Find all the first and second-order partial derivatives of the following function: Mathematics Assignment TCD, Ireland

QUESTION 1 

(a) Find all the first and second-order partial derivatives of the following function:

(b)If fixed costs are 10 and variables costs per unit are 2 and given the inverse demands function P=10 – 2Q_d:

            Find an expression for TR and TC functions in terms of Q

1. Find the maximum profit, and the value of Q and P, at which this is achieved. Verify your answer by checking second-order conditions.

1. i) the own-price elasticity of demand
2. ii) the cross-price elasticity of demand
3. iii)  the income elasticity of demand

(e)  Growth in an economy is given by GNP=75e 0.25t  Assume the initial value (i.e. first time period)  is t=0

1. i) How long would it take the economy to grow to €100m?
2. ii) What will GNP be after 5 years?  After 50 years?

QUESTION 3

A firm’s production function is given by Q= 100K 0.25L0.85.  Comment on the economic significance of your answers in each case.

1. i) Is this function homogenous? If so, to what degree?
2. ii) Comment on the returns to scale.
3. iii)  Find the marginal rate of technical substitution

• What combination of K, L will be chosen to maximize output if the firm’s total cost is constrained to €100, the price of labor is €4, and the price of capital is €10? Use the production function from part (a).

QUESTION 4

A consumer is going shopping with friends.  The consumer has the following utility function:                                                                
where x₁ represents the purchases of books and x₂ represents the purchases of shoes.

Show the following:

• Marginal utility for books and shoes.
• Does the law of marginal utility hold?
• Marginal rate of consumer substitution.

b). The price of books is €10 each and the price of shoes is €50 per pair. Using the same utility function, solve the following:
(i) If the consumer has a budget for the shopping trip of
€200, what will be the consumer’s utility level from the optimal combination purchases of books and shoes?
(ii) Suppose the consumer receives a present that doubles her income.  What will be the consumer’s utility from the optimal combination of books and shoes as a result of this windfall?