こんにちは、Frankです。
今回は限界代替率（Marginal Rate of Substitution [MRS]）
を求めます。技術的限界代替率を踏まえて理解を深めます。
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Hi there!
Today, I give it a shot to find a marginal rate of substitution based on
the knowledge of the marginal rate of technical substitution describ-
ed below.
The Marginal Rate of Technical Substitution is the number obtained
by multiplying the tangent to the identity curve at a point by minus one
under a given combination of labor and capital equipment and is called
the marginal rate of technical substitution (of labor for capital).
It represents the amount of capital that can be substituted for one unit
of labor under the condition that the amount of production is kept the
same.
Since the possibility of substitution of labor for capital decreases with
an increase in labor input, the technical marginal rate of substitution
diminishes with an increase in labor input on the same-equivalent curve.
I don’t have a 100 percent clear picture of the MRS, but I want to have a
knack for estimating the MRS by answering the subsequent problem.
Calculate the \(MRS_{x,y}\) at the points: \(A(2, 10)\) and \(B(10, 2)\), providing
the equation of an indifference curve is \(u(x, y) = xy = k\).
Here is my solution to the problem.
\(\frac{dy}{dx} = – \frac{MU_{x}}{MU_{y}} = – \frac{\frac{d(xy)}{dx}}{\frac{d(xy)}{dy}} = – \frac{y}{x}\)
\(MRS_{xy}(2, 10) = \frac{y}{x} = \frac{10}{2} = 5\)
\(MRS_{xy}(10, 2) = \frac{y}{x} = \frac{2}{10} = \frac{1}{5} = 0.2\)
For reference, an indifference curve is a geometric representation of
consumer preferences in microeconomics, a curve connecting com-
binations of goods that are equally desirable or provide the same uti-
lity. It is also called an equal utility line.
I am excited about posting my mathematical problems in English
though I am a novice-level mathematics blogger.
Stay tuned, and expect to see my next post.
Keep well.
Frank Yoshida
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【グローバルビジネスで役立つ数学】でもっと学習する b＾＾)
【参考図書】『もう一度高校数学』（著者：高橋一雄氏）株式会社日本実業出版社
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