Marginal rate of substitution

As marginal rate of substitution ( Abbreviation: GRS, English: marginal rate of substitution, MRS ) is called in economics at a two-goods viewing the absolute value of the slope of an indifference curve. The name derives from the property of the GRS, specify for each point on the indifference curve, in which the exchange ratio of the household would be willing to second the good versus the first exchange ( = substitute ).

Example

An example is outlined at this point based on the graphic besides. On an indifference curve are by definition all combinations of goods Good 1 (x1 units) and Well 2 (x2 units) that cause an identical level of utility.

Considering now the point B in the graph, so this corresponds to a specific amount of a combination of units of good 1 and units of good 2 Imagine that a female consumer of good 1, a small amount? X 1 is removed. To compensate she gets Δx2. As you can clearly make reference to the representation that a consumer is after this exchange with a sufficiently small choice of? X 1 still ( approximate ) on the same indifference curve, ie the exchange of? X 1 units Good 1 against Δx2 units Good 2 it provides ( approximate ) equal. The exchange ratio is Δx1/Δx2 and for? X 1 → 0, these are just around the ( absolute ) slope of the indifference curve.

Definition

The definition of the marginal rate of substitution of good 1 relative to good 2 is

With the function of the indifference curve. At a certain point on the indifference curve under the laws of the relationships apply trigonometric functions of a right triangle equivalently also

With the angle between the tangent to the indifference curve at the point under consideration and the ordinate ( see adjacent graphic example ).

Related to the utility function and properties

It is

That is, the GRS is the ratio of marginal utility.

This can be shown as follows: As in falls, and thus is also what the penultimate equation explains. Next is for a bundle of goods that the indifference curve lies in the plane so that they can record directly as a function for which. This can be seen as representing the bundle of goods and it is the definition of the indifference curve that (constant). The derivative of with respect to is now beyond that

( because it corresponds ), which together with exactly the listed equation of the GRS - which was to be shown.

This marginal rate of substitution is invariant under strictly positive monotonic transformation of the utility function.

The concept can also be used for a wider range of goods, in which case according to any goods:

The GRS is usually assumed to be strictly monotonically decreasing, which is ( at a twice differentiable utility function ) is equivalent to the statement that indifference curves are convex and corresponds directly to the Konvexitätsannahme of preferences in the preference- theoretic foundation. Intuitively, this means in the two-goods case that you have to be compensated for giving up a marginal unit of good 2 with the more units of good 1, the less one has of good 2.

Marginal rate of substitution factor

The marginal rate of factor substitution ( also marginal rate of technical substitution ( MRTS ) ) is used in the micro-economic production and cost analysis. The basic idea here is that a producer more factors of production (simply usually two) can use in the production of his estate. The ratio factor input is not uniquely determined in most cases, so that a production ratio can be replaced by another. The marginal rate of factor substitution () in this case indicates how many additional units of one factor (in the example work ) are needed for a unit less of the other factor (in the example capital ) to ensure the same output:

Here is the set of additional labor input, the less capital amount used. As the growth in a factor offset by a decrease in the other, the marginal rate of substitution factor assumes a negative value.

A role played by the marginal rate of substitution factor among others when comparing different production functions.

Intertemporal marginal rate of substitution

In the analysis of multi-period problems in macroeconomics is often also resorted to a form of marginal rate of substitution, which indicates the absolute value of the slope of the indifference curve of an intertemporal utility function; this indifference curve relies around in a two-period model of consumption in the first period ( "young", with earnings ) to that in the second ( " old ", no one was working ) in relationship.

Be about the intertemporal utility function of a representative agent with the intertemporal budget constraint ( with r the real ( world) interest rate and, the income in period t); the intertemporal utility function is again the sum of the benefit period, although the benefit is modified in period 2 by a constant discount factor. For simplicity, assume that the intertemporal budget constraint is satisfied with equality by reference to the standard assumption of strictly positive marginal utility of income.

Obtained for the budget constraint by changing

And thus the simplified utility maximization problem

With optimum condition

It denotes the expression as intertemporal marginal rate of substitution. It refers as opposed to standing on imported GRS on one and the same good (consumption), which, however, potentially in two periods can be " consumed " and this basically creates a different benefit.