Let f(x, y) denote the output of a country. If x units of...

Let $f(x, y)$ denote the output of a country. If $x$ units of its resources are spent on labor and $y$ units are spent on capital, then the level curve of $f$ with equation $f(x, y)=k,$ where $k$ denotes a constant output, is an isoquant. Each point $(x, y)$ on the isoquant corresponds to a level of investment in labor, $x,$ and investment in capital, $y,$ that results in the same level of output $k$. Suppose the output of a certain country is $$f(x, y)=100 x^{3 / 4} y^{1 / 4}$$ billion dollars when $x$ billion dollars are spent on labor and $y$ billion dollars are spent on capital. a. Find the output if the country spends $\$ 81$ billion on labor and $\$ 16$ billion on capital. b. Find an equation giving the relationship between $x$ and $y$ if the output remains constant at the level found in part (a). c. Complete the following table for the equation found in part (b). $$\begin{array}{llllll} x & 50 & 60 & 70 & 80 & 90 \\ \hline y & & & & \\ \hline \end{array}$$ d. Use the results of part $(\underline{c})$ to sketch the isoquant of $f$ corresponding to the constant output found in part (b).

Let $f(x, y)$ denote the output of a country. If $x$ units of its resources are spent on labor and $y$ units are spent on capital, then the level curve of $f$ with equation $f(x, y)=k,$ where $k$ denotes a constant output, is an isoquant. Each point $(x, y)$ on the isoquant corresponds to a level of investment in labor, $x,$ and investment in capital, $y,$ that results in the same level of output $k$. Suppose the output of a certain country is
$$f(x, y)=100 x^{3 / 4} y^{1 / 4}$$
billion dollars when $x$ billion dollars are spent on labor and $y$ billion dollars are spent on capital.
a. Find the output if the country spends $\$ 81$ billion on labor and $\$ 16$ billion on capital.
b. Find an equation giving the relationship between $x$ and $y$ if the output remains constant at the level found in part (a).
c. Complete the following table for the equation found in part (b).
$$\begin{array}{llllll}
x & 50 & 60 & 70 & 80 & 90 \\
\hline y & & & & \\
\hline
\end{array}$$
d. Use the results of part $(\underline{c})$ to sketch the isoquant of $f$ corresponding to the constant output found in part (b).

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Key Concepts

Production Function

A production function represents the relationship between input factors (like labor and capital) and the output produced. It shows how different combinations of inputs yield a certain level of output, making it a central tool in economic analysis to assess efficiency and growth.

Cobb-Douglas Production Function

The Cobb-Douglas production function is a specific type of production function where output is represented as a product of inputs raised to constant powers. Its form, often given by f(x, y) = A x^? y^?, helps illustrate how changes in the amounts of inputs affect overall production, with the exponents indicating the elasticity of output with respect to each input.

Isoquant

An isoquant is a curve that represents all combinations of inputs that yield the same level of output. In the context of production theory, isoquants are analogous to indifference curves in consumer theory, showing the rate at which one input can be substituted for another while keeping the overall output constant.

Input Substitution

Input substitution refers to the process of replacing one input with another in the production process while maintaining the same output level. This concept is illustrated by the slope of the isoquant, which indicates the marginal rate of technical substitution between the inputs.

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