What is MRS ? (#Economics)

Definition of MRS, Marginal rate of substitution 

- Definition of MRS 

: MRS is the rate at which the consumer is only just willing to exchange commodity 2 for a small amount of commodity 1.

This idea is so useful that it even has a name: the slope of an indifference curve is known as the marginal rate of substitution (MRS). The name comes from the fact that the MRS measures the rate at which the consumer is just willing to substitute one good for the other.

Thus the ratio defining the MRS will always describe the slope of the indifference curve: the rate at which the consumer is just willing to substitute a slightly little amount of the consumption of good 2 for a slightly little more less consumption of good 1. Marginal rate of substitution(MRS) measures the slope of an indifference curve.
[Practically, which means MRS is Derivative of Point on Utility Function U(＊), we would do persuasively mention about Marginal Utility(function) & Marginal Rate of Substitution on below chapter.]

Then, How can a MRS be calculated?

MRS at x' is the slope of the indifference curve at x'
MRS at x' is lim{Δx₂/Δx₁} = dx₂/dx₁ at x' (x' is intersect point between Any other function

Δx₁ → 0

dx₂ = MRS × dx₁ so, at x', MRS is the rate at which the consumer is only just willing to exchange commodity 2 for a small amount of commodity 1.

Marginal Utility(MU) and Marginal Rates-of-Substitution(MRS)

First, we define what is Marginal Utility means. Consider a consumer who is consuming some bundle of goods, (x₁,x₂). How does this consumer’s utility change as we give him or her a little more of good 1? This rate of change is called the marginal utility with respect to good 1. We write it as MU₁ and think of it as being a ratio,

MU₁ = ∂U / ∂x₁ = ΔU / Δx₁ 

MU₁ = u(x₁ +Δx₁,x₂)−u(x₁,x₂) / Δx₁

that measures the rate of change in utility (ΔU) associated with a small change in the amount of good 1 (Δx₁). 

Now, we research correlation between MU and MRS.
the Utility function U(x₁,x₂) can be used to measure the marginal rate of substitution (MRS). Recall that the MRS measures the slope of the indifference curve at a given bundle of goods. it can be interpreted as the rate at which a consumer is just willing to substitute a small amount of good 2 for good 1.

This interpretation gives us a simple way to calculate the MRS. Consider a change in the consumption of each good, (Δx₁,Δx₂), that keeps utility constant, that is, a change in consumption that moves us along the indifference curve. Then we must have

MU₁ Δx₁ + MU₂ Δx₂ = ΔU = 0 

[Upon statement equal as means as the general equation for an indifference curve is "U(x₁,x₂) = k", a constant]
Solving for the slope of the indifference curve we have

MRS = Δx₂ / Δx₁ = - MU₁ / MU₂

MRS = - (∂U / ∂x₁) / (∂U / ∂x₂) = (-1) × (Derivative of U by x₁) / (Derivative of U by x₂)

Budget Constrain(BC) ans Marginal Rates-of-Substitution(MRS)

And Next we have to mention about MRS and Budget Constrain(BC).
The Optimal choice happens when function line of Budget Constrain(BC), which is same as Rational Constrained Choice & Indifference Curve(later for MRS) intersect on some point (x₁',x₂')

This is the choice(x₁',x₂') of an optimal choice for the consumer which is most preferred in affordable bundle.
The Function of BC(Budget Constrain) looks like This

BC = (p₁,p₂,m) 

On the solving Test Question situation, if there mentioned as "Tangency Condition" then that means "The slope of the indifference curve at (x₁',x₂') equals the slope of the budget constraint.
The slope of the budget constraint is - p₁/p₂, and the slope of the indifference curve containing (x₁',x₂') is MRS and they are equal at (x₁',x₂')

Ex)

preference about product x₁ upon x₂ is slope of budget constraint. Ans slope of BC is - p₁/p₂.
and MRS is - ax₂ / bx₁ . but if, consider (x₁',x₂') is in Tangency Condition.
Then, - p₁ / p₂ is as same as - ax₂ / bx₁ 

    (1) If that case,

        x₂ = (a / b)(p₁ / p₂) x₁

    (2) Maximize the BC, then exhausts the budget remains

        so, (x₁',x₂') also exhausts the budget, which is p₁x₁' + p₂x₂' = m

substitute (1) in (2)

Then get, p₁x₁' + p₂ (a / b)(p₁ / p₂) x₁' = m

  x₁' = am / (a+b)p₁

  x₂' = am / (a+b)p₂

So we have discovered that the most preferred affordable bundle for a consumer with Indifference Curves (ex. Cobb-Douglas preferences)

(x₁',x₂') = [ am / (a+b)p₁ , am / (a+b)p₂ ]

Derivate of x₁' on U(x₁',x₂') is -a/b

Meaning x₁ is more preferred (a/b) times more than x₂

#Economics #Marginal Rates-of-Substitution #Indifference Curve #Marginal Utility #Budget Constrain