Hello, this is Frank.

Today, we will explore the important economic concept of the Marginal Rate of Substitution (MRS). Using high school-level calculus, we will work through a practical example to deepen our intuitive understanding of MRS.

The Marginal Rate of Substitution (MRS) is the rate at which a consumer can substitute one good for another while keeping the same level of utility along an indifference curve. In simple terms, it shows how much of y must be sacrificed to gain one additional unit of x.

We can also refer to the concept of the Marginal Rate of Technical Substitution (MRTS). MRTS measures how much labor can be substituted with capital while keeping output constant. Similarly, MRS represents the rate at which a consumer can substitute between goods while keeping their utility unchanged.

Problem

Find \(MRS_{x,y}\) at the points \(A(2, 10)\) and \(B(10, 2)\) for the indifference curve \(u(x, y) = xy = k\).

Solution Example

\(\frac{dy}{dx} = – \frac{MU_{x}}{MU_{y}} = – \frac{\frac{d(xy)}{dx}}{\frac{d(xy)}{dy}} = – \frac{y}{x}\)

\(MRS_{xy}(2, 10) = \frac{y}{x} = \frac{10}{2} = 5\)
\(MRS_{xy}(10, 2) = \frac{y}{x} = \frac{2}{10} = \frac{1}{5} = 0.2\)

From this calculation, we can see that MRS varies significantly depending on the point along the indifference curve. As a consumer increases x, the amount of y that must be sacrificed changes depending on the point, so MRS changes point by point.

By convention, the final value of MRS is expressed as a positive number without the minus sign. This is because MRS generally represents the substitution ratio as a positive quantity.

For reference, an indifference curve is a curve that connects combinations of goods that provide the same level of utility for a consumer. It is also called an equal-utility line.

Combining high school math and economics allows for a more intuitive understanding of MRS. Through these derivative calculations, you can firmly grasp the basics of consumer behavior.

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Learn more in Mathematics Useful for Global Business b＾＾)
Reference book: ‘High School Math Once Again‘ by Kazuo Takahashi / Nihon Jitsugyo Publishing
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