4. Ariel's T-shirt company hires labor at a rate of $8 per hour and rents capital (sewing machines) at a cost of $128 per machine per day. Throughout this question, you can think of Ariel renting fractions of sewing machines if necessary. Her production function is given by q = 6L3K3 where q measures the number of T-shirts produced each day, L measures the total number of hours worked by all employees in a day and K measures the number of sewing machines. (a) Using the Lagrangian technique, find the cost-minimizing combination of labor and capital for the production of q T-shirts per day. (b) Write down an equation for Ariel's long-run expansion path. (c) Find Ariel's long-run total, average, and marginal cost as a function of the number of T-shirts she produces each day. (d) Sketch Arieľ's long-run total expansion path in a diagram. Sketch at least two isoquants with the associated isocost lines. Using your diagram, determine how Ariel's long-run expansion path will shift if the cost of renting a sewing machine for a day decreases. Explain why this change occurs.
4. Ariel's T-shirt company hires labor at a rate of $8 per hour and rents capital (sewing machines) at a cost of $128 per machine per day. Throughout this question, you can think of Ariel renting fractions of sewing machines if necessary. Her production function is given by q = 6L3K3 where q measures the number of T-shirts produced each day, L measures the total number of hours worked by all employees in a day and K measures the number of sewing machines. (a) Using the Lagrangian technique, find the cost-minimizing combination of labor and capital for the production of q T-shirts per day. (b) Write down an equation for Ariel's long-run expansion path. (c) Find Ariel's long-run total, average, and marginal cost as a function of the number of T-shirts she produces each day. (d) Sketch Arieľ's long-run total expansion path in a diagram. Sketch at least two isoquants with the associated isocost lines. Using your diagram, determine how Ariel's long-run expansion path will shift if the cost of renting a sewing machine for a day decreases. Explain why this change occurs.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:4. Ariel's T-shirt company hires labor at a rate of $8 per hour and rents capital (sewing machines)
at a cost of $128 per machine per day. Throughout this question, you can think of Ariel renting
fractions of sewing machines if necessary. Her production function is given by q = 6LK
where q measures the number of T-shirts produced each day, L measures the total number of
hours worked by all employees in a day and K measures the number of sewing machines.
(a) Using the Lagrangian technique, find the cost-minimizing combination of labor and
capital for the production of q T-shirts per day.
(b) Write down an equation for Ariel's long-run expansion path.
(c) Find Ariel's long-run total, average, and marginal cost as a function of the number of
T-shirts she produces each day.
(d) Sketch Ariel's long-run total expansion path in a diagram. Sketch at least two isoquants
with the associated isocost lines. Using your diagram, determine how Ariel's long-run
expansion path will shift if the cost of renting a sewing machine for a day decreases.
Explain why this change occurs.
Expert Solution
Step 1
Given ,
Production function : q = 6K1/3L2/3
w = 8 /hour
r (cost of sewing machines ) = 128 per machine
Cost Constraint : C = wL + rK
C = 8L + 128K
Lagrangian Optimality setup is as follows :
Minimize (C = 8L + 128K ) subject to ( q - 6K1/3L2/3 )
L = 8L + 128K + (q - 6K1/3L2/3 )
Where , = Lagrangian Multiplier ,
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