Math Problem Statement
Cary raises peaches. Where L is the number of units of labor she uses and T is the number of units of land she uses, her output is f(L,T) = L½T½ bushels of peaches.
(a) On the graph below, plot some input combinations that give Cary an output of 4 bushels. Sketch a production isoquant that runs through these points. The points on the isoquant that gives her an output of 4 bushels all satisfy the equation T = _____________
Solution
We are given the production function:
We want to find input combinations (units of labor) and (units of land) that give an output of 4 bushels of peaches. This means:
Squaring both sides:
This is the equation of a hyperbola in terms of and . Now, to express in terms of , we solve for :
So, the points on the isoquant that give her an output of 4 bushels all satisfy the equation:
Plotting Input Combinations
To plot some input combinations, we can pick different values for and calculate the corresponding values for :
- If , then .
- If , then .
- If , then .
- If , then .
- If , then .
These combinations of and will form the points , , , , and on the graph.
Sketching the Isoquant
The isoquant is the curve that connects these points. It represents the set of all combinations of and that give an output of 4 bushels. Since the equation defines a hyperbola, the curve will be a downward-sloping convex curve.
Do you need further clarification on this or help with plotting the graph?
Related Questions:
- How would the isoquant change for an output of 8 bushels?
- What happens to the amount of labor required if Cary doubles the land ?
- How can we interpret the marginal rate of technical substitution (MRTS) along this isoquant?
- What does the shape of the isoquant tell us about the substitutability of labor and land in this production process?
- How would Cary's production function differ if she had increasing returns to scale?
Tip:
When interpreting an isoquant, remember that it shows all the combinations of inputs that result in the same level of output, offering insights into input substitution possibilities.
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Math Problem Analysis
Mathematical Concepts
Algebra
Production Functions
Isoquants
Economics
Formulas
f(L, T) = L^{1/2}T^{1/2}
L * T = 16
T = 16 / L
Theorems
Production theory (isoquants) in economics
Output-input relationship in production functions
Suitable Grade Level
Undergraduate Economics or Mathematics
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