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Constraint Equation? Multivariate calculus

  1. Apr 29, 2008 #1
    1. The problem statement, all variables and given/known data
    This is a second (university) year calculus problem dealing with calculus of multiple variables.

    In economics, utility is a measure of the relative satisfaction from, or desirability of, consumption of goods. A utility function u = u(a,b) gives the utility from consuming a units of a particular good and b units of another good.

    The utility function for Plike Mank's consumption of c units of cheese and m units of marbles is given by u(c,m) = 2(c^2)m.

    After this explanation there were some questions regarding the practical meaning of the partials, calculate the partials etc.

    Part (d)
    Plike's consumption of cheese and marbles is constrained by his salary. Write down a constraint equation if a unit of cheese cost $18, a unit of marbles cost $12 and he spends $216 on cheese and marbles.

    2. Relevant equations
    Don't know

    3. The attempt at a solution

    I'm not even sure what to do. I missed a couple of lectures this semester and I'm completely lost. Even if someone could point me to the relevant chapter in my textbook I'd be happy. I have never come across a constraint equation before.
  2. jcsd
  3. Apr 30, 2008 #2
    Assuming I'm interpreting the question correctly: Ignore the word 'constraint', and write the equation using common sense/intuition.

    The three values given are:
    1. cost of a unit of cheese
    2. cost of a unit of marbles
    3. amount spent on cheese and marbles

    So in the equation you would have 2 variables (which appear in the explanation about utility).

    I take it there's a part (e) to this question and so on that uses this equation as a constraint for utility?
  4. Apr 30, 2008 #3
    Thanks a lot for your help, I think I nailed it but I'd really appreciate if someone could give me some feedback on my work. Here's what I got:

    If marbles cost $12, cheese costs $18, total money is $216 then we set up the constraint equation
    18c - 12m - 216 = 0
    m = (216-18c)/12

    u(c,m) = 2(c^2)m = 2(c^2)(216-18c)/12
    = 36(c^2) - 3(c^3)

    Differentiating this gives
    u'(c) = 72c - 9(c^2) = 0
    c = 0 or 8
    Using c = 8, m=6

    Using these values gives a maximum utility of 4,608

    Does this look correct? Sorry about not using LaTex, I'm on a university computer with a 10 min time limit. Any feedback would be mucho appreciated.
  5. May 1, 2008 #4
    How come you have 18c - 12m and not 18c + 12m? It can help at first to add costs on one side and spendings on the other.
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