Feldman–Mahalanobis Model of Development Planning

Abstract

This chapter develops the theory of Feldman and Mahalanobis how to allocate resources in the optimal ways to maximize the growth rate with a stable planned economy. While Feldman provides numerical methods, Mahalanobis provides a closed-form algebraic method to arrive at the same solution of the optimal allocation of investible funds.

Appendices

Appendix A

Linearization

Suppose the production function is

$$X_{i} = \exp (at) \, K_{t}^{a} L_{t}^{b} \quad \left( {a + b \ge 1} \right)$$

We can write the above equation as

$$\dot{X}_{t} /X_{t} = a\left( {\dot{K}_{t} /K_{t} } \right) + b\left( {\dot{L}_{t} /L_{t} } \right)$$

where

$$\dot{X}_{t} = (\text{d}X/\text{d}T)$$

Linearizing using Taylor series expansion we get

$$\dot{X}_{t} = \lambda_{0} + \lambda_{1} X_{t} + \lambda_{2} \dot{K}_{t} + \lambda_{3} K_{t} + \lambda_{4} \dot{L}_{t} + \lambda_{5} L_{t}$$

Or in discrete time formulation

$$X_{t} = \frac{1}{{1 - \lambda_{1} }}\left[ {X_{t - 1} + \left( {\lambda_{2} + \lambda_{3} } \right)K_{t} - \lambda_{2} K_{t - 1} + \left( {\lambda_{4} + \lambda_{5} } \right)L_{t} - \lambda_{4} L_{t - 1} + \lambda_{0} } \right]$$

where

$$\lambda_{1} = \frac{{\partial \dot{X}}}{\partial X},\lambda_{2} = \frac{{\partial \dot{X}}}{\partial K},\lambda_{3} = \frac{{\partial \dot{X}}}{\partial K},\lambda_{4} = \frac{{\partial \dot{X}}}{\partial L},\lambda_{5} = \frac{{\partial \dot{X}}}{\partial L},$$

$$\lambda_{0} = \left[ {a \, \left( {\dot{K}/K)X + b \, (\dot{L}/L)X)} \right)} \right]_{t - 1}$$

For the purpose of linearization, we evaluate the production function term by term (i.e. to determine the parameters X) at the points t and $t - 1$. In the optimal control set-up every time we linearize the system, we have to change A, B and D matrices in the system equation.

Appendix B

Stochastic Simulation or the VAR Creation

The addition of the stochastically simulated disturbance term does not change the system equations significantly, but it would change the time path of the state and control variables and the value of the cost functional. The €_(t,j) vector comes from the estimation of the model within the same period. The generated errors should also be such as to satisfy a Gaussian process of zero mean and with a given positive semi-definite covariance matrix.

Appendix C

Experiment with Lower Level of Desired Paths

In this experiment, we have followed all the assumptions for Experiment 1 but have lowered the desired rate of growth for $aY$ and $mY$ from 6% to 3–5% to see the effect of that on the optimum rate of investment for the two sectors. The result is given in Table 5.2.

Table 5.2 Actual and optimum (stochastic) path for $mI/I$ (in percentage)

We can see that the ratio of industrial investment to the total investment is not very different from the actual ratio. This suggests that lowering of the desired goal for the economy would not affect the intersectoral allocation patterns given all other variables. In almost all the cases, we have seen the optimum allocation asks us to put more emphasis on the industrial sector (as the ratio given above implies).

Appendix D

Notations Used

• mY = industrial sector output

• aY = agricultural sector output

• aI = investment in the agricultural sector

• mI = investment in the industrial sector

• aK = gross capital stock in the agricultural sector

• mK = gross capital stock in the industrial sector

• aL = labour force in the agricultural sector

• mL = labour force in the industrial sector

• AS = annual acreage sown in the agricultural sector

• IMF = import of foodgrains from abroad

• MS = marketable surplus of foodgrains

• MD = demand for the marketable surplus of foodgrains

• P = intersectoral terms of trade (price of agricultural goods in terms of industrial goods)

• W = annual wage rate in the industrial sector

• A = annual acreage sown in the agricultural sector labour force in the agricultural sector

• t = time-trend