Marginal rate of substitution

Marginal rate of substitution (MRS). The MRS is higher in point B than in point A. It is in each case Δx ₂ / Δx ₁ (for illustrative purposes , the drawn Δx ₁ is the same in both cases).

The limit rate of substitution (abbreviation: GRS ) is the absolute value of the slope of an indifference curve in economics when considering two goods . The name gives the GRS the property of indicating for each point on the indifference curve the exchange ratio in which the household would be willing to exchange the second good for the first (= to substitute).

example

An example is sketched at this point using the graphic opposite. According to the definition, all product combinations of good 1 ( units) and good 2 ( units) that create an identical level of utility are on an indifference curve . ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle {\bar {u}}}$

If one now looks at the point in the graphic, it corresponds to a specific quantity combination of units of good 1 and units of good 2. Imagine that a small amount of good 1 is taken from a consumer . In compensation he receives . As one can see from the illustration, after this exchange, if the choice of is sufficiently small, the consumer is still on the same indifference curve (approximate), i.e. the exchange of -units good 1 for -units good 2 represents him (approximate) equal. The exchange ratio is and for this is precisely the (absolute) slope of the indifference curve. ${\displaystyle B}$ ${\displaystyle \Delta x_{1}}$ ${\displaystyle \Delta x_{2}}$ ${\displaystyle \Delta x_{1}}$ ${\displaystyle \Delta x_{1}}$ ${\displaystyle \Delta x_{2}}$ ${\displaystyle \Delta x_{2}\backslash \Delta x_{1}}$ ${\displaystyle \Delta x_{1}\to 0}$

definition

The MRS of good 1 with respect to good 2 at point A (B) corresponds to the tangent of the angle α (β). Since tan (α) <tan (β), the MRS in A is lower than in B.

The definition of the marginal rate of substitution of good 1 for good 2 is

${\displaystyle {\textrm {GRS}}_{1,2}(x_{1},x_{2})\equiv \left|f'(x_{1})\right|}$

with the function of the indifference curve. At a certain point on the indifference curve, the laws governing the relationships between trigonometric functions in a right-angled triangle also apply equivalently ${\displaystyle f(x_{1})}$

${\displaystyle {\textrm {GRS}}_{1,2}(x_{1},x_{2})=\tan(\phi )}$

with the angle between the tangent to the indifference curve in the point under consideration and the abscissa (see graphic example opposite). ${\displaystyle \phi }$

Relationship with the utility function and properties

It applies

${\displaystyle {\textrm {GRS}}_{1,2}(x_{1},x_{2})\equiv |f'(x_{1})|=-f'(x_{1})={\frac {\partial u(x_{1},x_{2})/\partial x_{1}}{\partial u(x_{1},x_{2})/\partial x_{2}}}}$.

That means the MRS corresponds to the ratio of the marginal utility :

${\displaystyle {\frac {\partial u(x_{1},x_{2})/\partial x_{1}}{\partial u(x_{1},x_{2})/\partial x_{2}}}={\frac {MU_{1}}{MU_{2}}}}$

This can be shown as follows: consider for a fixed bundle of goods which creates utility , the set of all bundles of goods, about which we are indifferent. Then it can be understood as the curve . This is strictly decreasing in the sense that ${\displaystyle (x_{1},x_{2})\in \mathbb {R} _{\geq 0}^{2}}$ ${\displaystyle {\bar {u}}}$ ${\displaystyle C=\{(x_{1}',x_{2}')\in \mathbb {R} _{\geq 0}^{2}\mid (x_{1}',x_{2}')\sim (x_{1},x_{2})\}}$ ${\displaystyle (x_{1},x_{2})}$ ${\displaystyle C}$ ${\displaystyle u(x,y)={\bar {u}}}$

${\displaystyle \forall (x_{1},x_{2}),(y_{1},y_{2}):(x_{1},x_{2}),(y_{1},y_{2})\in C\wedge x_{1}>y_{1}\implies x_{2}<y_{2}}$.

So we can always solve the equation , i.e. H. we find a such that (see also theorem of the implicit function ). So the simpler notation applies ${\displaystyle u(x_{1},x_{2})={\bar {u}}}$ ${\displaystyle f}$ ${\displaystyle f(x_{1})=x_{2}}$

${\displaystyle C=\{(x_{1},f(x_{1}))\mid x_{1}\in \mathbb {R} _{\geq 0}\}.}$

If we now derive the function , we get by means of a chain rule (if we use the notation again ): ${\displaystyle x_{1}\mapsto u(x_{1},f(x_{1}))\equiv {\bar {u}}}$ ${\displaystyle f(x_{1})=x_{2}}$

${\displaystyle \left({\frac {\partial u(x_{1},f(x_{1}))}{\partial x_{1}}},{\frac {\partial u(x_{1},f(x_{1}))}{\partial f(x_{1})}}\right)\cdot {1 \choose f'(x_{1})}={\frac {\partial u(x_{1},x_{2})}{\partial x_{1}}}+{\frac {\partial u(x_{1},x_{2})}{\partial x_{2}}}f'(x_{1})=0}$.

Note that the derivative of a constant function always vanishes. After further transformation, the assertion follows that we arrive at the exact equation of the GRS via via - which was to be shown. ${\displaystyle {\textrm {GRS}}=-f'(x_{1})}$

Note: since it falls strictly monotonic , what the above notation explains applies . ${\displaystyle f}$ ${\displaystyle |f'(x_{1})|=-f'(x_{1})}$

This marginal rate of substitution is invariant to positive, strictly monotonic transformation utility function .

The concept can also be used for a larger number of goods, in which case for any goods : ${\displaystyle a,b}$

${\displaystyle {\textrm {GRS}}_{a,b}(\mathbf {x} )={\frac {\partial u(\mathbf {x} )/\partial x_{a}}{\partial u(\mathbf {x} )/\partial x_{b}}}}$

The MRS is usually assumed to be strictly monotonically falling, which (with a twice differentiable utility function) is equivalent to the statement that indifference curves are convex and also corresponds directly to the convexity assumption of the preferences in the preference theory foundation. Intuitively, in the two-goods case, this means that if you do not have a marginal unit of good 2, you have to be compensated with more units of good 1, the less you have of good 2.

Marginal rate of factor substitution

The marginal rate of factor substitution (also marginal rate of technical substitution (GRTS) ) is used in microeconomic production and cost analysis. The basic idea here is that a producer can use several production factors (usually two for simplicity) in the production of his goods. In most cases, however, the factor input ratio is not clearly specified, so that one production factor can be replaced by another. The marginal rate of factor substitution ( ) indicates how many additional units of one factor (in the example work , ) are required to guarantee the same output with one unit less of the other factor (in the example capital , ): ${\displaystyle GRTS}$ ${\displaystyle L}$ ${\displaystyle K}$

${\displaystyle GRTS_{KL}=-{\frac {\Delta L}{\Delta K}}}$

It should be the amount used additionally work, the less amount used capital. Since the increase in one factor is offset by a decrease in the other, the marginal rate of factor substitution assumes a negative value. ${\displaystyle \Delta L}$ ${\displaystyle \Delta K}$

The marginal rate of factor substitution plays a role, among other things, when comparing different production functions .

Intertemporal marginal rate of substitution

When analyzing multi-period problems in macroeconomics , a form of the marginal rate of substitution is often used, which indicates the absolute slope of the indifference curve of an intertemporal utility function; using a two-period model, this indifference curve relates consumption in the first period (“young”, with earned income) to that in the second (“old”, without earned income).

Be about the intertemporal utility function of a representative agent with the intertemporal budget constraint (with the real (world) interest rate and , , income in period ); the intertemporal utility function is again the sum of the period utility, although the utility in period 2 is modified by a constant discount factor ( ). For the sake of simplicity, with reference to the standard assumption of a strictly positive marginal utility of income, assume that the intertemporal budget constraint is met with equality. ${\displaystyle U(c_{1},c_{2})=u(c_{1})+\gamma \cdot u(c_{2})}$ ${\displaystyle c_{1}+c_{2}/(1+r)\leq y_{1}+y_{2}/(1+r)}$ ${\displaystyle r}$ ${\displaystyle y_{t}}$ ${\displaystyle t=1,2}$ ${\displaystyle t}$ ${\displaystyle u(c_{t})}$ ${\displaystyle \gamma }$

You get for the budget restriction by changing

${\displaystyle c_{2}=(1+r)\left(y_{1}+{\frac {y_{2}}{1+r}}-c_{1}\right)}$

and with it the simplified utility maximization problem

${\displaystyle \max _{c_{1}}\;U(c_{1},c_{2})=u(c_{1})+\gamma \cdot u\left[(1+r)\left(y_{1}+{\frac {y_{2}}{1+r}}-c_{1}\right)\right]}$

with the optimality condition

${\displaystyle u'(c_{1})+\gamma \cdot (-1)\cdot (1+r)\cdot u'(c_{2})=0\Rightarrow {\frac {1}{1+r}}=\gamma \cdot {\frac {u'(c_{2})}{u'(c_{1})}}}$.

The expression is called the intertemporal limit rate of substitution. In contrast to the GRS introduced above, it relates to one and the same good (consumption), which, however, can potentially be "consumed" in two periods and in each of these generally provides a different benefit. ${\displaystyle \gamma \cdot u'(c_{2})/u'(c_{1})}$

See also

• Marginal rate of transformation
• Samuelson's condition

literature

• Geoffrey A. Jehle and Philip J. Reny: Advanced Microeconomic Theory. 3rd ed. Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 .
• Susanne Wied-Nebbeling and Helmut Schott: Fundamentals of microeconomics. Springer, Heidelberg a. a. 2007, ISBN 978-3-540-73868-8 .
• Harald Wiese: Microeconomics. An introduction. 5th edition. Springer, Heidelberg a. a. 2010, ISBN 978-3-642-11599-8 .

Remarks

1. ↑ See Jehle / Reny 2011, p. 12; Wied-Nebbeling / Schott 2007, p. 28 f.
2. ↑ See Wied-Nebbeling / Schott 2007, p. 30.
3. ↑ Jehle / Reny 2011, p. 18.
4. ↑ See Wied-Nebbeling / Schott 2007, p. 36.