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Marginal cost with two variables, confused.

  1. Jan 30, 2014 #1
    1. The problem statement, all variables and given/known data

    Your weekly cost to manufacture x bicycles and y tricycles is

    C(x,y)=20,000+60x+20y+50√(xy)

    a. What is marginal cost of manufacturing a bicycle.

    b. What is the marginal cost of manufacturing a tricycle.

    c. What is the marginal cost of manufacturing a bicycle at a level of 10 bicycles and 10 tricycles?

    d. What is the marginal cost of manufacturing a tricycle at a level of 8 bicycles and 8 tricycles?





    2. Relevant equations



    3. The attempt at a solution

    Normally I thought I would derive then plug in for x to get marginal cost.
    I derived x and got (25y^1/2)/x^(1/2) +60 but I dont know what to do next or if that is even right.

    I derived y and got (25x^1/2)/(y^1/2) +20 again I dont know what to do next
     
  2. jcsd
  3. Jan 30, 2014 #2

    Ray Vickson

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    Yes, that is (roughly) what is meant by marginal cost.

    For (a) and (b) that is all you can do, since a base pair (x,y) is not specified. However, in parts (c) and (d) you are given x and y values to work with. So, what do you think you should do next?

    In principle, the true marginal cost of 1 bicycle, starting from a base of ##(x_0,y_0)##, would be
    [tex]\text{actual marginal cost } = C(x_0+1,y_0) - C(x_0,y_0)[/tex]
    For a cost function that does not "curve" too much (or for values of ##x_0## and ##y_0## that are not "too small" we have a reasonable approximation if we take
    [tex] C(x_0+1,y_0) - C(x_0,y_0) \approx \frac{\partial C(x_0,y_0)}{\partial x},[/tex]
    so that is why I said "roughly" before. Basically, the partial derivative is what economists often use. To be absolutely sure of your usage, check the definitions in your textbook or course notes.
     
  4. Jan 30, 2014 #3
    Ok this is what I got so far.

    a. 60+25√y/x

    b. 20+25√x/y

    c. $85

    d $45
     
  5. Jan 30, 2014 #4

    haruspex

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    Yes, but be careful with parentheses. Strictly speaking, √x/y means (√x)/y, but to get those numbers you must mean (correctly) √(x/y).
     
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