Welfare economic marginal conditions

The welfare-theoretical analysis of exchange relationships results in several welfare-economic marginal conditions (mostly three, but sometimes also a subdivision into five or even seven) as necessary criteria for an optimum for society as a whole under aspects of efficiency. In detail, these are usually referred to as

Exchange optimum ,
Production optimum as well
simultaneous exchange and production optimum (optimal production structure ).

The analysis is carried out with a focus on aspects of Pareto efficiency , i.e. in the sense of a Paretian welfare economy: that is, distribution aspects are not taken into account here. It is also based on the four basic assumptions of the welfare economic theory:

An economic analysis starts with the decisions or the behavior of individual economic units ( methodological individualism ).

The individual individuals act selfishly. It is also assumed that the benefit levels cannot be compared and that they can only be measured ordinally.

The actors' actions are rational.

An exchange economy is considered, that is, there is a system of exchange relationships in the sense of performance and consideration.

Formally, the above Marginal conditions First order conditions for an efficiency maximum. If they are fulfilled, then the economic prosperity of a society cannot be increased further through marginal changes. However, the conditions are not sufficient insofar as they only indicate the presence of an extremum, but cannot be determined ad hoc whether this is actually a maximum and not a minimum and whether a possible maximum is a global maximum or just is a local. For this purpose, additional total conditions are required, which are also referred to as structural conditions and formally correspond to the conditions of the second order with regard to an efficiency maximum. If they are also fulfilled, then it is guaranteed that structural changes in the economic situation will no longer lead to increases in welfare. In the following only the marginal conditions will be discussed, since one can restrict oneself to their analysis when determining the efficiency maximum assuming normal curvature properties of the utility and production functions. The full competition model serves as the analysis framework . To simplify matters, it is assumed that there are only two consumers in the economy under consideration who consume two goods that are produced by two producers using two production factors. This model can easily be expanded as required.

The exchange optimum

If one first abstracts from production acts, then exchange acts are the only possible economic activities. The Pareto optimum is accordingly reached when no more welfare-increasing acts of exchange are possible. Situations of optimal exchange can be illustrated using an Edgeworth box . The points of the optimal exchange form the so-called contract curve , from which the (point) utility possibility curve can also be derived, which indicates the states of a Pareto-optimal distribution of two goods between two individuals. Since the indifference curves of the two individuals touch each other in the exchange optimum, their marginal rates of goods substitution are the same. The corresponding marginal condition is:

The division of two goods between two individuals is Pareto-optimal if the marginal rates of goods substitution (MRS) are the same for all individuals. All allocations must therefore be on the exchange contract curve. The GRS also agrees with the goods price ratio.
Reason: In a competitive equilibrium, the following applies from the consumer's point of view:

The indifference curves touch, which means that all MRS are the same.
Due to the utility maximization under budget constraints, each indifference curve also touches the price line (the slope of which corresponds to the goods price ratio), which is why the GRS corresponds to the goods price ratio.

Furthermore, the MRS corresponds to the reciprocal of the marginal utility ratio. Formally, the following applies with regard to the two goods and with the utility function and the price : ${\ displaystyle X}$ ${\ displaystyle Y}$ ${\ displaystyle U}$ ${\ displaystyle P}$

{\ displaystyle {\ frac {dY} {dX}} = {\ frac {\ frac {dU (X, Y)} {dX}} {\ frac {dU (X, Y)} {dY}}} = { \ frac {P_ {X}} {P_ {Y}}}}

In the optimum, the marginal utility ratios of the individuals also agree.

The production optimum

If one now abstracts from the exchange acts considered above and allows an increase in the stock of goods through production acts with an initially given stock of factors, one arrives at the Pareto optimum as a production maximum, the derivation of which is somewhat more extensive overall and is divided into three sub-points, namely

the optimal use of factors,
the optimal factor allocation and
the optimal degree of specialization.

The optimal use of factors

The optimal use of factors describes a situation in which the prosperity of the economy under consideration can no longer be increased by an alternative distribution of a production factor to the producers of a good. It is known from microeconomic theory that such a situation is reached when the marginal products of the production factor under consideration are the same for all producers with regard to the good under consideration. This means that the tangents of the respective indifference curves at this point have the same slope in the Edgeworth box. This optimal point is also called Pareto-efficient. No better position can be achieved for the two market participants through further exchange relationships.

The optimal factor allocation

This condition deals with the problem of how several (variable) factors of production can be optimally divided into the production of goods. Formally, the derivation of the optimal factor allocation corresponds to the derivation of the optimum exchange, in which two production factors and thus a production contract curve are mapped in a corresponding Edgeworth box instead of two goods. Analogous to the determination of the exchange optimum, it must apply here that in the optimum the marginal rates of factor substitution (GRFS) correspond, which in turn correspond to the ratio of the factor prices, which are the same for all companies.
Reason: In a competitive equilibrium, the following applies from the producers' point of view:

The production isoquants touch, whereby all GRFS are the same.
Due to the implementation of the minimum cost combination , the production isoquant touches the isocost line for all producers (the slope of which corresponds to the factor price ratio), which is why the GRFS corresponds to the factor price ratio.

The fact that the GRFS corresponds to the reciprocal value of the marginal products illustrates the connection with the optimal use of factors.

The optimal degree of specialization

The optimum degree of specialization (production optimum) is reached when the prosperity of the economy in question can no longer be increased by changing the degree of specialization, i.e. by changing the division of labor between the producers. This marginal condition aims at the same opportunity costs (corresponds to the slope of the tangent to the corresponding transformation curve and therefore the marginal rate of transformation (GRT)) and thus the same marginal cost ratios, i.e. in the optimum the ratios of the marginal costs of a pair of goods match with all producers who sell these goods generate, match.

Simultaneous exchange and production optimum

In the last section, the consumption and production sectors are brought together. This condition aims at an optimal production structure, i.e. a production according to the preferences of the individuals. It is a matter of determining the proportions in which the individual goods should be included in overall economic production. The starting point is a macroeconomic transformation curve, the points of which indicate a specific production structure. The optimal production structure is achieved when the marginal rate of transformation coincides with the marginal rate of substitution.
Reason: In a competitive equilibrium, the following must apply:

Equilibria in the goods markets can be represented as a transformation curve. The slope of this curve corresponds to the limit rate of the transformation at each point.
When creating the transformation curve, one assumes a given inventory of production factors (such as labor and capital). Production factors that are set free by reducing the production of one good are used entirely to produce other goods. For the factors of labor ( ) and capital ( ) and the goods and that is: ${\ displaystyle A}$ ${\ displaystyle K}$ ${\ displaystyle X}$ ${\ displaystyle Y}$

{\ displaystyle (-1) \ cdot dK_ {Y} = dK_ {X}}

{\ displaystyle (-1) \ cdot dA_ {Y} = dA_ {X}}

The change in the corresponding output is the product of the marginal product of the respective production factor and the change in its use, for example when changing the labor input while keeping the capital input constant for good X:

{\ displaystyle dX = dA_ {X} \ cdot GPA_ {X}}

(The same applies to all other cases.)

The GRT can thus be written as:

{\ displaystyle GRT = {\ frac {dY} {dX}} = {\ frac {dA_ {Y} \ cdot GPA_ {Y}} {dA_ {X} \ cdot GPA_ {X}}} = - {\ frac { GPA_ {Y}} {GPA_ {X}}}}

(Analogous for capital.)

When there is full competition, the producers are also price takers, which is why the following applies to them at maximum profit:

{\ displaystyle GK_ {X} = P_ {X}, \ quad GK_ {Y} = P_ {Y}}

In a competitive equilibrium, the last unit of a production factor used is rewarded according to its marginal value product . With regard to the labor factor, the following applies with a wage rate : ${\ displaystyle w}$

{\ displaystyle w_ {X} = P_ {X} \ cdot GPA_ {X}, \ quad w_ {Y} = P_ {Y} \ cdot GPA_ {Y}}

(Analogous for capital.)

With complete competition on the labor and capital markets and the same productivity of the individual labor or capital units, the prices of the individual production factors are the same for all companies, which is why:

{\ displaystyle w_ {Y} = w_ {X}}

(Analogous for capital.)

This results in:

{\ displaystyle GRT = - {\ frac {GK_ {X}} {GK_ {Y}}} = - {\ frac {P_ {X}} {P_ {Y}}}}

As already mentioned (see optimum exchange), the price ratio and the GRS are the same in terms of amount. Therefore:

{\ displaystyle GRT = - {\ frac {P_ {X}} {P_ {Y}}} = GRS}

As a rule, there are many different states with different utility distributions, which according to the above can be considered optimal for society as a whole. On the basis of the Pareto criterion, no statement can be made about which of these optima should be achieved (no interpersonal comparison of benefits possible).

Further marginal conditions

In addition to the marginal conditions mentioned above, there are two more that have a variable factor reserve on the topic.

literature

Fritsch, Michael / Wein, Thomas Ewers, Hans-Jürgen (2007): Market failure and economic policy. 7th edition, Munich, Verlag Franz Vahlen.
Pindyck, Robert S./Rubinfeld, Daniel L. (2005): Microeconomics. 6th edition, Munich and others, Pearson Studies.
Luckenbach, Helga (2000): Theoretical Foundations of Economic Policy. 2nd edition, Munich, Verlag Franz Vahlen.