On Optimal Social Investment in the Sciences and Humanities

Abstract

This essay examines the optimization of public, non-profit investment in the natural and social sciences and humanities, given that the individual project outcomes are spread over time and are uncertain. The more distant the results, the greater is the uncertainty, as measured by the success-or-failure variance. Diversification into many simultaneous projects reduces but does not eliminate this risk. By contrast, in a deterministic, no-risk model, there is no way to optimize undertakings in pure science other than to accept social or political consensus. The commonly-used planners’ discount rate of zero is shown in the Appendix to have the additional property of leading to the Golden Rule.

Appendix

The task here is to show how the discount rate of government decision makers affects the level of steady-state consumption per head. In particular, if this discount rate is set at zero, per capita consumption is maximized. This finding is modified for the case of Hicks-neutral progress, to refer to efficiency units rather than natural units of labor. This presupposes that social consensus accepts rising real income per head in step with rising output per worker. We begin by defining the Hamiltonian:

$$ H\left( {s,k,t,q} \right) = {E^{ - \delta \,t}}\left( {1 - s} \right)y(t) + \psi (t)\,\left( {sy(t) - \lambda k} \right) $$

(A1)

s is here defined as saving net of depreciation, and λ is the rate of growth in the labor force, n. In the steady state the ratio of capital to labor is constant so that capital must grow as fast as labor. This is true either in natural units or in efficiency units, as the two ratios are the same under the assumption of Hicksneutral progress, which is here assumed to be positive. The effectiveness of capital grows just as fast as the effectiveness of labor. Were we to assume Harrod-neutral progress, where only labor, and not capital, grows in effectiveness, λ would be redefined as n + m, where m is the rate of increase in labor’s effectiveness. This implies that the rate of savings would need to be higher in the steady state in order for the capital stock to keep up with the effective labor force. Here we continue with the Hicks-neutral assumption:

$$ y(t) = f(k) $$

(A2)

$$ k(t) = \begin{array}{*{20}{c}} {\frac{{K(t)}}{{L(t)}}} \hfill & { = \frac{{{K_e}(t)}}{{{L_e}(t)}}} \hfill \\{\psi (t)} \hfill & { = q{{(t)e}^{ - \delta t}}} \hfill \\\end{array} $$

(A3)

${e^{ - \delta \,t}}\left( {1 - s} \right)y(t)$ is the present discounted stream of benefits arising from consumption at time t. The second term on the right-hand side is the discounted stream of future benefits arising from investment at time t. y(t) is the product of the ratio of capital to labor (as the production function of Eq. 1 is homogeneous of first degree).

$$ \frac{{\delta H}}{{\delta k}}+\frac{{d\psi }}{{dt}} = 0 $$

(A4)

$$ \begin{array}{*{20}{c}} {Now,\,\frac{{d\psi }}{{dt}} = {e^{ - \delta \,t}}\left( {\frac{{dq}}{{dt}} - q(t)\delta } \right)} \hfill \\{and,\,\frac{{\delta H}}{{\delta k}} = {e^{ - \delta \,t}}\left( {1 - s} \right)\frac{{\delta y}}{{\delta k}} + \psi (t)\left( {\frac{{s\delta y}}{{\delta k}} - \lambda } \right)} \hfill \\{Hencee{\,^{ - \delta \,t}}\left( {\frac{{dq}}{{dt}} - q(t)\delta } \right) + {e^{ - \delta \,t}}\frac{{\left( {1 - s} \right)\delta \,y}}{{\delta \,k}} + \psi (t) \left( {\frac{{s\delta \,y}}{{\delta \,k}} - \lambda } \right) = 0} \hfill \\\end{array} $$

(A5)

Next we assume, as in the text, that planners choose a discount rate δ for future consumption of zero. With suitable cancellation of terms, this leaves:

$$ \frac{{dq}}{{dt}} + \frac{{\delta y}}{{\delta k}} - q(t)n = 0 $$

(A6)

But if δ = 0 then q(t) = 1, and so we are left with the conclusion:

$$ \frac{{\delta y}}{{\delta k}} = n $$

(A7)

This is the market discount rate which guides investment under the Golden Rule. One might see this as a more general foundation for the Golden Rule.