## (solution) ECONOMICS E/S-1010 HARVARD UNIVERSITY CONJECTURAL VARIATIONS

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Using the Conjectural Variations model, answer the following:

(See handout for details of this model.)

a) Cooperative Conjectural Variations: If both firms match one another?s quantity restrictions (dq2/dq1 = dq1/dq2 = +1), what will be the industry output (q1 + q2)?

If the firms colluded to form a monopoly, what will be the industry output (qm)?

How does q1 + q2 compare with qm?

b) Competitive Conjectural Variations: If each firm assumes that if it reduced output by 1 unit, the other firm will increase output by 1, (dq2/dq1 = dq1/dq2 = ?1), what will be the industry output (q1 + q2)? If the firms were in perfect competition with each other, what will be the industry output (qpc)? How does q1 + q2 compare with qpc?

c) Cournot Conjectural Variations: Between these two extremes, if each firm takes the other?s output to be given, (dq2/dq1 = dq1/dq2 = 0), what will be the industry output (q1 + q2)? If the firms were Cournot competitors, what will be the industry output (qc)? How does q1 + q2 compare with qc?

d) Consistent Conjectural Variations: If we don?t know how firms conjecture about each other?s decisions, what is an equilibrium set of conjectural variations (dq2/dq1 = dq1/dq2 =?). Does this match with any of the cases above (a, b, c)? Discuss the result.

ECONOMICS E/S-1010 HARVARD UNIVERSITY CONJECTURAL VARIATIONS MODEL

III. Market Structures ? Oligopoly (Duopoly) ? Competition on the basis of output (q): Consider a market with n firms, each producing qi (i = 1, ... ,n) output.

The Cournot model assumes that firm i treats firm j?s output as fixed in its decisions

(?qj/?qi = 0). But it is unrealistic that firms in a small industry take their rivals? actions as given.

In fact, this is incorrect everywhere except at equilibrium. In an actual market, a firm would

expect its rivals? behavior to change in response to the firm?s output decisions.

The Conjectural Variations model assumes that firm j?s output will respond to variations in

firm i?s output (?qj/?qi ? 0). Consider a linear demand curve with quantity intercept a and price intercept b:

q = a ? a/b p ? p = b ? b/a q Let there be 2 firms (n = 2), each of which produces a part of the total output (q = q1 + q2),

and each having a constant marginal cost of c. Thus, firm 1?s profit function is:

?1 = p q1 ? c q1 = [ b ? b/a (q1 + q2) ] q1 ? c q1

The Model:

Read the following pages from Brian R. Binger and Elizabeth Hoffman, Microeconomics with

Calculus (Glenview, IL: Scott, Foresman, 1988), pp. 418-422.

Note that the authors use x in place of q for quantity. 1 2 3 4 5 6

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