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  1. #1
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    Econ Bros : Help! Cobb Douglas Equation..

    I know there are some geniuses on this forum.. figured I'd ask this question here. My professor isn't responding to emails, and this homework is due in 5 hours. All my friends are idiots. I'm getting desperate lol.

    If you can provide an answer, and explain it, reps for life..

    1. Given the Cobb-Douglas utility function U = X^1/2 Y^1/2 and income (I) is $100, answer the following questions.

    A. Find the utility maximizing consumption bundle if the price of good X (Px) is $1 and the price of good Y (Py) is $1.

    B. Find the utility maximizing consumption bundle if the price of good X (Px) is $4 and the price of good Y (Py) is $1.

    C. What proportion of the consumers budget was spent of each of the goods in questions A and B.
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    Bump..
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  3. #3
    Registered User Meagzz's Avatar
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    There's a 'cheat' way to do Cobb Douglas. Lemme try and find my notes and i'll help. I did this stuff last year and I can't remember off the top of my head.
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    Originally Posted by Meagzz View Post
    There's a 'cheat' way to do Cobb Douglas. Lemme try and find my notes and i'll help. I did this stuff last year and I can't remember off the top of my head.
    Man, that would be a huge huge help.

    Greatly appreciated man!
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  5. #5
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    int'l econ student checking in. gonna take a stab at this, but take this with a grain of salt bc i haven't studied this part of cobb-douglas

    simplifying the cobb-douglas equation (googled this) we get
    x*, y* = [ (a/a+b)(m/px), (b/a+b)(m/py) ]
    x* and y* are optimal amounts of x and y
    a and b are the exponents in your function (1/2 in both cases)
    m is the budget constraint
    px and py are prices or x and y

    putting your equations into this:
    A: x*, y* = [ (.5/1)(100/1), (.5/1)(100/1) ]
    simplifying, x*,y* = [50,50]

    B: x*, y* = [ (.5/1)(100/4), (.5/1)(100/1) ]
    simplifying, x*,y* = [12.5, 50]

    C: just need to figure out the proportion. in A, the consumer spent 50% on both x and y, in B the consumer spent (12.5)(4)=50% on X and 50% again on y
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    Originally Posted by PLZGO View Post
    int'l econ student checking in. gonna take a stab at this, but take this with a grain of salt bc i haven't studied this part of cobb-douglas

    simplifying the cobb-douglas equation (googled this) we get
    x*, y* = [ (a/a+b)(m/px), (b/a+b)(m/py) ]
    x* and y* are optimal amounts of x and y
    a and b are the exponents in your function (1/2 in both cases)
    m is the budget constraint
    px and py are prices or x and y

    putting your equations into this:
    A: x*, y* = [ (.5/1)(100/1), (.5/1)(100/1) ]
    simplifying, x*,y* = [50,50]

    B: x*, y* = [ (.5/1)(100/4), (.5/1)(100/1) ]
    simplifying, x*,y* = [12.5, 50]

    C: just need to figure out the proportion. in A, the consumer spent 50% on both x and y, in B the consumer spent (12.5)(4)=50% on X and 50% again on y
    Thank you man, any other insight is helpful.

    I appreciate it!
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  7. #7
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    I believe you take the partial derivative of x and y, set each to 0 and plug it back in. Also don't forget about the budget constraint PxX + PyY=I

    So for the first one

    dx/dy= 1/2*x^(-1/2) + y^(1/2) = 0 y^(1/2) = 1/2 * (1/2x)
    dy/dx= x^(1/2)= 1*2 (1/2y)

    I don't feel like doing the rest but you should be able to figure it out from here


    EDIT: ^^ That post gives you the correct answer but the process is a lot more important. This is something you want to know how to do for your exams, it is one of the basic equations of micro econ
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  8. #8
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    http://i45.tinypic.com/s33db7.jpg
    http://i46.tinypic.com/maawzm.jpg

    My notes, maybe they can help. They're from an example where p1=5, p2=4 and m=200
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  9. #9
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    That's what I did (for no. A)

    PLZGO is correct.
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  10. #10
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    Originally Posted by yeezus View Post
    I believe you take the partial derivative of x and y, set each to 0 and plug it back in. Also don't forget about the budget constraint PxX + PyY=I

    So for the first one

    dx/dy= 1/2*x^(-1/2) + y^(1/2) = 0 y^(1/2) = 1/2 * (1/2x)
    dy/dx= x^(1/2)= 1*2 (1/2y)

    I don't feel like doing the rest but you should be able to figure it out from here
    Ahhh ok, that makes a little more sense.

    Interdasting..
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  11. #11
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    Originally Posted by Meagzz View Post
    Originally Posted by i1983 View Post


    That's what I did (for no. A)

    PLZGO is correct.
    Ahhhh.. thank you very much!
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  12. #12
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    Hey brah . you can either use lagrange method

    objective function U=x^1/2 y^1/2
    ST= Px *(x) + Py *(y)=100

    so set up the lagrange. substitute the given prices in for the constraint.

    To find the proportion of income just do Px*X/100 ---for proportion of income spent on good X
    Py*Y/100----for proportion of inccome spent on good y
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  13. #13
    Nonstop Rustlin' BMatt's Avatar
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    Originally Posted by repos View Post
    Hey brah . you can either use lagrange method

    objective function U=x^1/2 y^1/2
    ST= Px *(x) + Py *(y)=100

    so set up the lagrange. substitute the given prices in for the constraint.

    To find the proportion of income just do Px*X/100 ---for proportion of income spent on good X
    Py*Y/100----for proportion of inccome spent on good y
    Will rep on recharge, thank you good sir.
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    Originally Posted by yeezus View Post
    EDIT: ^^ That post gives you the correct answer but the process is a lot more important. This is something you want to know how to do for your exams, it is one of the basic equations of micro econ
    Yeah, that's what I'm afraid of.

    Not being able to replicate it come exam time.
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