50 PART • Introduction: Markets and Prices Our problem is to choose numbers for the constants a, b, c, and d This is done, for supply and for demand, in a two-step procedure: ț Step 1: Recall that each price elasticity, whether of supply or demand, can be written as E = (P/Q)(⌬Q/⌬P) where ⌬Q/⌬P is the change in quantity demanded or supplied resulting from a small change in price For linear curves, ⌬Q/⌬P is constant From equations (2.5a) and (2.5b), we see that ⌬Q/⌬P = d for supply and ⌬Q/⌬P = -b for demand Now, let’s substitute these values for ⌬Q/⌬P into the elasticity formula: Demand: ED = -b(P*/Q*) Supply: ES = d(P*/Q*) (2.6a) (2.6b) where P* and Q* are the equilibrium price and quantity for which we have data and to which we want to fit the curves Because we have numbers for ES, ED, P*, and Q*, we can substitute these numbers in equations (2.6a) and (2.6b) and solve for b and d ț Step 2: Since we now know b and d, we can substitute these numbers, as well as P* and Q*, into equations (2.5a) and (2.5b) and solve for the remaining constants a and c For example, we can rewrite equation (2.5a) as a = Q* + bP* and then use our data for Q* and P*, together with the number we calculated in Step for b, to obtain a Let’s apply this procedure to a specific example: long–run supply and demand for the world copper market The relevant numbers for this market are as follows: Quantity Q* = 18 million metric tons per year (mmt/yr) Price P* = $3.00 per pound Elasticity of suppy ES = 1.5 Elasticity of demand ED = - 0.5 (The price of copper has fluctuated during the past few decades between $0.60 and more than $4.00, but $3.00 is a reasonable average price for 2008–2011) We begin with the supply curve equation (2.5b) and use our two-step procedure to calculate numbers for c and d The long-run price elasticity of supply is 1.5, P* = $3.00, and Q* = 18 ț Step 1: Substitute these numbers in equation (2.6b) to determine d: 1.5 = d(3/18) = d/6 so that d = (1.5)(6) = ț Step 2: Substitute this number for d, together with the numbers for P* and Q*, into equation (2.5b) to determine c: 18 = c + (9)(3.00) = c + 27