# Math Help - Urgent Help Needed! Any Help Will Do

1. ## Urgent Help Needed! Any Help Will Do

Hi everybody,
This is my first post and already it's an urgent one.
If anyone can offer ANY help to ANY of these questions (preferably the latter parts of question 1 and question 3)

Hopefully someone has some answers for me, I'm TOTALLY stumped!

QUESTION 1

Assume the total cost function:

TC=500+10Q,

where Q = X+Y . The inverse demand functions for X and Y , respectively are:

Px = 110 - 10X
and
Py = 80 - 5Y

Find the profit maximising combination of output for the two products. Ensure you have found a maximum by checking the second-order sufficient conditions. Calculate the elasticity of demand for each product at these levels of output. Confirm that the lower elasticity product has the higher price. Explain why this should be the case.

QUESTION 2

Assume a production function

Q = 12L^2 - 0.75L^3

Derive expressions for the average product of labour (AP) and the marginal product of labour (MP), then:

(a) Find the optimum values for AP and MP.
(b) Verify that, at the critical points, the second-order sufficient condition holds.
(c) Check that average product equals marginal product at the AP critical point.

QUESTION 3

Assume the demand for a product q1 is a function of its own price (p1) and another product’s price (p2):

q1 = 10p1^-1p2^0.5

(a) Calculate the partial derivatives of the function for both prices.
(b) Using the result obtained in part (a), calculate the own price elasticity of demand.
(c) At what value of p1 is the own price elasticity unit elastic?
(d) Calculate the cross-price elasticity of demand. Is the other good a complement or a substitute? How can you tell?

QUESTION 4

(a) Find the marginal rate of technical substitution for the following production function:

Q = 10(0.2L(to the power of -0.5) +0.8K(to the power of -0.5))(to the power of -2)

(b) Find the marginal rate of substitution for the following utility function:

U = 10 + 0.2lnX1 + 0.8lnX2

Thank you in advance for any help!
Cheers,

Jack B

2. Originally Posted by jackbarendse
QUESTION 1

Assume the total cost function:

TC=500+10Q,

where Q = X+Y . The inverse demand functions for X and Y , respectively are:

Px = 110 - 10X
and
Py = 80 - 5Y

Find the profit maximising combination of output for the two products. Ensure you have found a maximum by checking the second-order sufficient conditions. Calculate the elasticity of demand for each product at these levels of output. Confirm that the lower elasticity product has the higher price. Explain why this should be the case.
Cost of producing $X$ units of the first product and $Y$ unitrs of the second is:

$C=500+10(X+Y)$

So the profit is:

$Profit(X,Y)=Revenue- Costs$

so:

$Profit(X,Y)=X P_X+Y P_Y- (500+10(X+Y))$

$Profit(X,Y)=X (110-10X)+Y (80-5Y)- (500+10(X+Y))$

This is the function you have to maximise

CB