Gerhard Tintner and N. S. Raghavan, Stochastic Linear Programming Applied to a Dynamic Planning Model for India (1970)

Tintner and Raghavan’s journal article extends the Mahalanobis two-sector planning model by treating sectoral output-investment coefficients as random variables. Its central contribution is to replace a single optimal national-income figure with an approximate probability distribution of optimal outcomes, and then examine how investment allocation changes that distribution. Planning under uncertainty consequently involves choosing among expected income, protection against low-income outcomes, and relative variability. The article proceeds from a deterministic benchmark through alternative approaches to uncertainty, a “passive” estimation of outcomes, and an “active” comparison of allocation policies, before qualifying its recommendations.

The initial model divides the economy into investment-goods and consumption-goods sectors. Investment enlarges productive capacity according to sectoral output coefficients and the proportions allocated to each sector. The authors replace the model’s conventional equalities with inequalities, impose a consumption floor and an overall investment constraint, and maximize national income in the terminal year. Using Indian statistics for 1949–1950, expressed in constant 1952–1953 prices, the deterministic calculation produces terminal income of 175.74 billion rupees. Although described as a five-year planning exercise, the numerical model runs from a base year, (t=0), through four subsequent periods.

The second section distinguishes chance-constrained programming, two-stage programming under uncertainty, and the authors’ distributional approach. The decisive move is to make the coefficients themselves uncertain and derive the resulting distribution of the optimized objective, rather than merely optimize with average coefficients:

All the data in linear programming problems (i.e., the elements of vectors $b, c$ and the matrix $A$) are given numbers. If this condition is abandoned, because it is unrealistic in practical applications, we obtain various problems.

The passive approach retains the allocation of one-third of investment to the investment sector and two-thirds to consumption goods. From sixteen observations for each sector, the authors fit independent gamma distributions using empirical means and variances. They select six coefficient values spanning each distribution’s fifth to ninety-fifth percentiles, calculate optimal national income at the resulting thirty-six combinations, and fit an outcome distribution by the method of moments. This converts technological uncertainty into a distribution of planning results, but its sparse numerical basis limits precision:

It should be pointed out that the distribution thus obtained will be only a rough approximation to the true distribution.

For the passive case, the reported mean terminal income is 196.632 billion rupees, the mode approximately 192.381, and the lower fifth-percentile level approximately 159.591. These quantities answer different questions: expected performance, the most probable outcome, and a low-outcome threshold. The authors suggest that such distributions could support comparisons between planning methods or countries. Their point is therefore broader than estimating an Indian income target: uncertainty changes what counts as an adequate comparison of plans.

The active approach makes the sectoral allocation of investment a policy variable while retaining the assumed coefficient distributions. Its conceptual advance is that policy affects the shape and location of the outcome distribution:

In economic planning, we might consider the effect of the allocation of investment to various industries upon the probability distribution of the objective function of the planner.

The authors compare constant investment-sector shares of one-third, one-half, and two-thirds. Among these tested policies, increasing the capital-sector share reduces expected terminal income, from approximately 196.632 to 187.672 and then 176.351 billion rupees. Conversely, variance falls and the lower fifth-percentile income rises, reaching approximately 164.724 under the two-thirds allocation. Thus a policy that performs worse by expected income can perform better by a downside-protection criterion. The authors recommend the larger capital-sector share when government wishes to prioritize safety, rather than present maximum expected income as the sole planning objective.

They then allow allocation to vary over time. One tested sequence raises the investment-sector share from one-third to one-half; another lowers it from one-third through one-quarter and one-fifth to one-tenth. The decreasing sequence gives the highest reported mean, approximately 204.733 billion rupees, and also the highest mode and lower fifth-percentile level. Its advantage therefore extends beyond average income. Yet the final comparison assigns it the greatest variance and a coefficient of variation of 13.4 percent, against 6.7 percent for the constant two-thirds policy. The policy ranking depends on the criterion: the declining sequence leads on mean, mode, and the lower-tail threshold, while the constant two-thirds allocation leads on minimum relative variability. These are comparisons among selected policies, not a demonstrated optimum over every possible allocation path.

The numerical presentation also warrants caution. The decreasing-share policy’s variance is reported as 408.653 in the earlier moment calculation but as 759.015 in Table IX; the concluding discussion follows the latter in describing it as the most variable. Table IV likewise labels the consumption share as one-third alongside an investment share of one-half, whereas the later comparison gives the complementary one-half share. These inconsistencies limit confidence in exact numerical recommendations without obscuring the article’s central distinction between alternative criteria of planning success.

The conclusion explicitly restricts the claims:

The assumption of statistical independence of the sectoral output-investment ratios is doubtful. In order to arrive at more useful results, a larger model should be constructed.

Greater sectoral detail and finer subdivision of the coefficient distributions are proposed as improvements. The article’s lasting relevance lies in its formulation of the planning problem: uncertain productive returns require evaluating distributions of outcomes, and different notions of security can favor different allocations. Its policy results remain tentative, but its methodological argument makes risk an explicit object of development planning rather than an incidental qualification to a deterministic target.