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N11124-E1 University of Nottingham BUSINESS SCHOOL A LEVEL 1 MODULE, AUTUMN SEMESTER 1999-2000 QUANTITATIVE METHODS 1A Time allowed ONE hour THIRTY minutes Candidates must NOT start writing their answers until told to do so Answer THREE questions Only silent, self-contained calculators with a single-line display are permitted in this examination. Dictionaries are not allowed with one exception. Those whose first language is not English may use a dictionary to translate between that language and English provided that neither language is the subject of this examination. 1 A firm faces the inverse demand function: P = 150 – 6Q And the average cost function AC 31 Q 2 5Q 35 10 Q where Q is the quantity produced and sold and P is the price of a unit of the good. (a) Derive and sketch the firm’s revenue function and explain its shape in economic terms. Find the levels of output and the corresponding prices where zero revenue is earned. (b) Find via differentiation the level of output and the corresponding price level at which the firm maximises total revenue. (c) Explain what is meant in general by a firm’s profit function. Derive the profit function for the firm in this question. (d) Find the level of output and the corresponding price level which maximise profit. (e) Explain, with the use of an appropriate sketch, why the firm produces more to maximise revenue than it does to maximise profit. N11124-E1 TURN OVER 2 2 N11124-E1 A firm has the production function: f(K, L) = 10K0.5L0.5 where K and L are the amounts of the capital and labour inputs employed respectively. 3 (a) Write down, in a form in which K appears as the dependent variable, the isoquant for an output level of 500. Hence find the quantity of capital which, in conjunction with a labour usage of 10 units, yields an output level of 500. (b) Find, using the method of Lagrange, the combination of labour and capital which minimise the cost of producing 500 units of output, given that the price of a unit of labour and capital are 10 and 5 respectively. (c) Without doing any extra work, state what is the marginal cost of producing 500 units for this firm. Justify your answer. In the following model of the market for free-range eggs Qd and Q8 represent quantities demanded and supplied respectively, P represents the market price of eggs, a, b, c and d are unknown constants. Qd = Qs Qd = a + bP Qs = c + dP (a) State the economic meaning of the constants b and c. (b) What restrictions should be placed on the values taken by a, b, c and d? (c) Solve, in terms of the unknown constants, the equilibrium market price and quantity traded. (d) Provide a sketch illustrating your answer to 3(c). (e) Use your answer to 3(c) to deduce the effect on market price and quantity traded of an increase in c. Briefly interpret your results. N11124-E1 3 4 N11124-E1 A price discriminating monopolist sells the same type of good to people in work and to the unemployed at different prices. Let x and y denote the amounts sold to the unemployed and employed respectively, and let px and py denote the respective prices charged to each market segment. The demand functions in each segment are x = 20 – 0.5px y = 10 – 0.125py The cost function faced by the firm is: C(Q) = Q2 + 3Q + 50 where Q is the total quantity produced. It may be assumed that the entire amount produced is also sold, so that Q = x + y 5 (a) Find the inverse demand functions for x and y, and hence write down the monopolist’s total revenue function in terms of x and y. (b) Find an expression for the monopolist’s profit as a function of x and y. (c) Determine the amounts the monopolist should sell in each market segment if profit is to be maximised. In addition, find the prices charged. Explain why the firm charges different prices in each segment. (d) Find the profit earned if the monopolist is adopting a profit maximising strategy. A firm’s Total Cost function is given by TC(Q) = 6Q2 + 25Q + 10 (a) What is the economic significance of the fact that TC(0) = 10? (b) Find the marginal cost function. (c) Find the average cost function. (d) Find the output level where average cost is minimised. (e) Find the marginal and average cost at this output level. Comment. N11124-E1 END