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4SSMN903 Advanced Mathematics
创建日期：2024-08-16 17:15:13
所属分类：数学mathematics
标签： 4SSMN903
Advanced Mathematics

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4SSMN903 Advanced Mathematics


Examination Period: Period 2

Time allowed: Students will have a 3 window to complete the paper, and 30 minutes to upload.

Question 1 [12 marks]

Consider a firm that sells a single product. The firm faces a demand

function q  =  a  ?  bp  +  Y, where q is the quantity demanded for the

product, p is the price of the product charged by the firm, and  is the

price of an alternative similar product from another firm in the market

(a, b, Y  >  0). The firm takes the price of the alternative product,  , 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    >  .   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   =  x?  + x?  ?  l? , where l  is the    labor supplied, x? and x? are the quantity of consumption for good 1 and 2    respectively. The price of good 1 and 2 are p? and p?  respectively (p? , p?  >     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

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 p?  > p? .                              [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 p?  > p? and w < 48p? .                                                   [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) = a + a? P(t) + a? P(˙)(t) + P(¨)(t) and S(t) = β + β? P(t). The market is in

equilibrium if D(t) = S(t). Assume that a = 30,  a?  = ?6,  a?  = 3, β =

12, β? = 4

(a) What is the coal price, P* when the coal market is in long run steady- state, 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?

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