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Math Help - Microeconomics- production function

  1. #1
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    May 2009
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    Post Microeconomics- production function

    Hi,
    I was wondering if anyone could please help me understand the following question?

    Thanks in advance!

    Firm A's production function is:
    Q = 5LK,
    where Q
    = output, L = labour measured in person hours, and K = capital measured in
    machine hours. The firm's labour cost
    is $20 per hour, while the firm uses $80 per hour
    as an implicit machine rental charge per hour. The firm's current budget is $64,000 per
    month to pay labor and capital.

    a) Given the information above, determine firm A's optimal capital/labor ratio.
    b) Set up and explain the constrained maximisation problem using the
    information given above.


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  2. #2
    Member
    Joined
    Sep 2007
    Posts
    127
    Quote Originally Posted by nikolaucl View Post
    Hi,
    I was wondering if anyone could please help me understand the following question?

    Thanks in advance!

    Firm A's production function is:
    Q = 5LK,
    where Q
    = output, L = labour measured in person hours, and K = capital measured in
    machine hours. The firm's labour cost
    is $20 per hour, while the firm uses $80 per hour
    as an implicit machine rental charge per hour. The firm's current budget is $64,000 per
    month to pay labor and capital.

    a) Given the information above, determine firm A's optimal capital/labor ratio.
    b) Set up and explain the constrained maximisation problem using the
    information given above.


    (I realise this is a bit late, but if you're still interested...)

    For (a)

    Here we're looking at the microeconomic principle of the marginal rate of technical substitution. If we let MPL = marginal product of labour, MPK = marginal product of capital, K = unit of capital, L = unit of labour, under certain conditions (which, due to the nature of the question, I'm assuming are satisfied), theory ensures us that the optimal capital labour ratio is:

    K*/L* = MPK/MPL

    MPL is nothing else but the price of labour. i.e. the wage, which we know is 20, and the MPK is the 'rent', which we know is 80.

    So the optimal ratio is  \frac{K*}{L*} = \frac{80}{20} = 4

    For (b)

    It's probably a good idea to understand, intuitively, what we're looking to do. We're looking to produce as much as we can, given a budget of 64000. We know that to produce one unit of the good takes 5LK, and we know that each unit of labour costs 20, and each unit of capital costs 80.

    So our constrained maximisation problem is:

    maximise 5LK subject to 20L + 80K \leq 64000

    Realising that since the production function is increasing in both labour and capital, the maximum would be attained where the constraint is effective, we could formulate this into a Lagrangian problem:

     \max Q(L,K, \lambda ) = 5LK + \lambda (64000 - 80K - 20L)
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  3. #3
    Newbie
    Joined
    Nov 2012
    From
    Poland
    Posts
    2

    Re: Microeconomics- production function

    Hello Guys! Does anyone of You know how to do this task? I have no idea what I can write about this in 500 words
    I will be so grateful if someone helps me.. please

    Labour force and production function...

    A company’s labour force is composed of three categories. Firstly, there are unskilled workers, described with the symbol L1, displaying low productivity, employed at packing and loading the baked pies. Then, the firm employs skilled workers, described with the symbol L2, displaying high productivity, employed and the actual making of the pies. Finally, the business is managed by skilled administrative personnel, described with the symbol L3, displaying moderate productivity, employed at all kinds of administrative tasks. Below you will find three possible, logarithmic formulae of the company’s production function. Choose the right one and write an argumentation of
    500 – 700 words, to justify your choice..


    a)ln(Y)=ln(K^a )+ln(L1^b1 )+ln(L2^b2 )+ln(L3^b3 )+ln(A) a>0;b2>b3>b1>0

    b)ln(Y)=ln(K^a )+ln(L1^b1 )+ln(L2^b2 )+ln(L3^b3 )+ln(A) a>0;b3>b2>b1>0

    c)ln(Y)=ln(K^a )+ln(L1^b1 )+ln(L2^b2 )+ln(L3^b3 )+ln(A) a>0;b1>b2>b3>0
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