Introduction to Economics II: Producer Theory Leslie Reinhorn Durham University Business School October 2014
Plan of the Lecture Introduction The Case of Perfect Competition pro t maximization problem producer surplus The Case of Monopoly pro t maximization problem welfare loss under monopoly price discrimination
Introduction A rm: a machine having the magical ability to convert one type of goods (Inputs) into another type of goods (Outputs) In the simple case of perfect competition, rms are price takers. In other cases, rms may have controls of prices to some degree, such as monopoly
Pro t Maximization: Perfect Competition production function: q = f (l, k) assumption 1: positive marginal output, i.e. f / l > 0 assumption 2: diminishing marginal output, i.e. 2 f / l 2 < 0 pro t maximization as price taker max fq;l,kg pq wl rk s.t. q = f (l, k) (1) equivalently, max fl,k g pf (l, k) wl rk the rst order conditions: p f l = w p f k = r (2) ) marginal revenue equals marginal cost (MR = MC) ) input demand functions: l (q; r, w) and k (q; r, w)
The Dual Problem: Cost Minimization Cost Minimization rst, to nd the cheapest way to produce q min fl,k g wl + rk s.t. q = f (l, k) (3) ) cost function c (q; r, w) = wl (q; r, w) + rk (q; r, w) second, to nd the optimal q to maximize the pro t max fqg pq c (q; r, w) (4) ) p = c(q;r,w ) q : the supply function
Cost Minimization: Iso-quantity Curve Two assumptions of f (l, k): Monotonicity: i.e. for l 0 l and k 0 k, we have f (l 0, k 0 ) f (l, k) Convexity: for (l 0, k 0 ) and (l, k) such that f (l 0, k 0 ) = f (l, k), then for λ 2 [0, 1] Remarks: f λl 0 + (1 λ) l, λk 0 + (1 λ) k f (l, k) the same assumptions as made for preferences) the iso-quantity curve for f (l, k) is like the indi erence curve for preference; we later will study non-convex production functions.
Cost Minimization: the Tangent Condition r w k c = wl + rk ) iso-cost curve l = c w MRTS (Marginal Rate of Technical Substitution) MRTS l,k = dl dk = ( f / k) ( f / l) = MP k MP l (5) Tangent condition: r w = MRTS l,k
An Example of C-D Production Function Example f (l, k) = k α l β, α, β > 0, α + β < 1. 1) cost minimization min fl,kg wl + rk s.t. q = f (l, k) the Lagrangian L (l, k, λ) = wl + rk λ [q f (l, k)] the rst order conditions are L l = w + λf l = 0, L k = r + λf k = 0 and L λ = q f (l, k) = 0, from which we get r w = f k f l = MRTS l,k = α β l k
An Example (cont d) Example (continued) substituting it into q = k α l β, we get the input demand functions l (q, r, w) = α β βr α+β 1 q α+β αw α+β 1 k (q, r, w) = q α+β αw βr and the cost function c (q; r, w) = wl (q; r, w) + rk (q; r, w). maximization max pq c (q; r, w) q the rst order condition is thus p = c (q; r, w) q (supply function) 2) pro t marginal revenue equals marginal cost.
Producer Surplus In above example, the supply curve is p (q) = MC (q) = c (q; r, w) q (6) Thus, integrating along the supply curve Z q 0 p (q) dq = Z q 0 c (q; r, w) dq = c (q ) q Producer Surplus is thus which is equal to the pro t of the rm. PS = pq c (q ) (7)
Producer Surplus
Introduction the other extreme case is monopoly A monopolist is the only supplier of a certain product in a market It is a price maker, yet subject to the constraint of market demand To set the pro t maximizing price / quantity on the demand curve Why does monopoly exist? Barriers to entry: not possible for other rms to enter a market. patent; government regulations, restricting market entry; Increasing returns to scale (natural monopoly): the more you produce, the lower the average cost the most e cient way of production is monopoly electricity supply (industries with large xed cost)
The Problem of a Monopolist Monopolist s pro t maximization max q p (q) q c (q) (8) the price p is no longer xed, comparing it with (4); p (q) is normally downward sloping: q %) p &. the rst order condition of (8) p (q) + p q q = c q (9) marginal revenue: MR (q) = p (q) + p q q marginal cost: MC (q) = q c MR = MC, exactly the same condition as in competitive case.
The Problem of a Monopolist (2) marginal revenue is lower than demand curve as p/ q < 0; monopoly outcomes (q m, p m ) and competitive outcomes (q e, p e ) q m < q e p m > p e marginal revenue monopoly outcomes
The Problem of a Monopolist (3) Welfare Implications: consumer+producer surplus competitive economy: A + B + C + D monopoly economy: A + B + D welfare loss in a monopoly economy
The Problem of a Monopolist (4) Example A linear example: demand p (q) = a pro t function is thus bq and cost c (q) = cq, a > c, the π (q) = p (q) q c (q) = (a bq) q cq the rst order condition π q = a 2bq c = 0 The monopoly price and output are q m = a c 2b p m = a + c 2
Price Discrimination: Introduction So far, we restrict to the case where the monopolist can just charge ONE price to all the consumers; However, in the case of monopoly, the monopolist can adopt various kinds of pricing strategies as price is now its control variable to explore more consumer surplus and achieve greater pro t We here examine some of the most common cases of price discrimination of a monopolist
1st Degree Price Discrimination 1st degree price discrimination: charge each consumer s willingness to pay for di erent unit of the product ) extract all consumer surplus It s perfect for the monopolist! What are the problems? perfect information about consumers preferences! strong implementation power of the monopolist (charge di erent prices to di units, di people) no re-sale markets.
2nd and 3rd Degree Price Discrimination Let s consider a less information demanding situation 2nd Degree Price Discrimination charge di erent unit prices for di erent quantities but all the consumers are treated the same way 3rd Degree Price Discrimination charge di erent prices to di erent groups of consumers