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:

f(L,T)=LT=L1/2T1/2f(L, T) = \sqrt{L} \cdot \sqrt{T} = L^{1/2}T^{1/2}

We want to find input combinations LL (units of labor) and TT (units of land) that give an output of 4 bushels of peaches. This means:

L1/2T1/2=4L^{1/2} T^{1/2} = 4

Squaring both sides:

LT=16L \cdot T = 16

This is the equation of a hyperbola in terms of LL and TT. Now, to express TT in terms of LL, we solve for TT:

T=16LT = \frac{16}{L}

So, the points on the isoquant that give her an output of 4 bushels all satisfy the equation:

T=16LT = \frac{16}{L}

Plotting Input Combinations

To plot some input combinations, we can pick different values for LL and calculate the corresponding values for TT:

  • If L=1L = 1, then T=161=16T = \frac{16}{1} = 16.
  • If L=2L = 2, then T=162=8T = \frac{16}{2} = 8.
  • If L=4L = 4, then T=164=4T = \frac{16}{4} = 4.
  • If L=8L = 8, then T=168=2T = \frac{16}{8} = 2.
  • If L=16L = 16, then T=1616=1T = \frac{16}{16} = 1.

These combinations of LL and TT will form the points (1,16)(1, 16), (2,8)(2, 8), (4,4)(4, 4), (8,2)(8, 2), and (16,1)(16, 1) on the graph.

Sketching the Isoquant

The isoquant is the curve that connects these points. It represents the set of all combinations of LL and TT that give an output of 4 bushels. Since the equation LT=16L \cdot T = 16 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:

  1. How would the isoquant change for an output of 8 bushels?
  2. What happens to the amount of labor LL required if Cary doubles the land TT?
  3. How can we interpret the marginal rate of technical substitution (MRTS) along this isoquant?
  4. What does the shape of the isoquant tell us about the substitutability of labor and land in this production process?
  5. 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