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

- 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

- Find the maximum profit, and the value of Q and P, at which this is achieved. Verify your answer by checking second-order conditions.
- i) the own-price elasticity of demand
- ii) the cross-price elasticity of demand
- 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

- i) How long would it take the economy to grow to €100m?
- 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.

- i) Is this function homogenous? If so, to what degree?
- ii) Comment on the returns to scale.
- 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 x_{1} represents the purchases of books and x_{2} 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?