#### Transcript 3.1 Ramsey Prices - Illinois State University

Ramsey Pricing Assumptions • Natural monopoly – Cost minimization requires one firm in the market • Strong natural monopoly – MC pricing crease deficit • MC with subsidies is not an option • Barriers to entry • Product demands are unrelated (we relax this assumption later in more complex models) Regulation • Objective is to maximize social welfare subject to the firm breaking even – Per unit charge (constant price) – Prices across markets can vary • First best pricing is not an option • Second best (Ramsey Prices) based on maximizing welfare subject to an acceptable level of profit Ramsey Idea • We can’t maximize Social Welfare • Must adopt a price different from MC • How much should price deviate from MC? – Raise prices until the marginal welfare losses across product markets are balanced. • Prices are raised in inverse proportion to the absolute value of the demand elasticities in each market and this minimized the welfare losses associated with higher prices. – Elastic, small raise – Inelastic, large raise Comparing Elasticities $ p2(q2) p1(q1) p΄ a f e Deadweight loss of market two d p1=p2 g b c o q q1΄ q2΄ q1=q2 The Math Behind Ramsey • Consider a multiproduct firm producing outputs q= (q1, … ,qn ), with cost function given by C(q). Demands for the n goods are represented by the inverse demand function, pi(qi), i=1, …, n. Consumer surplus in the ith market be given by CS i qi o pi ( xi )dxi pi (qi )qi (3.1) for i=1, …, n, and profit for the firm is n (3.2) pi (qi )qi C (q) i 1 • If we maximize a measure of welfare (W), it is the sum of all CSi plus profit, W CS i Max Welfare W CS i First Order Condition gives pi pi W C pi (q i ) qi pi (q i ) qi pi (q i ) 0 qi qi qi qi Implies C pi (q i ) qi Max Welfare s.t. Break even Profit L W ( ) First Order Condition gives pi C L C pi (q i ) ( pi (qi ) qi )0 qi qi qi qi Implies pi C pi (q i ) 1 qi qi q i What is Lambda? Let profit be constrained to 0 , then, using the envelope theorem, we have , or roughly L / W / speaking, if the firm’s profit is decreased by $1, which will mean a $1 deficit, then welfare will increase by $λ. Ramsey Prices • Rearrange (3.4), we obtain p (q ) MC (q)1 q pi(qi ) (3.5) i i i i 1 and then dividing both sides by pi (qi ) and yields (3.6) where pi (qi ) MCi (q) 1 pi (qi ) 1 ei ei pi (qi ) /qi pi(qi ) 0 , the elasticity of demand in market i . The price in (3.6) is called the Ramsey price in market i. Because (3.6) is true for all i, the formula states that the percentage deviation of price from marginal cost in the ith market should be inversely proportional to the absolute value of demand elasticity in the ith market. Ramsey Pricing and the Lerner Index • The Lerner Index is equivalent to the negative inverse of the price elasticity of demand facing the firm, when the price chosen is that which maximizes profits available because of the existence of market power. Results and Conclusions • From (3.6), and for all markets, the percentage deviation of price from marginal cost, times the price elasticity, sometimes called the Ramsey number, should be equal to –λ/(1+λ). • If λ is very small, the Ramsey number approaches zero, implying that prices will be very close to marginal cost. – The deficit under efficient pricing is small: – Reducing profit by $1 and recalculating prices will not increase welfare • If λ is large, the Ramsey number is close to 1; – such a situation occurs when an unconstrained monopolist barely breaks even – The price-cost deviations are so substantial that large welfare improvements are possible if a subsidy can be transferred to the firm to compensate for negative profit. More Results • Relatively inelastic demands mean smaller values of |ei| and higher percentage price deviations from marginal cost. • Relatively elastic markets, the percentage price deviation will be smaller, • Thus, for the ith and jth markets pi (qi ) MCi (q)/ pi (qi ) (3.6a) p (q ) MC (q)/ p (q ) j j j j j ej ei • Given initial Ramsey prices, if |ei| were to increase, ceteris paribus, pi and Mci would have to be brought closer together and /or pj and MCj be moved further apart for the Ramsey conditions to hold. Another Interpretation/Result • By adding pi qi pi MCi to both sides for markets i and j to obtain (3.7) p j (q j ) MC j (q) pi (qi ) MCi (q) MCi (q) MRi (q) MC j (q) MR j (q) Where MRi(q) is the marginal revenue in the ith market. By (3.7), if we were to increase output by Δqi and Δqj from the profit-maximizing solution, – the numerators would be the marginal gain from additional consumption, – the denominators the marginal loss in profit. – this marginal-benefit/marginal-loss ratio must be equal across markets at the point where profit is down to zero. Comparing Output Market Elasticities and Ramsey Pricing $ p2(q2) p1(q1) p΄ a f e Deadweight loss of market two d p1=p2 g b c o q q1΄ q2΄ q1=q2 Comparing Output Market Elasticities and Ramsey Pricing • Initially, prices and quantities demanded for both goods are equal at point d. • If prices must be increased to eliminate a deficit, which market should bear more of the burden in terms of a higher price? • If prices was raised to p΄, there would be a deadweight loss of adb in the market one and a loss of ecd in market two. The former is larger than the latter, indicating that price should increase more in market two than in market one. This is also called “what the traffic can bear” or “value-of-service pricing”.