Asked by sasis foods on Apr 29, 2024
Verified
A firm has the production function f(X, Y) X 1/2 Y 1/2, where X is the amount of factor x used and Y is the amount of factor y used.On a diagram we put X on the horizontal axis and Y on the vertical axis.We draw some isoquants.Now we draw a straight line on the graph and we notice that wherever this line meets an isoquant, the isoquant has a slope of 23.The straight line we drew
A) is vertical.
B) is horizontal.
C) is a ray through the origin with slope 3.
D) is a ray through the origin with slope 4.
E) has a negative slope.
Isoquants
A curve representing all combinations of inputs that give the same level of output.
Production Function
An equation that describes the relationship between inputs and the maximum output that can be produced.
Slope
The rate at which a line rises or falls, indicating the steepness of a line on a graph, often representing change in variables in economics.
- Decode the concept of isoquants and recognize their role within the framework of production theory.
- Interpret the function and significance of technical substitution rates concerning input-output relationships.
Verified Answer
SP
Shilvi PatelMay 03, 2024
Final Answer :
C
Explanation :
If the slope of the isoquant is $23$, then the marginal rate of technical substitution (MRTS) between factors X and Y is given by $MRTS = \frac{MP_x}{MP_y} = 2/3$. This means that if we increase the usage of one factor by 1, we need to decrease the other by 2/3 in order to keep the same level of output. In other words, the slope of any tangent to an isoquant at any point is equal to $-MRTS = -2/3$.
Now, the only straight line that can intersect isoquants with a slope of $-2/3$ is a ray through the origin with a slope of $3$ (or $-3$). This is because the slope of a straight line is constant, and all the isoquants in the production function have slopes less than $-1$, whereas the MRTS is $-2/3$. Thus, the only way to have an intersection with a slope of $-2/3$ is to have a ray with a slope of $3$ that intersects the isoquants at an angle of $30$ degrees. Therefore, the correct answer is (C).
Now, the only straight line that can intersect isoquants with a slope of $-2/3$ is a ray through the origin with a slope of $3$ (or $-3$). This is because the slope of a straight line is constant, and all the isoquants in the production function have slopes less than $-1$, whereas the MRTS is $-2/3$. Thus, the only way to have an intersection with a slope of $-2/3$ is to have a ray with a slope of $3$ that intersects the isoquants at an angle of $30$ degrees. Therefore, the correct answer is (C).
Learning Objectives
- Decode the concept of isoquants and recognize their role within the framework of production theory.
- Interpret the function and significance of technical substitution rates concerning input-output relationships.