First Theorem of Welfare Economics | Microeconomics

The first theorem of welfare economics is based on the two assumptions:

1. In the economy, all commodities are competitive. The equilibrium in the economy is Pareto efficient.

2. There is market for all commodities. Each commodity is produced in the economy and consumption of commodity ads to utility function.

In an economy, all markets are competitive. Consumers and producers believe that their decisions have no effect on prices. In order to reduce the complexities, we have assumed simple economy with two markets and two input markets. Each market has two demanders and one supplier. In both market prices of commodities is regarded as the parameter.

In both market, competition is possible. It is because of there are large numbers of traders of commodity from both sides of the market. Individual h gets income from selling input z_h. An individual’s own share in the profit of the two firms in the economy. It maximizes the utility function u_h (xh₁, xh₂, zh). It is subject to the budget constraint.

It is defined as follows:

Where

p_i: The price of commodity

I,w_h: The price of input h and

R^h: The non-wage income of h derived from share ownership.

The proportion of firm i owned by individual h is β_hi. The profit of firm i is π_i.

We can define the non-wage income as follows:

Where 0 ≤ β_hi ≤ 1, similarly Σ_h β_hi = 1(i = 1,2). Firm i seek to maximize its profit and it is subject to its price and output.

Such profit function is defined as follows:

The profit function is subject to its production function. It is defined as x_i = fⁱ(z_i1, z_i2). We can show the equilibrium of the economy with the help of is Pareto efficiency.

Consumer’s Choices:

In an economy, consumers choose the consumption bundles to maximize utility. The individual indifference curves for two goods are tangent to the budget line.

Therefore the utility out of two goods is given as:

Above equation clearly satisfies the condition of efficient consumption. Each consumer always compares the marginal rate of substitution between two goods. It is a price ratio between two goods. In the competitive economy, the market have same price. Therefore a marginal rate of substitution is always same.

Supply of Inputs:

Individuals h always compare the marginal rate of substitution. He compares between supply of input h and consumption of good i. It is considered as the ratio of the market prices of the input and good i.

Therefore:

Now the firm I which is supplying goods in the market maximizes (π_i) profit. Such firm chooses z_ih to satisfy the following equation.

The value of the extra output of good i is produced by an additional unit of input. Such input is equal to the cost of a unit of input h. Above equation 77 can be modified as follows.

Form equation 76, we further modify it as follows:

The efficient input supply satisfies the condition in the equation 78. It is mainly because consumers face the same relative prices for goods. Firms also face the relative prices of inputs.

Input Use:

The first principal of the firm is to maximize profit. Such profit maximization is possible by reducing the cost. But the cost minimization requires the firm to choose input mix. Sometimes it should prefer more labor or capital. While doing this adjustments firm must see that iso-quant is tangent to its iso-cost line.

Alternatively, if we divide the condition on z1 by the condition z2 for both firms in turn to get the following equation:

In the above equation, the efficient input use condition is satisfied. This is because the firm faces the same relative prices for inputs.

Output Mix:

If we use (equation77) and equation (75), then we get the following equation:

The above equation 77 shows fⁱ_h =w_h/p_i. The marginal rate of transformation between outputs is equal to the consumer’s marginal rates of substitution. Such substitution is observed between the two goods. It is satisfying condition for an efficient output mix. We have shown the equilibrium of this simple competitive economy. It satisfies the necessary conditions for Pareto efficiency. Suppose consumer’s utility functions are strictly quasi concave and the production function is convex, then the necessary condition is also sufficient. But for equilibrium, it will be efficient.

The efficiency of competitive equilibrium readily generalizes to an economy with many consumers, goods, inputs and firms. Such condition brings out very clearly the role of prices in achieving an efficient equilibrium. Such choices of the individuals are guided by the prices. Consumers face the fact that all relative prices are same.

It means that in equilibrium they all place the same relative valuation of goods and inputs. Therefore no reallocation of goods or inputs can achieve a Pareto improvement. Suppose we put it differently then all gains from mutually advantageous trade is the equilibrium and the equilibrium prices have been exhausted.

Limitations:

We have to assume that preferences and production possibilities are convex. The efficiency conditions are sufficient as well as necessary. This is a substantive assumption. There are many points on which such model is not valid. The equilibrium of a complete set of competitive markets is Pareto efficient. It does not imply that any particular market economy achieves a Pareto optimal allocation. First, the conditions of the theorem may not be satisfied. The market economy may not be efficient and therefore it cannot be optimal.

a. Firms and Consumers may not be Price Takers:

In market economy, there are monopoly sellers for different products. Consumers will not take prices as parameters. Therefore prices will not measure the marginal value of activities to all consumers. The efficiency conditions will be violated. This is because different consumers have different marginal values.

b. Incomplete Markets:

In the simple economy, there are markets for all commodities. There are various commodities that are regularly traded in the market. But still market for certain kind of commodities does not exist. For example, market for clean air does not exist. There are certain commodities which are demanded for future purposes but future markets are developing very slowly. In such market no prevailing prices exists or no clear guidance is available to consumers. The marginal valuations of activities are likely to be different. It will lead to inefficiency for commodities as well as market.

c. Disequilibrium in Markets:

In the market economy, if markets are not helping to set up the single relative price then it will not help individual for marginal valuations. It will affect the allocations and then it will be inefficient.

d. No Other Feasible Method:

In the market economies, if the prices are not at equilibrium then efficient mechanism is not possible. But at the same time there is no other feasible method exists in market.

e. Non Pareto Optimal:

We saw that if the FTWE condition is satisfied then it ensures the market allocation. Such market allocation is Pareto efficient. But such method is not Pareto optimal. The efficient allocation achieved by a market economy which may be highly inequitable. Therefore it may not maximize welfare functions. It is based on value judgments which favor equity.