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ECON6003 – Mathematical Methods of Economic Analysis

Mid-semester Exam: Semester 2, 2017

Question 1: [10 marks]

(a) At a given point in time, the marginal product of labor is 3 and the marginal product of capital is 2, the amount of capital is increasing by 2 per each unit of time and the rate of change of labor is +0.5. What is the rate of change of output?

(b) Firm 1 has a demand function given by q₁ = 1 - 2p₁ + p₂, where p₂ is the price charged by firm 2. The cost function of firm 1 is c(q₁) = (q₁)²+ q₁. Profit for firm 1 is given by p(p₁, p₂) = p₁ q₁(p₁, p₂) - c(q₁(p₁, p₂)). Use the product rule and the chain rule to determine the derivatives of firm 1 profit with respect to p₁ and p₂.

Question 2: [10 marks]

The revenue produced by a new oil well is $1 million per year initially (t=0), and it is assumed that it will rise uniformly to $6 million per year after 10 years. If we measure time in years and let f (t) denote the revenue (in millions of dollars) per unit of time at time t, it follows that f(t)=1+0.5t. If F (t) denotes the total revenue which accumulates over the time interval [0, t], then F'(t) =f (t).

i) Calculate the total revenue earned during the 10 year period (i.e. F(10)).

ii) Find the present value of the revenue stream over the time interval [0,10], if we assume continuously compounded interest at the rate r=0.05 per year.

Question 3: [10 marks]

Auslamb has a monopoly in the production of wool and mutton. The company buys sheep from farmers at a fixed price of $10 per head. Each sheep is then sheared to produce 1 kg of wool and slaughtered to produce 25 kg of mutton. The wool is sold in a market where the price per kilogram of wool, p_w, is given by

where q_w is the quantity (in kilograms) of wool sold. The mutton is also sold in the market where the price per kilogram of mutton, p_m, is given by

where q_m is the quantity (in kilograms) of mutton sold.

(a) If Auslamb buys x sheep and sells the wool and mutton it obtains in the respective markets, write the expression for its profit as a function of x, assuming that paying for the sheep input is the only cost of production.

(b) Write out the first order conditions (FOC) for profit maximisation.

(c) Verify that the point x=100 satisfies the first order conditions. Is it the only stationary point? Check the second order sufficiency conditions. Is x=100 a maximum? If so, is it a local or global max.

(d) Suppose the government imposes a tax t per kilogram of mutton sold. What is the expression for the firm’s profit? Re-express the first order conditions (FOC) for profit maximisation.

(e) Find the implicit derivative dx*/dt by using the FOC from (d), where x* is the optimum input of sheep and t is the tax rate on mutton and sign it. Evaluate this derivative at the point where t=0 and interpret the result you obtain from an economic viewpoint.

(f) Will the market price for wool change with changes in the tax rate t on mutton and if so how? Does this result make sense from an economic viewpoint?

Question 4: [10 marks]

Suppose that there are three different types of food: food 1, food 2 and food 3. All three types of food yield quantities of vitamins A, C and E but each yields the vitamins in different amounts. The amounts of the three vitamins yielded per unit of each food type can be represented in vector form if we let the first, second and third components of the vector represent the amounts of vitamins A, C and E respectively per unit of the food. Assume that the vectors are f₁ =(1,3,2)^T,

f₂ =(3,1,2)^T and  f₃ =(2,2,2)^T. Thus a unit of food 2 yields 3 units of vitamin A, 1 unit of vitamin C and 2 units of vitamin E.

i) Show that the three vectors are linearly dependent.

ii) Suppose that a consumer wishes to consume the quantities of vitamins given by the following vector (9,7,8)^T. Let (x₁, x₂, x₃) denote the quantities consumed of the three types of foods. Solve for x₁ and x₃ as functions of x₂ such that the consumer achieves the desired quantities of vitamins.

iii) Within what interval must x₂ lie in order for all three values of x_i to be non-negative? If the consumer was obliged to consume 5 units of food 3, would it be feasible for the consumer to exactly achieve the desired quantities of the three vitamins?