can you explain how they got M=y^2/2 and L=2y^2 from part b of this question 15. Apple uses labour and machines to produce iPhones according to the production functiony = LM, where L is the number of units of labour used and M is the number of machines.The cost of labour is w₁ = 4 per unit and the cost of using a machine is wM = 16. Applefaces a fixed cost of F = 25 for renting its factory. Given these input prices, Apple chooseshow much labour and machines to employ in order to minimize the cost of producing anygiven amount of iPhones y. Apple is therefore solving a cost-minimization problem.(a) If Apple is cost-minimizing, how many workers will it use for each machine? Thatis, find the number of workers per machine that Apple will employ. (Hint: setup thetangency condition.)(b)Continuing to assume that Apple produces any given number of iPhones y in thecheapest way possible, use the equation defining the number of workers per machinefrom part (a) and Apple's production function to find expressions for the cost-minimizingchoices of labour and machines as functions of output produced y.=b) We have y LAMA and L = 4M. Solving them together gives: Mand L = 2y².

The production function describes the relationship between inputs used in production and the output…

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can you explain how they got M=y^2/2 and L=2y^2 from part b of this question

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter11: Profit Maximization

Section: Chapter Questions

Problem 11.9P

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can you explain how they got M=y^2/2 and L=2y^2 from part b of this question

15. Apple uses labour and machines to produce iPhones according to the production function
y = LM, where L is the number of units of labour used and M is the number of machines.
The cost of labour is w₁ = 4 per unit and the cost of using a machine is wM = 16. Apple
faces a fixed cost of F = 25 for renting its factory. Given these input prices, Apple chooses
how much labour and machines to employ in order to minimize the cost of producing any
given amount of iPhones y. Apple is therefore solving a cost-minimization problem.
(a) If Apple is cost-minimizing, how many workers will it use for each machine? That
is, find the number of workers per machine that Apple will employ. (Hint: setup the
tangency condition.)
(b)
Continuing to assume that Apple produces any given number of iPhones y in the
cheapest way possible, use the equation defining the number of workers per machine
from part (a) and Apple's production function to find expressions for the cost-minimizing
choices of labour and machines as functions of output produced y.
=
b) We have y LAMA and L = 4M. Solving them together gives: M
and L = 2y².

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