Economic problem based on constraint of scarce product

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August 1st 2010, 06:36 PM
seanP

Economic problem based on constraint of scarce product

Hi, so i'm trying to figure out how to do this question but I'm a bit lost.

I assume the answer to a) is 3P1 + P2 = K
after that, I'm not sure what to do.
Any help is greatly appreciated! Thanks in advance!!
p.s I'm not trying to make anyone do this for me, just a point in the right direction would be nice :)

Consider a two product firm with a profit function:

$projection (q1,q2) = 50 + 5q1 - q1^2 + 4q2 - q2^2 - q1q2$

where q1 and q2 are the output levels of products 1 and 2 respectively. The
manufacturing of the two products uses a scarce resource: one unit of product 1 uses 3 units of the resource and one unit of product 2 uses one unit of the resource. The firm has K units of this scarce resource.

a) Write down the firm's constraint that involves the use of this scarce product.

b) Assume the constraint is binding, i.e., that K is sufficiently small that all
of the scarce resource will be used. Solve the constrained maximization
problem of the rm using the substitution method.

c) What is the profit of the firm at the optimal values of q1 and q2?
Hint: the answer will be a function of K.

d) What is the marginal value of the scarce resource to the firm (in terms of
increased profit)?

e) For what value of K would the resource no longer be scarce? , i.e., how high
must K be for the firm to choose not to use all of this resource?
August 2nd 2010, 12:50 PM
SpringFan25

a) your answer is more or less right. It should be

$3q_1 + q_2 \leq K$

The inequality is there because the firm does not have to use all of its available resources. I dont understand why you wrote P instead of Q. Typo?

b)
Can you write the maximisation problem?

it looks like: Maximise ((profits)) subject to ((constraint))

You found the constraint in part a

Spoiler:

$\displaystyle \max_{q_1,q2} ~~ 50 + 5q1 - q_1^2 + 4q_2 - q_2^2 - q_1q_2 - \lambda \left( 3q_1 + q_2 - K \right)$

solve using the usual methods
(differentiate with respect to q1, q2 and lambda. Set all partial derivatives to zero)
August 2nd 2010, 03:05 PM
applesian

(Rock) yar we're classmates

i have the same question, could you give some hints on what should be done for part e)?

i'm guessing maybe i'm supposed to sub in the constraint into the profit function:
π ( q1, K – 3q1) = 50 + 5q1 – q1^2 + 4 * (K – 3q1) – (K – 3q1) ^2 – q1* (K – 3q1)

am i very far off? what would void the constraint for those two quantities? does making it not scarce means it's no longer constrained?

in that case.... that would just make the two quantities equal to themselves without the K and q2? i probably don't make sense.. sheesh
August 3rd 2010, 07:48 AM
SpringFan25

Part (d) is trying to give you a hint on how to find the answer to part (e). You want the value of k at which the marginal value of the scarce resource is 0.

An easier way to think about is this:

In part (c), you found the profit of the firm assuming it consumes exactly K resources $\pi(K)$

So, find the value of K at which profits no longer increase by using more resources $\frac{d \pi(K)}{dK} = 0$

Because this is the point at which profits fall by using more resources, it follows that any resources available after this point would not be consumed. So this is point the question is looking for.
August 3rd 2010, 09:15 AM
applesian

wooo! i actually did it like that afterwards

thank you!