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Egwald Economics: Microeconomics Oligopoly / Government Firm | Differentiated Oligopoly | Monopolistic Competition Imperfect Competition Models
Egwald's popular web pages are provided without cost to users. The many firms in a monopolistically competitive industry produce differentiated yet similar products. New firms can easily enter the industry. Textbooks often give retail trade or the hotel industry as examples. A monopolistically competitive firm's own demand curve is highly elastic, permitting it to vary its price within a narrow range of prices. The other firms' products are either very close substitutes or, a large number of other firms' products are substitutes (not necessarily very close substitutes). The diagram below shows the industry and firm's demand and marginal revenue curves. The slopes of the firm's demand and marginal revenue curves are greatly exaggerated. In reality, they are almost coincident horizontal lines. The firm's own demand schedule intersects the "industry's (average) demand schedule" at the firm's "equilibrium level of output and price," given the output levels of the other firms in the industry. These concepts are explained below.
Almost every day I get "flyers" in the mail informing me about the products of a new or existing restaurant, clothing store, or laundry, and their terms of trade (relative prices). Monopolistically competitive firms advertise (unlike perfectly competitive firms). Advertising seems necessary for a firm to enter and to remain in the industry. Retail firms pay taxes on the value of products sold (in Canada - G.S.T. and provincial sales taxes). The effect of varying these taxes on firms will be built into the model of a monopolistic industry. Suppose there are n firms in the industry. If q is the output of product and v is the amount of advertising of a typical firm, then the price, p, of its product is: p(q,v) = A / (n * q + B) + ad(v) A and B are positive constants. Since we are modelling the typical firm, in equilibrium each of the n firms will sell the same amount of product. Each firm's advertising function ad(v) shifts the demand curve. It is given by: ad(v) = c1*v - c2*v2 where c1 and c2 are positive constants. The negative sign ensures diminishing returns to advertising. Without advertising, a firm's sales prospects are diminished. Including the term, B, as a positive constant in the "industry's (average) demand schedule" ensures a "finite price," even when n = 0, or q = 0. The Monopolistic Competition model on this page is in some ways similar to the model of monopolistic competition of Henderson and Quandt (235-239). It uses a functional form for the demand functions of the firms that can be found in R.G.D. Allen (114). The cost function for the typical firm is: c(q,v) = a*q2 + b*q + c + s*v where a, b, c, and s are positive constants. Fixed costs are (c + s*v) because they do not vary with q.
ac(q,v) = a*q + b + (c + s*v) / q With an ad valorem tax at rate t (sales tax rate = t / (1-t)), the revenue to the firm is: r(q,v) = q * p(q,v) * (1 - t) At given levels of output, q, and advertising, v, the after tax profit of the typical firm is given by: prof(q,v) = r(q,v) - c(q,v) If the firm sets q and v so as to maximize its profit, the following first order marginal conditions must obtain:
profq = rq - cq = 0
where the subscripts represent the partial derivative of the function with respect to the specified variable. These first order conditions plus their associated Jacobian are used for each set of parameters of the model to solve for the optimal value of q and v. Suppose, first, that all firms in the industry behave (in concert) as the typical firm. Each firm will attempt to satisfy the above first order conditions by selecting appropriate levels of q and v. Firms will enter (or exit) the industry until the after tax price equals the average cost at the equilibrium level of q, the typical firm's output. At this equilibrium q and equilibrium number of firms n, the average cost curve will be tangent to the average revenue curve, p(q,v), and profits ~= 0. Such an equilibrium is shown below.
However, the typical firm's own demand schedule is given by: p(q,x,v) = A / ((n-1) * q + x + B) + ad(v) where the other (n-1) firms are at their equilibrium level of output q, while the typical firm varies its own output x. In the diagram above, we can see that the firm's own average revenue curve, p(q,x,v), intersects the ac(q,v) and p(q,v) curves at their point of tangency. Here the firm's own marginal revenue, mr(q,x,v,) curve lies above its marginal cost curve, mc(x,v). Thus, it may try to increase its sales by cutting its price slightly. If the other firms follow suit, this price cutting will lead to an actual loss in revenue and an increase in the firm's costs with profit turning negative. Some firms might exit, or firms might increase prices to their former level, restoring the zero profit condition. If the industry's firms are aggressive enough, we might expect, provided it is feasible, another "equilibrium" to obtain: with sales at that level of ouput where each firm's own average revenue curve, p(q,x,v), is tangent to its own average cost curve, ac(x,v), and where its own marginal revenue, mr(q,x,v), equals marginal cost, mc(x,v). This is the equilibrium below. You can adjust the values of the parameters of the model with the form below if you so desire to see how the equilbium changes.
The first order conditions for profit maximization consist of the following two nonlinear equations:
L1 = profq = rq(q,v) - cq(q,v) = 0
where
rq(q,v) = [p(q,v) + q * pq(q,v)] * [1 - t]
The Jacobian matrix of these equations is (differentiating wrt x when appropriate):
The gradient and Newton methods use the first order equations and the Jacobian matrix to solve for the equilibrium - profit maximizing - values of q and v.
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