Microeconomics 3200/4200: Part 1 P. Piacquadio p.g.piacquadio@econ.uio.no August 17, 2017 P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 1 / 16
Outline Market demand 1 Equilibrium 2 P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 2 / 16
Defining the market demand From individual demands to the market demand : I graphically; I Algebraically: X 1 ( p 1 , p 2 , m A , m B ) = D 1 A ( p 1 , p 2 , m A )+ D 1 B ( p 1 , p 2 , m B ) . P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 3 / 16
The inverse demand function The market demand function defines quantity as a function of prices. The inverse demand function P ( X ) defines prices as a function of quantity. It answers the following question: what is the price P ( X ) such that the demand is X ? Example: D A = max [ 20 � p , 0 ] ; D B = max [ 10 � 2 p , 0 ] . P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 4 / 16
The price elasticity The slope of the market demand function is not very informative. The elasticity tells how quickly quantity q react to changes in prices p (or income m ). The price elasticity is: ε ⌘ ∆ q / q ∆ p / p , or ε ⌘ ∆ q / ∆ p . q / p Example: D A = max [ a � bp , 0 ] . P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 5 / 16
Elasticity and revenue Define the revenue R as the product of price p and quantity q : R = pq . Change prices and quantities, i.e. p 0 = p + ∆ p and q 0 = q + ∆ q . Then, R 0 = p 0 q 0 or: R 0 = ( p + ∆ p )( q + ∆ q ) = = pq + q ∆ p + p ∆ q + ∆ p ∆ q Let ∆ R = R 0 � R . For small changes, ∆ p ∆ q ≈ 0. Then ∆ R = q ∆ p + p ∆ q . Dividing by ∆ p (and taking q as a function of p ): ∆ R ∆ p = q ( p )( 1 � | ε ( p ) | ) . P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 6 / 16
Marginal revenue Instead of dividing by ∆ p , we can divide ∆ R by ∆ q : ✓ 1 ◆ ∆ R ∆ q = p ( q ) 1 � . | ε ( q ) | When ∆ q is small, this is the formula of the marginal revenue . What happens if ε ( q ) = � 1? What if ε ( q ) ? � 1? What does it mean? P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 7 / 16
Income elasticity The income elasticity is the ratio between the % change in quantity and the % change in income: ε m ⌘ ∆ q / q ∆ m / m . An inferior good has negative income elasticity; a luxury good has income elasticity larger than one; a normal good... P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 8 / 16
Income elasticity... Income elasticities are interdependent! To show this: p 1 x 1 + p 2 x 2 = m p 1 x 0 1 + p 2 x 0 2 = m 0 . p 1 ∆ x 1 + p 2 ∆ x 2 = ∆ m , divide both sides by m and add x 1 x 1 and x 2 x 2 near each price: p 1 x 1 ∆ x 1 + p 2 x 2 ∆ x 2 = ∆ m m . m x 1 m x 2 Let s 1 ⌘ p 1 x 1 and s 2 ⌘ p 2 x 2 m . Substitute and dividing by ∆ m m gives: m s 1 ε m 1 + s 2 ε m 2 = 1 . P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 9 / 16
Market supply At the beginning of this course, we defined the equilibrium principles as: I Prices adjust and ensure that the “demand” meets its “supply.” We studied and defined the market demand. We instead postpone the market supply. For the time being, we just assume (as we did for the supply of apartments for rent) that there is some market supply curve: S ( P ) , that is the sum of the quantity supplied by each firm (or consumer) for each level of prices. P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 10 / 16
The competitive market A market is competitive if a number of conditions hold: 1 the market is atomistic: buyers and sellers are small so that every agent is price taker; 2 the commodity is homogeneous, privatly appropriable, and infinitely divisible; 3 perfect information, no externalities, no transaction costs. P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 11 / 16
The competitive equilibrium In a competitive market, the competitive equilibrium consists of: I equilibrium price p ⇤ such that D ( p ⇤ ) = S ( p ⇤ ) ; I equilibrium quantities such that each agent satisfies the optimization principle. Graphical examples: I horizontal and vertical supply functions. P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 12 / 16
Example with linear demand and supply Assume D ( p ) = a � bp and S ( p ) = c + dp . What is the equilibrium price p ⇤ ? D ( p ⇤ ) = a � bp ⇤ = c + dp ⇤ = S ( p ⇤ ) , and, solving, bp ⇤ + dp ⇤ = ( b + d ) p ⇤ = a � c , implying that p ⇤ = a � c b + d . The equilibrium quantities demanded and supplied are: b + d = ad + bc D ( p ⇤ ) = S ( p ⇤ ) = a � b a � c b + d . P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 13 / 16
Introducing taxes When adding taxes, the price paid by the buyer might di ff er from the price received by the seller: I a quantity tax is a tax that depends on the units of commodity bought or sold; I a value (ad valorem) tax is a tax that depends on the price of the commodity bought or sold. A quantity tax implies that, at the equilibrium, D ( p ⇤ D ) = S ( p ⇤ B ) and: p D = p S + t . A value tax implies that, at equilibrium, D ( p ⇤ D ) = S ( p ⇤ B ) and: p D = ( 1 + τ ) p S . P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 14 / 16
The e ff ect of taxes What is the e ff ect of taxes on prices and quantities? I it depends. Who pays the cost of the tax? I it depends. The elasticities of demand and supply curve play a crucial role. See graphs! What is the e ff ect on consumers’ surplus? See graphs. P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 15 / 16
Taxation with linear demand and supply Assume D ( p D ) = a � bp D and S ( p S ) = c + dp S . What is the equilibrium price with a quantity tax t ? D ( p ⇤ D ) = a � bp ⇤ D = a � b ( p ⇤ S + t ) = c + dp ⇤ S = S ( p ⇤ S ) , and, solving, bp ⇤ S + dp ⇤ S = ( b + d ) p ⇤ S = a � c � bt , S = a � c � bt D = a � c + dt implying that p ⇤ and, since p ⇤ D = p ⇤ S + t , p ⇤ b + d . b + d To compare with the previous solution write: S = a � c bt bt b + d = p ⇤ � p ⇤ b + d � b + d D = a � c dt dt b + d = p ⇤ + p ⇤ b + d � b + d P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 August 17, 2017 16 / 16
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