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Thread: Utility functions

  1. #1
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    Utility functions

    Hi!

    I've got the following utility function:

    $\displaystyle U = (y_1 - 2) (y_2 - 4) = y_1y_2 - 4y_1-2y_2-8$

    prices are

    $\displaystyle p_1 = 2 , p_2=4$

    $\displaystyle p_1$ is the price of good 1
    $\displaystyle p_2$ is the price of good 2

    $\displaystyle y_1$ is the quantity of good 1
    $\displaystyle y_2$ is the quantity of good 2

    The budget costraint is:

    $\displaystyle 100 = 2y_1 + 4y_2$

    Now, we know that we have equilibrium when we have

    $\displaystyle \frac { \delta U / \delta y_1} {p_1} = \frac { \delta U / \delta y_2} {p_2}$

    because weighted marginal utilities are equal.

    Ok, now this is where i get stuck! The book "says" that

    with our data, we have:

    $\displaystyle \frac { \delta U / \delta y_1} {p_1} = \frac {y_2 - 4} {2}$ and $\displaystyle \frac { \delta U / \delta y_2} {p_2} = \frac {y_1 - 2} {4}$
    My question is: why are $\displaystyle \delta U / \delta y_1$ equal to $\displaystyle y_2 - 4$ and $\displaystyle \delta U / \delta y_2$ equal to equal to $\displaystyle y_1 - 2$ ?

    Thank you in advance!
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  2. #2
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    $\displaystyle \displaystyle \frac{\delta U}{\delta y_1}$ is the partial differential of U with respect to y1. This means you differentiate the function with respect to y1, treating all other variables as constants

    $\displaystyle \displaystyle \frac{\delta U}{\delta y_1}=\frac{\delta (y_1 y_2)}{\delta y_1} + \frac{\delta (-4 y_1) }{\delta y_1} + \frac{\delta ( -2 y_2) }{\delta y_1} + \frac{\delta (-8)}{\delta y_1} $

    Taking the first term, remembering to treat y2 as a constant when differentiating, this is
    $\displaystyle \displaystyle \frac{\delta (y_1 y_2)}{\delta y_1} = y_2$



    Now all the other terms
    $\displaystyle \displaystyle \frac{\delta U}{\delta y_1}=\frac{\delta (y_1 y_2)}{\delta y_1} + \frac{\delta (-4 y_1) }{\delta y_1} + \frac{\delta ( -2 y_2) }{\delta y_1} + \frac{\delta (-8)}{\delta y_1} $

    $\displaystyle \displaystyle \frac{\delta U}{\delta y_1}= y_2 - 4 + 0 + 0$

    $\displaystyle \displaystyle \frac{\delta U}{\delta y_1}= y_2 - 4$

    You try the other one.
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  3. #3
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    $\displaystyle \displaystyle \frac{\delta U}{\delta y_2}=\frac{\delta (y_1 y_2)}{\delta y_2} + \frac{\delta (-4 y_1) }{\delta y_2} + \frac{\delta ( -2 y_2) }{\delta y_2} + \frac{\delta (-8)}{\delta y_2} $

    $\displaystyle \displaystyle \frac{\delta (y_1 y_2)}{\delta y_2} = y_1$

    $\displaystyle \displaystyle \frac{\delta (-4 y_1) }{\delta y_2} = 0$ (because $\displaystyle y_1$ is a constant and its derivative is 0, multiplied by -4 gives 0, right?)

    $\displaystyle \displaystyle \frac{\delta ( -2 y_2) }{\delta y_2} = -2 $ (because $\displaystyle y_2$'s derivative is 1, multiplied by -2 gives -2, right?)

    $\displaystyle \displaystyle \frac{\delta (-8)}{\delta y_2} = 0$ (because 8 is a constant and its derivative is 0?)

    $\displaystyle \displaystyle \frac{\delta U}{\delta y_2} = y_1 - 2 $

    Ok, thank you. Please, tell me if I did something wrong!

    Thank you again!
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  4. #4
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    perfect!
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  5. #5
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    Quote Originally Posted by SpringFan25 View Post
    perfect!
    Ok, thanks again
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  6. #6
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    Hi, again. I've got another problem here. I have to maximize the following Utility function: $\displaystyle max U = U(y_1, y_2)$

    subject to the following budget costraint: $\displaystyle R = y_1 p_1 + y_2 p_2$

    Of course $\displaystyle \displaystyle y_1=\frac {R - y_2 p_2} {p_1}$

    Then $\displaystyle max U = U(\frac {R - y_2 p_2} {p_1}, y_2)$

    Clear till here... now he says:

    By differentiating this expression and making it equal to zero, we get

    $\displaystyle \displaystyle \frac {\delta U}{\delta y_1} \frac {\delta y_1}{\delta y_2} +\frac {\delta U}{\delta y_2} = 0$
    Of course I don't want to take advantage of your helpfulness again, but could you please just tell me if I have to do the same things we've seen in my first post? Thanks again!
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