The second theorem of welfare economics has certain advantages over first theorem of welfare economics. It explains that if all consumers have convex preferences and all firms have convex production possibility sets then Pareto efficient allocation can be achieved. The equilibrium of a complete set of competitive markets are suitable for redistribution of initial endowments.

In the second welfare theorem, Pareto efficient allocation is A*. In such A* allocation, individual h has consumption xh*. Firm j produces the output yj*. We know that at A* is a point where all consumers will have the same marginal rates of substitution between all pairs of commodities. Let’s assume that pi denote the consumers’ marginal rate of substitution between commodity i and commodity 1.

We can represent it through equation as follows:

We can define above equation differently because right hand side of the equation is assumed as 1.

We can interpret = 1(1, p2,….pn). It is the set of relative prices, with commodity 1. Suppose if we redistribute the individual’s initial endowments into commodities and shareholdings then in terms of equation, it can be written as,

Where,

xh: h’s initial endowment after the redistribution,

βhj: The post redistribution shareholding in firm j

p yj: The profit earned by firm j.

In the above equation, there are two apparent problems with this redistribution. In the original explanation, individual’s initial endowments x̅h are consisting only of labor time. That it is not possible to transfer such endowments from one individual to another. They are inseparable and indivisible from one individual to another individual. Individual utilizes time for productive purposes.

Firms also produce the goods with constant returns to scale. Firm has the breakeven point at yj*. It also faces price and input cost. If we redistribute the shareholding of firm then it will not affect on budget constraints. If the firm earns zero profit then also budget constraint will not get affected.

For solving the problem of individual and firm, we need to use transfer Th where Σh T = 0. It is measured in terms of the number. Suppose Th > 0 then individual h must pay tax of Th of good 1. There is difference between holding and initial holding of good.

If the stock is available, then holding of good 1 will get automatically reduced. Suppose,

Sometimes, individual pay tax after selling of some of his holding of other goods. Suppose Th < 0, then he receives lump sum subsidy by holding his good 1. Such goods allowed to get increased.

If we assume that such phenomena of subsidy is exists then the lump sum transfers are written as follows:

Where,

Rh: Individual’s full income at the original initial holdings.

Now we can modify the above function as follows:

The lump sum transfer approach is equivalent to redistribution of initial holding of all goods and shares. If we assume that equation 83 or equation 84 is holding the same value. Therefore the value of h’s has full income in terms of a number. Such number is equivalent to the cost of the Pareto efficient commodity bundle xh* at the prices.

Suppose h is maximized uh(xh) then it is subject to budget constraints that is p xh ≤ Rh. Individual would set his/her marginal rate of substitution between commodity i and 1 which is equal to the relative price pi. His/her demand for good i at relative prices P would be equal to the amount of good i. Therefore individual receives A* as the Pareto efficient allocation.

Suppose firm j face the relative prices p and it would choose to produce yj*output. It will produce optimal quantity to maximize profit. Therefore Pareto efficiency required that firm to have marginal rate of transformation. The marginal rate of technical substitution and marginal products are equal to the relative commodity prices. The relative prices of pi are equal to the consumer’s marginal rates of substitution at the Pareto efficient commodity bundles. The profit of the firm pyj is maximized at the Pareto efficient output yj*.

The relative prices are the basis for demand and supply of goods. Such relative prices are identical to the required Pareto efficient allocation which we have defined at A*. The supply and demand decisions are based on relative prices p and they are compatible. In the market economy, the equilibrium is based on relative price p. it is a suitable choice of endowment. It achieves the desired Pareto efficient allocation and it is the equilibrium price in competitive economy.

Criticism:

i. Incomplete and Uncompetitive Markets:

We know that in real world market are neither competitive nor complete. We specified in the above paragraph that the redistribution of the initial endowments and allow market to do allocate resources efficiently. It is desired to have efficient allocation. Suppose a firm produces output based on the prices in the market. It uses prices as parameter for production. But it is invalid if the firm has monopoly power. Most of the time firm realizes that prices are affected by the production decisions. There are charges in production function and demand. The supply play important role than only prices.

ii. Convex Technology:

The preferences and technology for a firm are non-convex. The relative prices p* does not support the desired efficient allocation. It is expected in a competitive economy.

iii. Redistribution:

It may not be possible to make the kind of initial redistributions required for this theorem. It is essential that the redistributions are lump sum. The individuals may not able to alter the amounts paid or received. The taxes which should not affect their behavior at the margin otherwise it will play reverse role.

If it is possible for an individual to alter the amount paid or received under the distribution by changing their demands or supplies. The effective prices individuals face are not the market prices and will differ across individuals. If prices do not adjust to the same set of relative prices then efficiency conditions will be violated.