(a) Write down an expression for the profit of the firm. Obtain and provide an economic interpretation for the first-order profit-maximising conditions. (b) Find the quantities q1 and q2 that satisfy the first order conditions. (c) Find the second-order conditions for profit maximisation. For what values of α are the second-order conditions satisfied?
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Answer all parts (a) to (e) of this question. A profit-maximising firm produces quantities q1 and q2 of two goods, with total costs given by: C = q 2 1 + 2q 2 2 − αq1q2. The goods are sold in competitive markets at prices p1 and p2. (a) Write down an expression for the profit of the firm. Obtain and provide an economic interpretation for the first-order profit-maximising conditions. (b) Find the quantities q1 and q2 that satisfy the first order conditions. (c) Find the second-order conditions for profit maximisation. For what values of α are the second-order conditions satisfied? (d) Assuming that the second-order conditions are satisfied, obtain the firm’s profitmaximising outputs of the two goods. (e) Use calculus to determine how the optimal quantities vary with an increase in p1
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