B.Sc. (Hons) Degree in Social Sciences B.Sc. (Hons) Degree in Mathematical Science with Finance and Economics B.Sc. (Hons) Degree in Banking & International Finance Part II Examination SS2.06 Economic Theory Treated Mathematically Friday 2 June 2000 5.00 pm – 8.00 pm Answer FOUR of the ten questions INTERNAL EXAMINER: Professor D. Glycopantis EXTERNAL EXAMINER: Professor S. Thompson Page 1 of 6 1. The utility function of an individual is given by U [ x11 x 21 ] 1 where xi denotes the quantity of Good i. The individual is a price taker and has a fixed income, M. The price of Good 1 is denoted by p1 and the price of Good 2 by p2 . 2. (i) Give first and second order conditions for utility maximisation subject to the budget constraint. (ii) Calculate the demand functions and show that they are homogeneous of degree zero in prices and income. (iii) Obtain the form of the indirect utility function. homogeneous of any degree in prices and income. (iv) Suppose now that the individual has to pay a tax t1 per unit of Good 1 bought and receives a subsidy s2 per unit of Good 2 bought, where s2 p2 . Obtain the indirect utility function and explain for which combination of t1 and s1 the individual remains as well off when the tax and subsidy are introduced as before. Explain whether it is The minimum cost function of a firm is given by 1 1 1 C* 2 y 4 p1 2 p2 2 p1 p2 where y denotes the quantity of output and pi the price of Input i. (i) Obtain the production function of the firm and show the shape of the isoquants on an appropriate plane. Explain also whether the production function exhibits constant, increasing or decreasing returns to scale. (ii) Calculate the expressions for x1 x2 , and where xi is the quantity demanded of p2 p1 Input i and give an intuitive explanation of their signs. (iii) Verify the usual relation between the average variable cost and the marginal cost functions of the firm. Page 2 of 6 3. A firm produces two outputs using two factors of production. The production functions are given by Y1 min( L1 , K1 ) and Y2 min( L2 , K2 ) where Yi denotes the quantity of Output i , Ki the quantity of capital in the production of Output i and Li the quantity of labour in the production of Output i, and is a positive constant. The total quantity of capital at the firm's disposal is 1 unit and the total quantity of labour is 1 unit. The firm can sell its products at fixed positive prices p1 and p2 where pi denotes the price of Good i. The firm's objective is to maximise its revenue. 4. (i) Derive the transformation frontier of the firm and explain how its shape varies with . (ii) For =2, explain whether it is possible that the firm specialises in the production of one good only. (iii) Suppose that technological change takes place in the production of Good 1 so that any combination of labour and capital produce now 2 times as much as before. Derive the new transformation frontier of the firm. Consider the following neoclassical growth model: 1 Y 3K 2 L 1 2 2L dK sY K dt 0 s 1 L Loent where Y denotes the quantity of output, K the quantity of capital and L the quantity of labour. s, , n and Lo are positive constants. The initial quantity of capital, K(0) Ko is also given. (i) Interpret the relationships above and explain whether the marginal productivity theory of distribution holds. (ii) Given the initial condition K(0) Ko 0, discuss the questions of existence, uniqueness and stability of the steady-state growth paths. (iii) Calculate the golden rule capital-labour ratio and, without engaging in any calculations, give a condition which must be satisfied so that the steady state capital-labour ratio and the golden rule capital-labour ratio will be equal. Page 3 of 6 5. 6. There are two duopolists in a market for a particular product. Their profit functions are 1 (16 2(q1 q 2 ))q1 and 2 (20 (2q1 3q 2 ))q 2 where qi is the quantity of the good produced by Firm i. (i) Calculate the collusion solution. (ii) Calculate the duopolists' reaction functions and the Cournot-Nash equilibrium. Explain whether it is stable. (iii) Prove that the Cournot solution is inefficient and that the collusion solution is efficient. Give an intuitive explanation of these results. A particular economy consists of two traders and two goods. The utility function of 2 1 x Trader 1 is given by U (1 x11 )e 12 and of Trader 2 by U log x 21 log x 22 , where xij denotes the quantity of Good j consumed by Trader i. The initial endowments of Trader 1 are ( x11, x12 ) (1, 0) and those of Trader 2 are ( x21, x22 ) (0,1). (i) Prove that the contract curve of this economy is given by x12 2 x11 and show 1 x11 it on an Edgeworth-box diagram. (ii) Calculate the competitive equilibrium and show it on in your diagram. Check that the competitive allocation is Pareto efficient. (iii) Explain why in calculating the competitive prices only relative prices can be determined. Page 4 of 6 7. (a) The indirect (maximum) profit function of a firm is given by * Py2 2( PL PK ) where Py is the price of output, PL the price of labour and PK the price of capital. Obtain the form of the production function and show the isoquants on an (L, K) plane. Explain whether the production function exhibits constant, increasing or decreasing returns to scale. (b) A monopolist's total cost function is given by TC 8q q 2 and his demand function by p 10 2q where p denotes price and q quantity. The objective of the firm is to maximise its profit. 8. (i) Calculate the solution of the monopolist's profit maximisation problem. (ii) Suppose the government taxes the monopolist according to the formula 2 T To tq where T is total taxes and To , t are fixed constants. Calculate the effect of a change in t on the level of profits. The indirect utility function of a consumer is given by 1 M U* logp1 logp2 3 p1 3 where M is income, pi the price of Good i, is a positive constant and M 3 p1 . (i) Check that U* is homogeneous of degree zero in prices and income. (ii) Obtain the demand functions of Goods 1 and 2 and show that they are homogeneous of degree zero in prices and income. (iii) Calculate (iv) Suppose an extra unit of income becomes available. Explain how it will be allocated between the two goods. (iv) Obtain the consumer's utility function and show on a graph what the indifference curves map will look like. x1 and give an explanation of its sign. p2 U U Page 5 of 6 9. Consider the following linear programming problem: Minimise C 3 p1 2(2 ) p2 6 p3 Subject to 2 p1 2 p2 0 p1 4 0 p1 2 p2 4 p3 6 p1 , p2 , p3 0 10. (i) For 0 1 , obtain the solution of the above problem and the solution of its dual. (ii) Give a possible economic interpretation to the primal problem and the dual problem. (iii) Explain what happens to the solution of the dual when takes values greater or equal to 1. Consider the following macroeconomic system: (1) (2) (3) (4) Y C I G C Co 0. 75Y d Y d Y To tY (5) M kPY (6) YN (7) Y ' ( N ) W and P W N P (8) I Io r 1 r 1 where Y is real output, Y d is disposable real output, C is consumption, I is Investment, r is the rate of interest, P is the price level, and M,G, are, respectively, the exogenously determined supply of money and government expenditure. Co , To , t , Io , k , , and are positive constants with t<1. (i) Interpret the relations above and explain which is the liquidity trap rate of interest. (ii) Obtain from the relations (1) to (5) above the IS-LM curves and calculate the effect on real output of an increase in G. Calculate also the effect on real output of an increase in M and compare the two effects. (iii) Consider now the complete system. Explain briefly whether the system decomposes into a real subsystem and a money subsystem. 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