Monetary Growth Theory:
Tobin versus Sidrauski

a dark satanic mill


James Tobin (1965) presented a simple model of monetary growth which follows the Solowian model (which Tobin had independently developed in a 1955 model) in all respects plus one: the existence of government debt (net "outside" wealth). For our purposes, let us assume there is only one such type of outside wealth: money. Let money yield a certain rate of return which we shall denote as i (we are concerned with a broad aggregate for money).

In the original Solow model, savings, as a constant proportion of income, translated immediately into the accumulation of physical capital (investment). This implied, of course, that physical capital was the only form of wealth that existed. However, money is an important alternative store of wealth. Tobin (1965) proceeded to place his portfolio propositions in a growth context. How does behavior change under these circumstances?

The Tobin proposition is based on a quite simple closure with the equation of exchange, MV = PY. Dynamizing, and assuming constant velocity, we get (dM/dt)/M = (dP/dt)/P + (dY/dt)/Y or simply:

g(M) = g(P) + g(Y)

where g(x) represents the proportional growth rate of a variable x. Rewriting this, we see that g(P) = g(M) - g(Y), thus the rate of inflation (g(P)) is merely the difference between the rate of money growth and the rate of output growth. The greater the difference, the higher the inflation rate. Note the important implication that if money stock is constant (g(M) = 0), then prices will fall at the rate of growth of output (g(Y)) and real money balances, M/P, will grow at that same rate.

Let real wealth (A) be held in two assets, real money (M/P) and physical capital (K), in a proportion such that:

A = B(M/P) + (1-B)K

where b lies between 0 and 1.

The proportion B is thoroughly "Tobinesque" - to be endogenously determined by portfolio allocation. Money yields a real rate of interest defined as r = i - g(P), where i is the nominal interest rate and g(P) is the inflation rate. Capital, on the other hand, gains a return equivalent to its marginal productivity (fK). So the portfolio allocation decision basically resolves into:

B = h(r - fK)

where h(.) is some positive function. Thus, if the return on money is greater than the return on capital, (r > fK) so the portion of the portfolio allocated to money climbs (B rises). If, on the contrary, r K, then B falls and more wealth is held in the form of capital. Inflation, as is obvious, will decrease the demand for money and increase the allocation of wealth to capital.

There is, however, a second behavioral modification. Primarily, we have two sorts of income for each individual: from per capita output (f(k)) and the growth in the valuation of money, d(M/P)/dt. Thus income per capita, y, is given by:

y = f(k) + [d(M/P)/dt]/L

where L is the number of individuals, d(M/p)/dt is, then, is the growth in the value of money which constitutes an "increase" in wealth. Let us denote "real balances per person" as m ( = M/pL). The growth of real per capita money balances, dm/dt, can be easily derived. Given that m = M/pL, then, taking the derivative of the logs of this term:

g(m) = g(M) - g(P) - g(L)

where g(m) = (dm/dt)/m is the growth rate of the real money-labor ratio. Assuming full employment, then g(L) is merely the population growth rate, i.e. g(L) = n. Thus multiplying through by m:

dm/dt = m[g(M) - g(P) - n]

Thus, an increase in inflation (g(P)) will obviously result in lower real money balances per person. Similarly, an increase in population growth necessarily means that real money per capita is less (since there are more people to share it). Obviously, an increase in nominal money supply will increase per capita money holdings.

Let us now take simply M/P. Taking the time derivative of the logs, we obtain g(M/P) = g(M) - g(P). Multiplying through by M/P, we get d(M/P)/dt = M/P[g(M) - g(P)] so that dividing through by L:

[d(M/P)/dt]/L = m(g(M) - g(P))

since M/PL = m as stipulated earlier. Now total wealth, W, is merely K + M/P. Thus, in per capita terms, total wealth becomes (k + m). Now, as we know, m is a fraction B of total per capita wealth, i.e. m = B(k + m). Thus, solving this for m:

m = Bk/(1-B)

then, plugging into our equation:

[d(M/P)/dt]/L = (g(M) - g(P))Bk/(1-B)

so inputing our term for [d(M/p)/dt]/L into the per capita income equation y = f(k) + [d(M/P)/dt]/L, we get:

y = f(k) + (g(M) - g(P))Bk/(1-B)

Since savings are a portion (s) of income, then savings per capita are now:

sy = sf(k) + s(g(M) - g(P))Bk/(1-B)

Now comes the great departure from the traditional Solowian formulation. Solow had posited that savings equals investment but now a portion of savings is dedicated to the accumulation of real money balances. Therefore, S is no longer equal to I. Rather:

S = I + d(M/p)/dt

aggregate savings are applied either to the accumulation of capital (I = dK/dt) or the accumulation of real money balances (d(M/p)/dt) - depending, of course, on the relative returns of the alternative assets. Therefore, we can rewrite the system as:

dK/dt = S - d(M/p)/dt

Now, let us do as before. Recall that k = K/L so g(k) = g(K) - g(L). As g(L) = n, then g(K) = g(k) + n or, simply dK/dt = K(dk/dt)/k + nK. Cancelling terms and rearranging dK/dt = L(dk/dt) + nK thus equating with our earlier term:

L(dk/dt) + nK = S - d(M/p)/dt

and reorganizing a bit:

dk/dt = S/L - [d(M/p)/dt]/L - nk

Now, S/L is merely per capita savings, and we know this to be sf(k) + s(g(M) - g(P))Bk/(1-B) from before. Similarly, we know that [d(M/P)/dt]/L = (g(M) - g(P))Bk/(1-B) from before. Thus, plugging all this in:

dk/dt = sf(k) + s(g(M) - g(P))Bk/(1-B) - (g(M) - g(P))Bk/(1-B) - nk

or, cleaning up a bit:

dk/dt = sf(k) - (1-s)(g(M) - g(P))Bk/(1-B) - nk

which is similar to our Solowian differential equation, except that instead of dk/dt = sf(k) - nk, we now subtract an extra term in the middle, (1-s)(g(M) - g(P))Bk/(1-B). Now, in Solowian steady-state, dk/dt = 0 so that sf(k) = nk which was shown in our previous diagram (and is reproduced in our diagram at point A), as the intersection between the nk line and the sf(k) curve at A - with per capita output y* and capital-labor ratio k*.

However, under the Tobin (1965) formulation, we would have instead as a steady state condition that dk/dt = 0 and dm/dt = 0. The second condition, steady per capita money balances, implies that (from our previous equation):

dm/dt = m(g(M) - g(P) - n) = 0

where, since m > 0, this implies that g(M) - g(P) - n = 0, or, expressing for g(P):

g(P) = g(M) - n

so plugging that in for g(P) in our other steady-state condition (dk/dt = 0), we get:

dk/dt = sf(k) - (1-s)(g(M) - (g(M) - n))Bk/(1-B) - nk = 0


sf(k) - (1-s)Bnk/(1-B) - nk = 0


(1-B)sf(k) = [(1-B) + (1-s)B]nk

so, solving for nk:

sf(k)(1-B)/[(1-B) + (1-s)B] = nk

which is our steady-state condition for the Tobin model. As is obvious, if B = 0 (all allocation of wealth goes into capital and none into money), then this reduces to sf(k) = nk. However, if B > 0 , then we see immediately that:

sf(k)(1-B)/[(1-B) + (1-s)B] savings are lower because (1-s)B > 0. Thus, in comparison with sf(k), the savings function is smaller: some savings are being siphoned off into money holdings instead of being invested. Therefore, as shown in our figure, we get smaller savings function with an equilibrium at B - as compared with the regular Solow equilibrium at A. [Note: we really should have budged y = ? (k) and hence sf(k) slightly upwards to account for d(M/p)/dt, growth in money values]. Note, immediately, that the steady-state capital-labor ratio under the Tobin model (k**) is lower than the steady state capital-labor ratio under Solow (k*). Similarly, there is lower per capita output under Tobin (y**) than under Solow (y*). Thus, the equilibrium settles at a lower capital-labor ratio and lower output-labor ratio with the presence of money - although the rates of growth of the aggregate variables are the same (both at n).

We can see, then, that with the existence of money, output per capita collapses. Thus, the greater the money demand, the smaller savings available for investment, and the lower the rate of growth. This is why Tobin (1965) recommended that moderate levels of inflation be instituted to eat away at the rate of return on money (r = i - p), increasing the relative attractiveness of capital as a store of value (as fK would rise far above r plus any equity premium). As such, then, people would prefer to store their wealth in capital (B declines) -leading to higher rates of growth. For instance, if the rate of return on money falls to zero so that no money is held, the new equilibrium will be at the Solow equilibrium, k*. If we had started from the Tobin level, k*, then there will be capital-deepening and higher rates of growth as k rises above k** towards k*. Thus, inflation can lead to higher rates of growth - what is commonly known as the "Tobin Effect".

It follows, then, from this that money is not neutral. Changing the growth of money supply will change the rate of return on money which will then change the steady-state capital-labor ratio. Money, then, has real effects.

Finally, we should note that Tobin (1965) constructed this model in order to address a particular problem originally contained in the Harrod-Domar model. Namely, if the warranted rate of growth is greater than the natural rate of growth, then capital accumulation will occur without bound and the marginal product of capital will fall continuously - possibly even negative. However, if the marginal product of capital falls below the rate of interest, this implies that investment (which responds to ? K) will lie always below savings (which responds to interest). Thus, we obtain stagnation. But, by placing a floor on fK, continual stagnation might be averted. This floor, adequately enough, could very well be the rate of return on money. Thus, when fk hits r, then accumulation will tend to halt as people switch from accumulating into accumulating money. Now, if there is a surge of inflation, then r will fall and, in essence, the "floor" to fK will be lower, so further capital accumulation can occur. However, if inflation is very high, then the value of money (M/p) approaches zero. Thus, there is little point of switching into low value money. Thus, the lesson from Tobin is that a little inflation can avert stagnation but too much will be self-defeating.

D. Levhari and D. Patinkin (1968) and H.G. Johnson (1967) objected to Tobin's model on the basis of the fact that it treats money solely as a store of value and, in effect, ignores the services it performs in overcoming transactions costs, etc. As a result, they implicitly argued that Tobin should really have developed, at some point, the idea of placing money in the utility function.

This is what Miguel Sidrauski (1967) set out to do. Sidrauski complained that Tobin's assertion of the non-neutrality was not justified. In an effort to upstage Tobin's conclusions, Sidrauski attempted to construct a growth model with money which explicitly took long-run neutrality (even super-neutrality) into the system but allowed for short-run non-neutrality as the system moved towards its steady-state values.

Sidrauski's (1967) model is similar to Tobin's in that money is treated as an alternative store of wealth to capital and is created by deficit spending on the part of the government. Following Patinkin's (1956) service arguments, let us place money into a utility function so that utility at time t is U(ct, mt), i.e. utility is gained from per capita consumption of goods (ct) and per capita real money balances (mt). Therefore, following the representative agent framework of Cass-Koopmans, the object of the agent is to maximize intertemporal utility:

where p is the subjective rate of time preference. As in Tobin (1965), Sidrauksi proposes that wealth can be held in the form of either money or capital, i.e. total assets A = K + M/p. So, normalizing in per capita terms:

a = k + m

where a = A/L, k = K/L and m = M/pL where L is the labor supply. We can establish a per capita production function, y = f(k). Savings can be resolved either into capital investment or increasing real money holdings. We impose the condition that money is supplied by the government and the debt is financed that way - thus money growth is identical to the current deficit which we shall call Q. So, in per capita terms, then:

S/L = I/L + Q/L = i + q

where i is gross per capita capital accumulation and q is gross per capita money growth (the meaning of "gross" shall be given subsequently). We assume that all government spending consists of transfer payments, e, so that the budget each individual faces each period is:

f(k) + e

per capita income plus net per capita transfer payments e. Since all income is either consumed or saved, then the budget constraint, then per capita savings resolves to:

S/L = f(k) + e - c

So equating our two per capita savings terms:

f(k) + e - c = i + q

Now, i, gross per capita investment, assuming zero depreciation, can be resolved into i = dk/dt + nk where dk/dt is net capital accumulation for existing members of society and nk is the new accumulation necessary to endow new members (n is the population growth rate). Similarly, gross money growth can be resolved into:

q = dm/dt + (g(P) + n)m

where dm/dt is net money accumulation for existing members of society and (g(P) + n)m is the money accumulation necessary to endow new population (nm) and keep existing assets intact in real terms (mg(P)). This is no different than what we saw in Tobin. Plugging all back into our savings function:

f(k) + e - c = dk/dt + nk + dm/dt + (g(P) + n)m


dk/dt = f(k) + e - c - nk - (g(P) + n)m - dm/dt

which is the equation for capital accumulation. Note that dm/dt, money accumulation, is being subtracted. Recall that a = k + m, and also da/dt = dk/dt + dm/dt, thus rearranging terms:

da/dt = dk/dt + dm/dt = f(k) + e - c - na - mg(P)

which states that accumulation (of both types of assets) is equal to the difference between income (from f(k) as well as transfer payments, e) and consumption - with accumulation accounted for to endow new members, na, and money expansion to keep up with inflation (to keep real balances intact), mg(P).

Following Cass-Koopmans, we maximize intertemporal utility (where, recall, utility is a function of consumption and real money per capita) with a pair of stock and flow constraints:

a = k + m

da/dt = dk/dt + dm/dt = f(k) + e - c - na - mg(P)

the second flow constraint establishes that investment cannot be greater than savings both into physical assets a stock constraint, which will establish that the value of assets must be left unchanged. We can plug the first into the second constraint so we end up with only one constraint:

da/dt = dk/dt + dm/dt = f(k) + e - c - n(k + m) - mg(P)

Note the important difference, in this model, that the objective of the consumer is not only to maximize intertemporal consumption but also to maximize intertemporal money holdings. If all wealth is held in the form of money, then he is reducing future growth (by inhibiting capital accumulation) and hence lower future wealth. However, if all is held in the form of capital, then he is ignoring the presence of money balances in the utility function. The object, then, is to find some convenient balance between money and capital which also involves a balance between savings and consumption. Therefore, there are two control variables: money holdings and consumption.

Setting up the current-value Hamiltonian then:

H = U(c, m) + v[f(k) + e - c - n(k + m) - mg(P)]

where v is the present-value costate. The necessary conditions for a dynamic maximum are merely the first order conditions with respect to consumption and money holdings:

(1) dH/dc = dU/dc - v = 0

(2) dH/dm = dU/dm - v(n + g(P)) = 0

(3) - dH/dk = dv/dt - pv = -v(fk - n)

(4) dH/dv = dk/dt = f(k) + e - c - n(k+m) - mg(P)

Note that the first condition yields dU/dc = Uc = v and dU/dm = Um = v(n + g(P)). Thus:

Um/Uc = n + g(P)

which simply states that the marginal rate of substitution between money and consumption is equal to the nominal rate of interest (real rate plus inflation). [if we conceive of this as somehow proxying for present/future consumption relation, then this condition is similar to the old Fisher result of equating the rate of interest with the marginal rate of time preference.] Further, note that third condition yields:

dv/dt = - v(fk - n - p)

So, at the steady-state, g(v) = (dv/dt)/v, it must be true that fk = n + p, i.e. the rate of return on capital (i.e. the rate of interest) is equal to the natural rate of growth (n) plus the rate of time preference (p) - which is the Golden Utility condition once again. Thus, the rate of interest has absolutely no monetary variables. Furthermore, since the real rate of interest determines optimal capital levels, then inflation, which does not affect fk, has therefore no effect on capital accumulation - contradicting, then, the Tobin Effect.

The main lesson of the Sidrauski story, then, is that the steady-state rate of interest is, again, independent of any monetary variables. This implies that not only is money neutral but it is also superneutral. Its rate of growth (and inflation) does not affect the rate of growth of real variables. However, its neutrality is not maintained during the process of adjustment towards the steady state. But neutrality is only meaningful in a long-run sense to begin with.


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