Maximisation of Social Welfare
Prof. Bator in his paper “The Simple Analytics of welfare maximisation” has presented a more thorough and systematic analysis of the problem of social welfare maximisation. It is a summary of the static long run general equilibrium conditions of a perfectly competitive economy. It combines the Pareto optimality conditions with the social welfare function and provides a determinate and unique solution to the problem of maximisation of social welfare.
The analysis of maximisation of social welfare of social welfare is based on the following.
- There are two standardised and perfectly divisible inputs, labour (L) and capital (K). The two are supplied in fixed quantities.
- Only two homogeneous commodities X and Y are produced in the economy. The production function for each commodity is given and does not change. Each production function is smooth, shows constant returns to scale and diminishing marginal rate of technical substitution along any Isoquant which means that the Isoquants are convex to the origin.
- There are two individuals, A and B, in the economy. Each has a set of smooth indifference curves convex to the origin which reflect consistent ordinal preference functions.
- There is a social welfare function that is based on the positions of A and B in their own preference scale i.e. W = W(WA,WB). It presents a unique preference ordering of all possible situation.
Given these postulations, the problem is to determine the welfare maximising values of
- the input of labour into the production of X and Y,
- the input of capital into the production of X and Y,
- the total amount of X and Y produced and
- the distribution of X and Y between the two individuals A and B. these steps are analysed as under:
From the Production Function to the Production Possibility Curve
The diagram describes the general equilibrium of production. There are fixed amounts of two inputs, labour (L) and Capital (K) available to the economy for the production of two commodities X and Y. Ox is the origin of input labour which is measured along the horizontal axis and Oy of input capital which is measured along the vertical axis. The horizontal sides of the two axes, Ox and Oy represent commodity X and the vertical sides commodity Y. The production function for each commodity is given by smooth Isoquants which are characterised by constant returns to scale and diminishing marginal rates of technical substitution MRTS. These Isoquants are X1, X2 and X3 for commodity X for which Ox is the origin and Y1, Y2 and Y3 for commodity Y for which Oy is the origin.
At points P1 Q1 and R1 an Isoquant of commodity X is tangent to an Isoquant of commodity Y and so satisfies the condition xMRTSLK = yMRTSLK. By joining these tangency points leads to the production contract curve Ox P1Q1R1Oy in input space. The various points on this contract curve are of efficiency locus where an increase in the production of X implies a necessary reduction in the output of Y.
From this production contract curve, we can trace the production possibility curve or transformation curve in the output space from the input space. The production possibility curve associated with the contract curve OxP1Q1R1Oy of the diagram is plotted as TC. This curve shows the various combinations of X and Y that can be produced with fixed amounts of labour and capital. Consider point P1 on the contract curve and input space of diagram.
If the Isoquant Y3 represents 300 units of input Y and X1 and 50 units of X they are mapped in the output space as point P. Likewise points Q1 and R1of the diagram are traced in the output space as points Q and R respectively in the diagram. By joining points P, Q and R we derive the production possibility curve TC for commodity X and Y. With given amounts of labour an capital and fixed technology, the economy cannot attain any point above TC curve.
Nor can it have a point inside the TC curve for that will remain underutilisation of the two factor endowments. The economy must therefore be on the TC curve to maximise the community welfare. Further, the slope of any point on the production on the production possibility curve of diagram reflects the marginal rate of transformation MRT of X into Y. in other words, it indicates by how much the output of Y must be reduced by transferring enough capital and labour to produce one more unit of X.
Grand Utility Possibility Curve to Constrained Bliss Point
In order to find out which of the Paretian optimum points on the grand utility possibility curve represents, the maximum social welfare, we have to draw a social welfare function. Diagram represents W, W1 and W2 as three social welfare functions or social indifference curves of the society.
Each social welfare function is represents the carious combinations of A’s utility and B’s utility which give the same level of satisfaction. But a movement along a social welfare function involves interpersonal comparisons of utility.
Assuming that W, W1 and W2 are the social welfare curves which exist for the society, social welfare will be maximised where the grand utility possibility curve is tangent to a social welfare curve diagram. F is the point of maximum social welfare as determined by that tangency of W1 curve and GU curve.
This is known as the point of “constrained bliss” for the reason that a movement away from point F along the GU curve will reduce total social welfare. Take point P or R on the grand possibility curve GU. They represent a lower level of welfare curve W. All points which are below the point of constrained bliss F are of non-Pareto optimality.
And all points above this point such as C on the W2 curve are beyond the reach of the society for the reason of given factor endowment and technology. Thus point F is of maximum social welfare where the general equilibrium conditions of production, exchange and production and exchange are simultaneously satisfied.
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