Technological change and the U.S. real interest rate

Abstract

Economic theory predicts that the rate of technological growth exerts a positive influence on the real rate of interest. To test this hypothesis, I examine the relationship between the inflation-adjusted yield of the 90-day Treasury-Bill and two measures of innovation: the rate of growth of the stock of patents per worker and R&D spending relative to GDP. As theory predicts, from 1963 to 2008 the rate of interest responds positively to an increase in either measure. The point-estimates imply an elasticity of intertemporal substitution between one and two. The findings suggest that a change in the stance of monetary policy and a wave of innovation both contributed to the rise of real interest rates in the 1980s.

Appendix

In Aghion and Howitt (1998), a single final good (Y) is produced according to:

$$ {Y_t}=Q_t^{{\alpha -1}}\left( {\int_0^{{{Q_t}}} {{A_{it }}x_{it}^{\alpha }di} } \right)L_t^{{1-\alpha }}, $$

(A.1)

where Q _t is a measure of the variety of intermediate durables at time t, L _t is the labor force, x _i is the quantity of durable i, and A _i is the productivity of the latest vintage of durable i. Time is continuous. Each durable is produced in its own sector according to:

$$ {x_{it }}=\frac{{{K_{it }}}}{{{A_{it }}}}, $$

(A.2)

where K _i is the capital used in sector i. Aghion and Howitt assume that each worker has a constant propensity to imitate, so the variety of durables converges to a level proportional to the population. In math:

$$ {L_t}=l\cdot {Q_t}. $$

(A.3)

Since all sectors face identical cost functions each sector ends up producing the same quantity, which means that:

$$ \int_0^{{{Q_t}}} {x_{it}^{\alpha }di} ={Q_t}\cdot x_t^{\alpha }. $$

(A.4)

Assuming that the capital market clears, so that ${Q_t}{x_t}={{{{K_t}}} \left/ {{{A_t}}} \right.}$, the production function can be expressed in the Cobb-Douglas form:

$$ {Y_t}=K_t^{\alpha }{{\left( {{A_t}{L_t}} \right)}^{{1-\alpha }}}. $$

(A.5)

Output per worker is therefore equal to

$$ \frac{{{Y_t}}}{{{L_t}}}={A_t}\cdot k_t^{\alpha }, $$

(A.6)

where $k\equiv {K \left/ {{\left( {A\cdot L} \right)}} \right.}$ is capital per effective worker, and A _t is the average productivity of capital at time t. As in the Solow model, the economy converges to a steady state at which capital per effective worker is constant, and economic growth is sustained by the discovery of improved vintages of capital.

To establish a relationship between patent counts and the rate of technological growth, I let ϕ _t denote the flow of innovations in each sector at time t, and p _t denote the stock of patents. If a constant fraction 1/b of all innovations is patented, then

$$ {{\dot{p}}_t}=\frac{1}{b}\cdot {Q_t}\cdot {\phi_t}, $$

(A.7)

where a dot denotes the change per unit of time. Aghion and Howitt assume that each innovation raises average productivity by a constant factor equal to θ / Q _t , so

$$ {{\dot{A}}_t}=\frac{\theta }{{{Q_t}}}{A_t}\cdot b\cdot {{\dot{p}}_t}. $$

(A.8)

Then by (A.3), and as in Kortum (1993), productivity grows at a rate proportional to the flow of patents, since:

$$ \frac{{{{\dot{A}}_t}}}{{{A_t}}}\equiv {g_A}=\psi \cdot \frac{{{{\dot{p}}_t}}}{{{L_t}}}, $$

(A.9)

where $\psi =l\cdot b\cdot \theta$. Because the types of capital proliferate as the population expands, the flow of patents is scaled by the labor force.

If 0 / Q _t is an increment rather than a factor, then productivity growth is proportional to the flow of patents per effective worker, since:

$$ {g_A}=\theta \cdot b\cdot \frac{{{{\dot{p}}_t}}}{{{A_t}{Q_t}}}=\psi \cdot \frac{{{{\dot{p}}_t}}}{{{A_t}{L_t}}} $$

(A.10)

Unfortunately there is no direct measure of A _t . However, the stock of patents is given by:

$$ {p_t}\equiv \int_0^t {{{\dot{p}}_s}ds} =\frac{1}{{b\cdot \theta }}\cdot \int_0^t {{Q_s}{{\dot{A}}_s}ds} =\frac{{{Q_t}{{\dot{A}}_t}}}{{b\cdot \theta }}\cdot \int_0^t {{e^{{-\left( {{g_A}+n} \right)\left( {t-s} \right)}}}} ds. $$

(A.11)

The second equality uses that if the stock of technical knowledge is growing at a constant rate g _A , and variety is growing at the same rate n as the labor force, then ${{\dot{A}}_t}={{\dot{A}}_s}{e^{{{g_A}\left( {t-s} \right)}}}$, and ${Q_t}={Q_s}{e^{{n\left( {t-s} \right)}}}$. Equation (A.11) reduces to:

$$ {p_t}=\frac{{{Q_t}{{\dot{A}}_t}}}{{b\cdot \theta \cdot \left( {{g_A}+n} \right)}}\left( {1-{e^{{-\left( {{g_A}+n} \right)t}}}} \right). $$

(A.12)

As the economy moves further away from the initial date, the second term in the bracket approaches zero. If enough time has elapsed, then Eq. (A.12) becomes:

$$ {p_t}=\frac{{{Q_t}{A_t}}}{{b\cdot \theta }}\cdot \frac{{{g_A}}}{{{g_A}+n}}. $$

(A.13)

Solving (A.13) for Q _t A _t and plugging the solution into (A.10) yields:

$$ {g_A}=\frac{{{{\dot{p}}_t}}}{{{p_t}}}-n. $$

(A.14)

The rate of growth of the stock of patents is equal to the rate of growth of GDP. In the absence of population growth it is also equal to the rate of growth of income per worker. The perpetual inventory method described in the text generates growth in the stock of patents of 3 % per year during the sample period, very close to the 3.1 % rate of growth of GDP during the same period.