# Why the Rate of Return Exceeds the Growth Rate

I have been scratching my head over the elementary inequality $r>g$ for some time. As you probably know, this inequality is central to the “actuarial” mechanism that drives stratification in Piketty’s work. Basically, if the rate of return (net of depreciation) outpaces the growth of the overall economy, the capital-to-output ratio, $\beta=K/Y$, or equivalently, the wealth-to-income ratio, increases. Since the upper tail of the wealth distribution obeys the Pareto law, this implies relentlessly increasing stratification under fairly normal conditions. In other words, in light of the demographic-actuarial logic, the only thing standing between patrimonial capitalism and us, is war and taxes.

But how can $r>g$ in perpetuity? This looks like an impossibility: If capital income grows at a faster rate than the economy, then the share of capital income in GDP, $a$, should rise until it is 100%, and then $r$ could not exceed $g$ anymore. I will show how this is a fallacy in the baseline model of economic growth. It is also true of generic models when we replace key exogenous variables in the Harrod-Domar-Solow-Cobb-Douglas model by their micro-theoretic, endogenously arrived at, equilibrium values. For instance, inter-temporal optimization in dynastic models yield the same results as the basic model. But I will not attempt to show that $r>g$ holds under very general conditions. Instead, I will show how this sheds light on global imbalances and the international economy.

In the baseline model, the output $Y_{t}$ given capital $K_{t}$ and effective labor $L_{t}$ is

$Y_{t}=K_{t}^aL_{t}^{1-a}$.

Assuming zero population growth, effective labor grows at the same rate as productivity, say $g$. Suppose that the savings rate is $s$, and the rate of return in global markets is $r$. Then, the Harrod-Domar-Solow-Cobb-Douglas model implies that in the steady-state,

$r\times s = a\times g$,

where $a$ is the capital share in national income, which is mathematically determined by the elasticity of substitution of capital and labor. (It is the exponent of capital if one assumes the Cobb-Douglas production function as we have.) Another immediate implication is that the capital-to-income ratio converges to the ratio of the savings rate and the growth rate of the economy. That is,

$\beta:=K/Y \longrightarrow s/g$,

which is why low growth regimes imply increasing stratification. Where is $r$ hiding? Well, $r$ must satisfy $r\times s =a\times g$. Under conditions of global capital mobility, it is $r$ is that given exogenously. At the global level,

$r= a\times g/s$,

in the steady-state. For commonly observed values of the parameters, the implied rate of return, $r$ is reasonable. For instance, for current global ballpark estimates, $a=32\%, g=3\%$, and $s=24\%$, which implies that $r=4\%$; a good ballpark for long-term interest rates. The only way for $r$ to fall below $g$ is if the share of capital in national income is less than the savings rate, i.e., $a. But this is almost never the case. The share of capital in national income hovers around a third, whereas the global savings rate stays close to a quarter. Also, if $r, agents’ inter-temporal optimization requires them to borrow as much money as possible; an implausible result. Under fairly normal conditions then, the rate of return exceeds the rate of growth of the economy.

So where did we go wrong in our apparent contradiction? We went wrong in assuming that $r>g$ implied an unbounded share of capital in income. This does not happen because the marginal return to capital falls as capital intensity rises. In the standard model, capital-to-output ratio stabilizes at $s/g$, and capital’s share in income, $a$, determines the rate of return of capital. (Capital’s share in income, $a$ is, in turn, determined by the elasticity of substitution between capital and labor). This implies that that the rate of return on capital may easily exceed the rate of growth of productivity, and hence the overall economy, even if capital’s share in national income is held constant!

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The thing to understand is this: The savings rates and growth rates of all countries are codetermined. We can take $a$ to be technologically determined outside the model. Also, under perfect capital mobility, the rate of return is the market-clearing price of capital that equates global savings and investment. Suppose that the capital share is constant across countries. Then the ratio of savings rate to growth rate must be equal across countries as well. Of course, the ratios may be different in the short run, but they will tend to their steady-state values over time. That is, savings-to-growth ratios are equal across national economies in equilibrium.

In the note on global imbalances, my claim was that the high-savings strategy of China, Japan, and Germany works through the capital account to suppress the US savings rate. One can use the straightforward model to understand that dynamic mathematically. In what follows, we will basically mine the steady-state equality $g/s=r/a$.

Consider a bipolar open international economy where both the poles are price-takers in the global capital market. Suppose that the capital share in every national economy is 30%. Let’s compare two scenarios.

For the reference scenario, suppose that the market clearing rate of return is 6%. Then the ratio of growth rate to the savings rate of every national economy must be 20%. (Which implies that all national economies must converge to the capital-income ratio of 500%). Let’s say in equilibrium we have one pole, say the United States, saving at the rate of 20% (and hence growing at 4%); and the other, say China, saving 30% (and thus growing at 6%).

For the second scenario suppose that China follows a high-savings strategy. By a combination of wage suppression and financial repression, it raises its savings rate to 50%. Let’s say this depresses the market-clearing global real interest rate to 4%. The ratio of growth rate to the savings rate of every national economy must now be 12.5%. So China grows at 6.25%. The United States has to lower its savings rate to say 16%. The US thus grows at 2%.

In light of the lowering of trend growth from 4% to 2% at the center, we may use the current account frame-of-reference and see the second, prevailing, scenario as one of secular stagnation. Equivalently, we may use the capital account frame-of-reference and see the second scenario as one of a global savings glut. This is basically what is going on in the world economy (see chart of American and Chinese savings rate below). Ben Bernanke has inaugurated his blog by arguing that global interest rates are low because of a global savings glut; pointing out that central banks cannot control long-term rates (see picture above of the yield curve courtesy of the Grey Lady). Summers responded by defending his secular stagnation thesis.

Guys, it’s the same thing. And it is the result of the high-savings strategies of US’ trade partners, especially China. Boosting demand in the United States, whether through fiscal or monetary means is not the answer. This is a US foreign economic policy issue. The right question to ask is this: How can the United States persuade China, Germany, and Japan to lower their savings rate? For unless these states pursue alternate strategies to secure their economic interests there is no hope for resolving global imbalances.