Science:Math Exam Resources/Courses/MATH105/April 2010/Question 04

MATH105 April 2010

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q1 (f)  •  Q1 (g)  •  Q1 (h)  •  Q1 (i)  •  Q1 (j)  •  Q1 (k)  •  Q1 (l)  •  Q1 (m)  •  Q1 (n)  •  Q2  •  Q3  •  Q4  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q6  •  

   Other MATH105 Exams

  •  April 2009  •  April 2011  •  April 2010  •  April 2012  •  April 2013  •  April 2014  •  April 2015  •  April 2016  •  April 2017  •  April 2018  •  

Question 04

The production function for a firm is $\displaystyle f(x,y)=5x^{2/3}y^{1/3},$ where $x$ and $y$ are the number of units of labour and capital utilized respectively. Suppose that labour costs $108 per unit and capital costs $2 per unit and that the firm decides to produce 600 units of goods. Use Lagrange multipliers to determine the amounts of labour and capital that should be utilized in order to minimize cost. You need not show that your solution minimizes the cost.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
What are the objective function and the constraint function?

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We are given that labour costs $108 per unit and capital costs $2 per unit, and we want to minimize cost. Thus the objective function is $\displaystyle C(x,y)=108x+2y$. The firm wants to produce 600 units, so using the production function we want $\displaystyle 5x^{2/3}y^{1/3}=600$. Thus, after dividing both sides by 5, the constraint function is $\displaystyle g(x,y)=x^{2/3}y^{1/3}-120$, and the constraint is g(x, y) = 0. The Lagrange multiplier method says that the gradients of the objective function and the constraint function should be proportional, or $\nabla C(x,y)=\lambda \nabla g(x,y)$. The gradient of the objective function is $\nabla C(x,y)=\langle 108,2\rangle$. and the gradient of the constraint function is $\nabla g(x,y)=\langle {\frac {2}{3}}x^{-1/3}y^{1/3},{\frac {1}{3}}x^{2/3}y^{-2/3}\rangle$ Then we have three equations that must be satisfied $108=\lambda {\frac {2}{3}}x^{-1/3}y^{1/3}$, $2=\lambda {\frac {1}{3}}x^{2/3}y^{-2/3}$, $\displaystyle x^{2/3}y^{1/3}-120=0$. Solving the second equation for $\lambda$ gives $\lambda =6x^{-2/3}y^{2/3}$, and substituting into the first equation to eliminate $\lambda$ gives $108=6x^{-2/3}y^{2/3}{\frac {2}{3}}x^{-1/3}y^{1/3}$. Simplifying, we have $\displaystyle 108=4x^{-1}y$, and then we solve for y to get $\displaystyle y=27x$. Now substituting into the third equation, the constraint equation, gives $\displaystyle x^{2/3}(27x)^{1/3}-120=0$. Simplifying, we have $\displaystyle 3x=120$, and thus $\displaystyle x=40$. We then have that $\displaystyle {}y=27x=27(40)=1080.$ Therefore the cost is minimized when using 40 units of labour and 1080 units of capital.

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Lagrange multipliers, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag