - (n+d)k

This is the general version of the no government, no taxes case.

Adding taxes make the the following adjustments (I'm not going to run through the derivation again):

Lump-Sum Tax:

The constraint becomes kdot = f(k) - c - (n+d)k -TL

n + p + d = f'(k)

c = f(k) - (n+d)k - TL (TL is the amount of the tax)

Income Tax:

constraint: kdot = f(k) - c - (n+d)k - TI*f(k) (TI being the income tax rate)

(n + p + d) / (1-TI) = f'(k)

c = (1-TI)*f(k) - (n+d)k

Consumption Tax:

constraint: kdot = f(k) - c - (n+d)k - TC*c (TC = consumption tax rate)

n + p + d = f'(k)

c = [f(k) - (n+d)k] / (1 + TC)

Capital Tax:

constraint: kdot = f(k) - c - (n+d)k - TK*k (TK = kapital tax rate)

n + p + d + TK = f'(k)

c = f(k) - (n+d+TK)k

Comprehensive Asset tax (C Post's favorite):

constraint: kdot = f(k) - c - (n+d)k - TA*(c+k)

n + p + d + TA = f'(k)

c = [f(k) - (n+d+t)k] / (1+TA)

To make specific dollar comparisons we need an actual production function and some estimates of N, n, p, and d. Estimates of US values will be used for these numbers. The equilibrium capital stock, k, and the specific T* needed to raise a given amount of money will be determined endogenously. Total government spending will be arbitrarily set at $500 billion, asset and capital taxes can't possibly raise real world amounts in equilibrium.

For all models:

the production function is:

GNP = 1500 * N^0.7 * K^0.3

per capita that's:

Y = 1500 * k^.3 (this is what I'll actually be using)

the labor supply, N, is 123,000,000.

n is 1%

p is 6%

d is 3%

Using $21 trillion for the current US capital stock ($170,731.71 per worker), the production function would work out as follows for the US:

1500 * 123M^.7 * $21T^.3 = $6.85 trillion

And average individual income would be:

$6.85T / 123M = $55,691 (seems high, but what you take home doesn't include benefits or taxes)

Now for the economic effects of different tax systems all trying to raise $500B, or $4065.04 per worker: (I was going to use actual government spending, but capital and asset taxes can't possibly raise that much in equilibrium. Yep, I had to redo everything because of that)

In order of efficiency (high to low):

1

Lump-Sum Tax: (I'll just be posting the equations and the T*, k, and c figures from here on out)

450 * k^-.7 = .01 + .06 + .03

c = 1500 * k^.3 - (.01 + .03)k - 4065.04

TL = $4065.04

TL = $4065.04

k = $165,529.35

c = $44,490.24

Tied with

Consumption Tax:

450 * k^-.7 = .01 + .06 + .03

c = [1500 * k^.3 - (.01+.03)k] / 1+TC

TC * c = $4065.04

TC = 9.137%

k = $165,529.35

c = $44,490.24

3

Income Tax:

450 * k^-.7 = (.01+.06+.03) / (1-TI)

c = (1-TI) * 1500 * k^.3 - (.01+.03)k

TI * 1500*k^.3 = $4065.04

TI = 7.622%

k = $147,804.42

c = $43,355.96

4

Comprehensive Asset tax (C Post's favorite):

450 * k^-.7 = .01 + .06 + .03 + TA

c = [1500 * k^.3 - (.01+.03+TA)k] / (1+TA)

TA * (k+c) = $4065.04

TA = 2.52%

k = $120,072.27

c = $41,242.03

5

Capital Tax:

450 * k^-.7 = .01 + .06 + .03 + TK

c = 1500 * k^.3 - (.01+.03+TK)k

TK * k = $4065.04

TK = 3.952%

k = $102,861.76

c = $39,657.82

To reiterate the final results:

Lump-sum: $44,490.24 Consumption: $44,490.24

Income: $43,355.96

Asset: $41,242.03

Capital: $39,657.82

Which, boys and girls, is why we use income taxes rather than asset taxes to fund government activities (consumption taxes would be better still).

I would like to express my deepest gratitude to the people of Texas Instruments for their invaluable inclusion of a 'solver' function in their TI-85.