In economics, a production model is a mathematical relationship between the output of a company or a country and the labor and capital equipment required to produce that output. Much of the pioneering work in the field of production models occurred in the 19205 when Paul Douglas of the University of Chicago and his collaborator Charles Cobb proposed that the output P can be expressed in terms of the labor L and the capital equipment K by an equation of the form P = c L α K β where c is a constant of proportionality and α and β are constants such that 0 < α < 1 and 0 < β < 1. This is called the Cobb-Douglas production model. Typically, P , L , a n d K are all expressed in terms of their equivalent monetary values. These exercises explore properties of this model. (a) Consider the Cobb-Douglas production model given by the formula P = L 0.75 K 0.25 . Sketch the level curves P L , K = 1. P ( L , K ) = 2 , and P L , K = 3 in an LK - coordinate system ( L horizontal and K vertical). Your sketch need not be accurate numerically, but it should show the general shape of the curves and their relative positions. (b) Use a graphing utility to make a more extensive contour plot of the model.
In economics, a production model is a mathematical relationship between the output of a company or a country and the labor and capital equipment required to produce that output. Much of the pioneering work in the field of production models occurred in the 19205 when Paul Douglas of the University of Chicago and his collaborator Charles Cobb proposed that the output P can be expressed in terms of the labor L and the capital equipment K by an equation of the form P = c L α K β where c is a constant of proportionality and α and β are constants such that 0 < α < 1 and 0 < β < 1. This is called the Cobb-Douglas production model. Typically, P , L , a n d K are all expressed in terms of their equivalent monetary values. These exercises explore properties of this model. (a) Consider the Cobb-Douglas production model given by the formula P = L 0.75 K 0.25 . Sketch the level curves P L , K = 1. P ( L , K ) = 2 , and P L , K = 3 in an LK - coordinate system ( L horizontal and K vertical). Your sketch need not be accurate numerically, but it should show the general shape of the curves and their relative positions. (b) Use a graphing utility to make a more extensive contour plot of the model.
In economics, a production model is a mathematical relationship between the output of a company or a country and the labor and capital equipment required to produce that output. Much of the pioneering work in the field of production models occurred in the 19205 when Paul Douglas of the University of Chicago and his collaborator Charles Cobb proposed that the output P can be expressed in terms of the labor L and the capital equipment K by an equation of the form
P
=
c
L
α
K
β
where c is a constant of proportionality and
α
and
β
are constants such that
0
<
α
<
1
and
0
<
β
<
1.
This is called the Cobb-Douglas production model. Typically,
P
,
L
,
a
n
d
K
are all expressed in terms of their equivalent monetary values. These exercises explore properties of this model.
(a) Consider the Cobb-Douglas production model given by the formula
P
=
L
0.75
K
0.25
.
Sketch the level curves
P
L
,
K
=
1.
P
(
L
,
K
)
=
2
,
and
P
L
,
K
=
3
in an LK-coordinate system (L horizontal and K vertical). Your sketch need not be accurate numerically, but it should show the general shape of the curves and their relative positions.
(b) Use a graphing utility to make a more extensive contour plot of the model.
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Correlation Vs Regression: Difference Between them with definition & Comparison Chart; Author: Key Differences;https://www.youtube.com/watch?v=Ou2QGSJVd0U;License: Standard YouTube License, CC-BY
Correlation and Regression: Concepts with Illustrative examples; Author: LEARN & APPLY : Lean and Six Sigma;https://www.youtube.com/watch?v=xTpHD5WLuoA;License: Standard YouTube License, CC-BY