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Deadweight Loss Problem Assume a straight-line, downward sloping inverse demand curve: p = 100 – q. Marginal Cost = 20. I What is the allocatively efficient price? 2. What is the profit maximizing price? Solution 1. The ALLOCATIVELY EFFICIENT price is always where price = marginal cost. That is the point at which all profitable transactions have taken place. (Pareto Optimal) P = MC means that Price = 20. 2. There are two steps to finding the profit maximizing price. STEP 1: Determine the Marginal Revenue (MR) Function STEP 2: Equate MR to MC Step 1: Determine the Marginal Revenue Function. 1a. First, determine the revenue function (See 1a above). Revenue = price x quantity. Price = 100 – q. Therefore, 3. What is the efficiency loss (deadweight loss) that results from charging the higher profit-maximizing price? Revenue = p x q = (100 – q) x q = 100q – q2 1b. Take the derivative to determine MARGINAL revenue. Factor Rule d/dx Cx = C. Therefore d/dq (100q) = 100 Power Rule d/dx xn = nxn-1 Therefore d/dq( q2) = 2q. Marginal Revenue (MR) = 100 – 2q Step 2: Use the Profit Maximizing Rule, MR = MC MC = 20 = 100 – 2q. Therefore q = 40. Calculating the profit maximizing price by using the demand equation: p = 100 – q = 100 – 40 = 60, the profit maximizing price 3. To calculate the efficiency (or welfare) loss due to pricing at the profit maximizing level, draw diagram representing the consumer surplus when the P = MC and when the P is set profit maximizing point of MR = MC. Use the values for P and Q at both the allocatively efficient level and profit maximizing level. Then calculate the area of lost due to profit maximizing pricing. That’s the deadweight loss (DWL) Allocatively Efficient Level: P = 20, Q = 80 Profit Maximizing Level = P = 60, = 40 Price Deadweight Loss is defined by the Yellow Triangle 60 40 DWL D At MC = P; Q = 80 and P = 20 At MC = MR; Q = 40 and P = 60 20 40 0 40 DWL = the triangle Base = 40, Height = 40 Area = 40 x 40 x ½ = 800. 80 Quantity Scale Economies Problem A firm has a cost curve, C = 50 + 2q + 1/2q2. Solution 1. Average Cost is simply Total Costs divided by Quantity (q). Therefore: 1. Write the equation for Average Costs (AC): C/q = 50/q + 2q/q + 1/2q2/q = 50/q + 2 + 1/2q 2. What are the AC values for q = 4, 8, 10 and 12 respectively? 3. Take the derivative of C and determine the marginal cost curve. Use that function to estimate the value of MC at points q = 4, 8, 10 and 12 respectively. 2. Plug in the q values into the AC formula to calculate the various AC values. q = 4; 50/4 + 2 + ½(4) = 16.5 q = 8; 50/8 + 2 + ½(8) = 12.25 q = 10; 50/10 + 2 + ½(10) = 12 q = 12; 50/12 + 2 + ½(12) = 12.17 3. C = 50 + 2q + 1/2q2 dC/dq = d/dq (50) + d/dq(2q) + d/dq(1/2q2 ) 4. With the AC and MC values in 2 and 3, compute the index of scale economies, S. At what values of q are economies of scale present? What about diseconomies of scale? And at what value are constant returns to scale exhibited? Therefore the estimated values of MC is dC/dq = 0 + 2 + 2*(1/2q1) = 2 + q q = 4; 2 + 4 = 6 q = 8; 2 + 8 = 10 q = 10; 2 + 10 = 12 q = 12; 2 + 12 = 14 4. The Index of Scale Economies = AC/MC. If the Index > 1, scale economies exist. If it is <1, then diseconomies of scale exist. If it = 1, then there are constant returns to scale present. When the Index > 1, Average Cost is greater than Marginal Cost, which means at that point, for each additional unit produced average cost is falling. This is technically what is meant by scale economies: average cost is declining as more quantity is being produced. When the Index is < 1, the MC is greater than AC. At this point, for each addition unit produced, average cost is rising. This is the technical definition of diseconomies of scale. q = 4; AC/MC = 16.5/6 = 2.75 > 1. Therefore scale economies are present. q = 8; AC/MC = 12.25/10 = 1.225 > 1. Therefore scale economies are present. q = 10; AC/MC = 12/12 = 1. Therefore constant returns to scale are present. q = 12; AC/MC = 12.17/14 = .869 < 1. Therefore diseconomies of scale are present.