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Math Help - Math help wanted :)

  1. #1
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    Math help wanted :)

    A firm’s cost function and the demand functions are
    C ( x ) = 5x and p = 25 - 2x respectively, where x is the amount produced and/or demanded.

    (a) Find the output level that will maximize the firm’s profits. What is the
    maximum profit?

    (b) If a tax of
    t per unit is imposed, which the firm adds to its cost, find the output level that will maximize the firm’s profits. What is the maximum profit? (Note that the optimal output and profits will both be functions of the tax rate t rather than some fixed values.)

    (c) Determine the tax
    t per unit that must be imposed to obtain the maximum tax revenue. (Hint: Use the solution of output as a function of t from 2(b) to formulate the tax revenue function.)

    (d) Given the solution for
    t in (c), find the optimal output and profits using their optimal choice functions in (b). (This time, the solutions will be some fixed values.)

    I have absolutely no clue how to do these - could someone show me how to do these please?
    Last edited by marie7; September 10th 2009 at 06:38 PM.
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  2. #2
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    It has been awhile... but..
    Profit maximisation occurs at MR = MC.
    The price is P=25-2x. So total revenue is TR=X*P = 25x-2x^2,
    So marginal revenue is MR=25-4x. And from the cost function, we know MC=5. So setting MR=MC, 25-4x = 5, so x=5.
    Profit is \pi=TR-TC=(25x-2x^2)-(5x)=25(5)-2(5)^2-5(5)=50

    Still working through the other parts of this question

    Which uni are you at?
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  3. #3
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    Hey I'm at Macquarie uni (Sydney) doing a Bachelor of Economics o_O lol
    Thanks for helping mate! Are you at uni too?
    Hey, I'm going offline now, but I'd still appreciate anyone's help Be on tomorrow
    Last edited by marie7; September 10th 2009 at 09:26 PM.
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  4. #4
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    Quote Originally Posted by marie7 View Post
    Hey I'm at Macquarie uni (Sydney) doing a Bachelor of Economics o_O lol
    Thanks for helping mate! Are you at uni too?
    Hey, I'm going offline now, but I'd still appreciate anyone's help Be on tomorrow
    Seem to have same homework as my uni
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  5. #5
    Member garymarkhov's Avatar
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    Quote Originally Posted by marie7 View Post
    (b) If a tax of t per unit is imposed, which the firm adds to its cost, find the output level that will maximize the firmís profits. What is the maximum profit? (Note that the optimal output and profits will both be functions of the tax rate
    Robb rocked (a), so I'll get you started on (b).

    Starting information:

    R=xp = 25x-2x^2

    C=5x

    When a tax is levied on the firm, its cost function becomes C=5x + tx.

    As in (a), optimal output is where MR = MC.

    So 25-4x=5+t. Now just solve for x.

    Once you've got your new x, plug it into the profit function (profit = revenue-cost) to find what profit will be at the optimal quantity produced. I think you can take it from here...

    Let me know what you get and then we can deal with (c) and (d).
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  6. #6
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    I need help with this exact same question. But im stuck on c.

    I thought that MC was TC' , which would be 5 not 5+t since t is a constant ?


    Edit: ok i get what you did.


    So i worked out


     x = \frac{-20+t}{-4}

    Is that right?
    Last edited by mr fantastic; September 12th 2009 at 01:26 AM. Reason: Merged posts
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  7. #7
    Member garymarkhov's Avatar
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    Quote Originally Posted by el123 View Post
     x = \frac{-20+t}{-4}
    Is that right?
    That's what I get, too. So if I'm right, then you're right. Let's hope we're both right!

    Still stuck on (c)?
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  8. #8
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    yeah still stuck on c.

    I worked out if x =0 then t = 20.

    But not really sure what to do next.

    Take the profit function then put t in then solve for x?
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  9. #9
    Member garymarkhov's Avatar
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    Quote Originally Posted by el123 View Post
    yeah still stuck on c.

    I worked out if x =0 then t = 20.

    But not really sure what to do next.

    Take the profit function then put t in then solve for x?
    Nope. We need a new term called GR for government revenue. GR=tx because the amount of the tax times the quantity sold by the firm is equal to the revenue.

    In part (a), we solved MR = 20-4x = 5+t to get x=5-\frac{1}{4}t.That expression told us how much the firm would want to produce for any given t.

    And now we know that the government (or some tax-levying authority) wants to maximize GR=tx. If we know what x is no matter what t is, we can write GR=tx=t(5-\frac{1}{4}t). I think you know how to maximize that function, right?
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  10. #10
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    um not really , but i think
     GR = tx = 5t-(\frac{1}{4}t)

    so Marginal gov rev would be GR'

    GR' = MGR =  5 - \frac{1}{2}t

    Then we find derivative of total cost? which is

    TC =  5x+5t-(\frac{1}{4}t)

    and now im lost
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  11. #11
    Member garymarkhov's Avatar
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    Not quite, but you're on the right track.

    GR=t(5-\frac{1}{4}t)

    GR=5t-\frac{1}{4}t^2. Don't forget to distribute the t all through the stuff in the brackets!

    Next, take the first derivative of GR.

    GR'=5-\frac{1}{2}t

    Set the result equal to zero (because we want to find the max, which will occur when the slope is zero).

    5-\frac{1}{2}t=0

    t=10

    Man, that's a big ass tax! And they're training us economists to help make them as big as possible! Gosh, governments suck
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  12. #12
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    Thanks mate, legend!

    Do you think you could show me the steps of how to simplify this. I keep screwing up.

     20(5-\frac{t}{4})-2(5-\frac{t}{4})^2+t(5-\frac{t}{4})
    Last edited by mr fantastic; September 12th 2009 at 01:27 AM. Reason: Merged posts
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  13. #13
    Member garymarkhov's Avatar
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    Quote Originally Posted by el123 View Post
    Do you think you could show me the steps of how to simplify this. I keep screwing up.

     20(5-\frac{t}{4})-2(5-\frac{t}{4})^2+t(5-\frac{t}{4})
    Start by carefully expanding everything. Keep in mind that a(b + c) = ab + ac.

    So, for the first item you'll encounter, expand 20(5-\frac{t}{4}) to (20)(5)-(20)(\frac{t}{4})=100-\frac{20t}{4}=100-5t

    Then move on to the next item. Show your attempted solution and I'll let you know where you're going wrong.
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  14. #14
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     100-5t-10^2+\frac{2t}{4}^2-5t-\frac{t^2}{4}

    I know the answer is

     -\frac{3t^2}{8}+5t+50

    But no idea how they get that , all my things cancel each other out.
    Last edited by mr fantastic; September 12th 2009 at 01:28 AM. Reason: Merged posts
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  15. #15
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    so final answer shuld be  \frac{1}{8}(t-20)^2
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