Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due
4SSMN903 – SAMPLE EXAM PAPER
Module Code and Title: 4SSMN903: Advanced Mathematics
Examination Period: Period 2
Question 1 [12 marks]
Consider a firm that sells a~ single product. The firm faces a demand function q = a − bp + yp, where q is the quantity demanded ~for the product, p is the price of the product charged by the firm, and p is the price of an alternative similar product from another firm in the ma~rket (a, b, y > 0 ) . The firm takes the price of the alternative product, p, as given . The cost of producing each unit of the product is C (C > 0 ) and the firm pays no other cost.
a) Write down the profit function of the firm in terms of q, a, b, C, y and 
. . [2]
b) Assume that cb a . What is the profit maximising quantity of production for the firm? [4]
c) If the price of the alternative product increases, what is its impact on the quantity produced by the firm? [2]
d) What are the economic assumption(s) that are implicit in the model? Discuss if they are reasonable assumptions .** [4]
Question 2 [28 marks]
Consider a consumer with utility function U = x1 + x2 − l2 , where l is the labor supplied, x1 and x2 are the quantity of consumption for good 1 and 2 respectively . The price of good 1 and 2 are p1 and p2 respectively (p1, p2 >
0 ) . The consumer cannot spend more than its earning from working, and has no other source of income . The consumer has an hourly wage rate of w.
In the short-run, the consumer has signed a work contract that the work hour is fixed at l(̅) > 0 . The consumer cannot consumer negative amount of quantity of goods .
a) Write down the constraints faced by the consumer in the short run . [2]
b) Find the utility maximising quantity of consumption for good 1 and 2 in the short run . Assume p1 > p2 . [10)
c) In the long run, the consumer could choose the labour supply l , i.e . l is not fixed and is a choice for the consumer . The consumer cannot work more than 24 hours a day, and cannot work negative amount of hours . Find the utility maximising quantity of consumption and labour supply . Assume p1 > p2 and w < 48p2 . [12]
d) What are the economic assumption(s) implicit in the model? Discuss if they are reasonable assumptions . ** [4]
Question 3 [20 marks]
Let the continuous change of the capital stock, k , for any period, t, depend positively on new investment I, and negatively to the depreciation of the
existing capital stock. Let I = 40 and let the depreciation rate d = 0.4 . The capital stock level is initially 20.
a) What will be the capital stock in ten years? What will it be as t , tends towards infinity? What is the significance of your results? [8]
b) What is the relationship between the steady state level of capital, investment, and the depreciation rate? [3]
c) Show that the rate of change in the capital stock is initially positive . Does it increase indefinitely? [3]
d) The level of investment has been assumed to be constant through time, at 40. Is this plausible? What would happen to the steady state level of capital should investment levels increase through time? ** [6]
Question 4 [20 marks]
Let p(t) represent the average price of combine harvester at any given point in time . The change in p(t) over the time is given by 1 + 2t plus 20% of the pre-existing price at time t. The average price of a combine harvester at time t = 0 is £50,000 .
a) Solve for the expression showing the average price of a combine harvester as a function of time. [7]
b) Characterize the time path of the price index . [6]
c) Is it plausible for the average price combine harvesters to be characterized by your answers as in a) and b)? ** [7]
Question 5 [20 marks]
The coal market is described by the following demand and supply functions D(t) = a0 + a1 P(t) + a2 P˙(t) + P(¨)(t) and S(t) = β0 + β1 P(t) . The market is in equilibrium if D(t) = S(t) . Assume that a0 = 30, a1 = −6, a2 = 3, β0 = 12, β1 = 4
(a) What is the coal price, P* when the coal market is in long run steadystate, whereby P(¨) = 0 andP˙ = 0 ? [4]
(b) Find the time path of the coal price for P(0) = 10, P˙(0) = 2 . [6]
(c) Does a steady-state actually exist given the supply and demand equations given in the question, if so, at what point in time does it arise? [4]
(d) Are the supply and functions you were given a reasonable approximation to supply and demand within the coal market? ** [6]