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Market Demand Molly W. Dahl Georgetown University Econ 101 – Spring 2009 1 From Individual to Market Demand Functions Consumer i’s demand for commodity j x*ji (p1 , p2 , mi ) Market demand for commodity j n *i X j (p1 , p2 , m ,, m ) x j (p1 , p2 , mi ). i 1 1 n 2 From Individual to Market Demand Functions The market demand curve is the “horizontal sum” of the individual consumers’ demand curves. Suppose there are only two consumers, i = A,B. 3 From Individual to Market Demand Functions p1 p1 p1’ p1” p1’ p1” 20 x*A 1 15 *B x1 4 From Individual to Market Demand Functions p1 p1 p1’ p1” p1 p1’ p1” 20 x*A 1 15 *B x1 p1’ x*1A xB 1 5 From Individual to Market Demand Functions p1 p1 p1’ p1” p1 p1’ p1” 20 x*A 1 15 *B x1 p1’ p1” x*1A xB 1 6 From Individual to Market Demand Functions p1 p1 p1’ p1” p1 p1’ p1” 20 x*A 1 15 *B x1 The “horizontal sum” of the demand curves of individuals A and B. p1’ p1” 35 x*1A xB 1 7 Elasticities Elasticity measures the “sensitivity” of one variable with respect to another. The elasticity of variable X with respect to variable Y is % x x,y . % y 8 Own-Price Elasticity What is the own-price elasticity pi of demand in a very small interval of prices centered on pi’? * pi ' dXi X* ,p pi’ i i Xi ' dpi is the elasticity at the point ( Xi ', pi ' ). Xi ' Xi* 9 Own-Price Elasticity dX*i X* ,p * i i dpi Xi pi Consider a linear demand curve. If pi = a – bXi then Xi = (a-pi)/b and * dXi 1 . dpi b Therefore, pi 1 pi X* ,p . i i ( a pi ) / b b a pi 10 Own-Price Elasticity pi pi = a - bXi* pi X* ,p i i a pi a a/b Xi* 11 Own-Price Elasticity pi a pi X* ,p i i a pi pi = a - bXi* p 0 0 0 a/b Xi* 12 Own-Price Elasticity pi a a/2 pi X* ,p i i a pi pi = a - bXi* a a/2 p 1 2 aa/2 1 0 a/2b a/b Xi* 13 Own-Price Elasticity pi = a - bXi* pi a a/2 pi X* ,p i i a pi a pa aa 1 0 a/2b a/b Xi* 14 Own-Price Elasticity pi pi X* ,p i i a pi pi = a - bXi* a own-price elastic a/2 1 own-price unit elastic own-price inelastic 0 a/2b a/b Xi* 15 Revenue, Price Changes, and Own-Price Elasticity of Demand Inelastic Demand: Raising a commodity’s price causes a small decrease in quantity demanded Sellers’ revenues rise as price rises. Elastic Demand: Raising a commodity’s price causes a large decrease in quantity demanded Seller’s revenues fall as price rises. 16 Revenue, Price Changes, and Own-Price Elasticity of Demand * Sellers’ revenue is R(p) p X (p). 17 Revenue, Price Changes, and Own-Price Elasticity of Demand * Sellers’ revenue is R(p) p X (p). * dR dX So X* (p) p dp dp 18 Revenue, Price Changes, and Own-Price Elasticity of Demand * Sellers’ revenue is R(p) p X (p). * dR dX So X* (p) p dp dp * p dX * X (p )1 * X (p ) dp 19 Revenue, Price Changes, and Own-Price Elasticity of Demand * Sellers’ revenue is R(p) p X (p). * dR dX So X* (p) p dp dp * p dX * X (p )1 * X (p ) dp X* (p)1 . 20 Revenue, Price Changes, and Own-Price Elasticity of Demand dR X* (p)1 dp dR 0 so if 1 then dp and a change to price does not alter sellers’ revenue. 21 Revenue, Price Changes, and Own-Price Elasticity of Demand dR X* (p)1 dp dR 0 but if 1 0 then dp and a price increase raises sellers’ revenue. 22 Revenue, Price Changes, and Own-Price Elasticity of Demand dR X* (p)1 dp dR 0 And if 1 then dp and a price increase reduces sellers’ revenue. 23 Revenue, Price Changes, and Own-Price Elasticity of Demand In summary: Own-price inelastic demand: 1 0 price rise causes rise in sellers’ revenue. Own-price unit elastic demand: 1 price rise causes no change in sellers’ revenue. Own-price elastic demand: 1 price rise causes fall in sellers’ revenue. 24 Marginal Revenue, Quantity Changes, and Own-Price Elasticity of Demand A seller’s marginal revenue is the rate at which revenue changes with the number of units sold by the seller. dR( q) MR( q) . dq 25 Marginal Revenue, Quantity Changes, and Own-Price Elasticity of Demand p(q) denotes the seller’s inverse demand function (i.e., the price at which the seller can sell q units). Then R( q) p( q) q so dR( q) dp( q) MR( q) q p( q) dq dq q dp( q) p( q) 1 . p( q) dq 26 Marginal Revenue, Quantity Changes, and Own-Price Elasticity of Demand q dp( q) MR( q) p( q) 1 . p( q) dq and so dq p dp q 1 MR( q) p( q) 1 . 27 Marginal Revenue, Quantity Changes, and Own-Price Elasticity of Demand 1 MR( q) p( q) 1 says that the rate at which a seller’s revenue changes with the number of units it sells depends on the sensitivity of quantity demanded to price; i.e., upon the of the own-price elasticity of demand. 28 Marginal Revenue, Quantity Changes, and Own-Price Elasticity of Demand 1 MR(q) p(q)1 If 1 then MR( q) 0. If 1 0 then MR( q) 0. If 1 then MR( q) 0. 29 Marginal Revenue, Quantity Changes, and Own-Price Elasticity of Demand If 1 then MR( q) 0. Selling one more unit does not change the seller’s revenue. If 1 0 then MR( q) 0. Selling one more unit reduces the seller’s revenue. If 1 then MR( q) 0. Selling one more unit raises the seller’s revenue. 30 Marginal Revenue, Quantity Changes, and Own-Price Elasticity of Demand An example with linear inverse demand. p( q) a bq. Then R( q) p( q)q ( a bq)q and MR( q) a 2bq. 31 Marginal Revenue and Own-Price Elasticity of Demand p a p( q) a bq a/2b a/b q MR( q) a 2bq 32