16 Equilibrium

From Individual to Market Demand Functions The market demand curve is the “horizontal sum” of the individual consumers’ demand curves. E.g. suppose there are only two consumers; i = A,B.

From Individual to Market Demand Functions p1 p1 p1’ p1’ p1” p1” 20 15 p1 p1’ The “horizontal sum” of the demand curves of individuals A and B. p1” 35

Elasticities Elasticity measures the “sensitivity” of one variable with respect to another. The elasticity of variable X with respect to variable Y is

Arc and Point Elasticities Elasticity computed for a single value of pi is a point elasticity:

Marginal Revenue 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 so

Point Own-Price Elasticity pi a own-price elastic a/2 (own-price unit elastic) own-price inelastic a/2b a/b Xi* Revenue increases

Marginal Revenue and Own-Price Elasticity of Demand $ a/2b a/b q R(q) q a/2b a/b

Market Equilibrium A market is in equilibrium when total quantity demanded by buyers equals total quantity supplied by sellers. D(p*) = S(p*) or D-1(q*) = S-1(q*)

Market Equilibrium Market demand Market supply p q=S(p) D(p*) = S(p*); the market is in equilibrium. p* q=D(p) q* D(p), S(p)

Market Equilibrium Market price must fall towards p*. Market demand Market supply p q=S(p) D(p’) < S(p’); an excess of quantity supplied over quantity demanded. p’ p* q=D(p) D(p’) S(p’) D(p), S(p) Market price must fall towards p*.

Market Equilibrium Market price must rise towards p*. Market demand Market supply p q=S(p) D(p”) > S(p”); an excess of quantity demanded over quantity supplied. p* p” q=D(p) S(p”) D(p”) D(p), S(p) Market price must rise towards p*.

Market Equilibrium An example of calculating a market equilibrium when the market demand and supply curves are linear:

Market Equilibrium What are the values of p* and q*? p Market supply S(p) = c+dp p* Market demand D(p) = a-bp q* D(p), S(p) What are the values of p* and q*?

Market Equilibrium At the equilibrium price p*, D(p*) = S(p*). That is,

Market Equilibrium At the equilibrium price p*, D(p*) = S(p*). That is, which gives and

Market Equilibrium p Market supply S(p) = c+dp Market demand D(p) = a-bp D(p), S(p)

Market Equilibrium Can we calculate the market equilibrium using the inverse market demand and supply curves? Yes, it is the same calculation.

Market Equilibrium the equation of the inverse market demand curve. the equation of the inverse market supply curve.

Market Equilibrium At equilibrium, D-1(q*) = S-1(q*). D-1(q), S-1(q) Market inverse supply S-1(q) = (-c+q)/d p* Market inverse demand D-1(q) = (a-q)/b q* q At equilibrium, D-1(q*) = S-1(q*).

Market Equilibrium and At the equilibrium quantity q*, D-1(p*) = S-1(p*). That is, which gives and

Market Equilibrium D-1(q), S-1(q) S-1(q) = (-c+q)/d D-1(q) = (a-q)/b q

Market Equilibrium Two special cases: quantity supplied is fixed, independent of the market price, and quantity supplied is extremely sensitive to the market price.

Market Equilibrium Market quantity supplied is fixed, independent of price. p S(p) = c+dp, so d=0 and S(p) º c. q* = c q

Market Equilibrium Market demand Market quantity supplied is fixed, independent of price. p S(p) = c+dp, so d=0 and S(p) º c. D-1(q) = (a-q)/b q* = c q

Market Equilibrium with d = 0 give Market demand Market quantity supplied is fixed, independent of price. p S(p) = c+dp, so d=0 and S(p) º c. p* = (a-c)/b p* = D-1(q*); that is, p* = (a-c)/b. D-1(q) = (a-q)/b q* = c q with d = 0 give

Market Equilibrium Market quantity supplied is extremely sensitive to price. p S-1(q) = p*. p* q

Market Equilibrium Market quantity supplied is extremely sensitive to price. p Market demand S-1(q) = p*. p* D-1(q) = (a-q)/b q

Market Equilibrium Market demand Market quantity supplied is extremely sensitive to price. p S-1(q) = p*. p* = D-1(q*) = (a-q*)/b so q* = a-bp* p* D-1(q) = (a-q)/b q* = a-bp* q

Quantity Taxes A quantity tax levied at a rate of $t is a tax of $t paid on each unit traded. If the tax is levied on sellers then it is an excise tax. If the tax is levied on buyers then it is a sales tax.

Quantity Taxes What is the effect of a quantity tax on a market’s equilibrium? How are prices affected? How is the quantity traded affected? Who pays the tax? How are gains-to-trade altered?

Quantity Taxes A tax rate t makes the price paid by buyers, pb, higher by t from the price received by sellers, ps: When a tax is presented in a market, there are 2 prices of interest: price the demander pays and price the supplier gets. These prices differ by the amount of tax:

Quantity Taxes Even with a tax the market must clear. I.e. quantity demanded by buyers at price pb must equal quantity supplied by sellers at price ps.

Quantity Taxes and describe the market’s equilibrium. Notice that these two conditions apply no matter if the tax is levied on sellers or on buyers. Hence, a sales tax rate $t has the same effect as an excise tax rate $t.

Quantity Taxes & Market Equilibrium p Market demand Market supply p* No tax q* D(p), S(p)

Quantity Taxes & Market Equilibrium Market demand Market supply p An excise tax raises the market supply curve by $t $t p* q* D(p), S(p)

Quantity Taxes & Market Equilibrium Market demand Market supply p An excise tax raises the market supply curve by $t, raises the buyers’ price and lowers the quantity traded. pb $t p* ps qt q* D(p), S(p) And sellers receive only ps = pb - t.

Quantity Taxes & Market Equilibrium p Market demand Market supply p* No tax q* D(p), S(p)

Quantity Taxes & Market Equilibrium p Market demand Market supply An sales tax lowers the market demand curve by $t p* $t q* D(p), S(p)

Quantity Taxes & Market Equilibrium Market demand Market supply p An sales tax lowers the market demand curve by $t, lowers the sellers’ price and reduces the quantity traded. pb pb pb p* ps $t qt q* D(p), S(p) And buyers pay pb = ps + t.

Quantity Taxes & Market Equilibrium Market demand Market supply p A sales tax levied at rate $t has the same effects on the market’s equilibrium as does an excise tax levied at rate $t. pb pb pb $t p* ps $t qt q* D(p), S(p)

Quantity Taxes & Market Equilibrium Who pays the tax of $t per unit traded? The division of the $t between buyers and sellers is the incidence of the tax.

Quantity Taxes & Market Equilibrium p Market demand Market supply pb pb pb p* ps qt q* D(p), S(p)

Quantity Taxes & Market Equilibrium Market demand p Market supply Tax paid by buyers pb pb pb p* Tax paid by sellers ps qt q* D(p), S(p)

Quantity Taxes & Market Equilibrium E.g. suppose the market demand and supply curves are linear.

Quantity Taxes & Market Equilibrium and With the tax, the market equilibrium satisfies and so and Substituting for pb gives

Quantity Taxes & Market Equilibrium and give With tax: -The price paid by the demander increases and the price received by the supplier decreases. -The amount of the price change depends on the slope of the demand and supply curves.

Quantity Taxes & Market Equilibrium The tax paid per unit by the buyer is The tax paid per unit by the seller is

Tax Incidence and Own-Price Elasticities The incidence of a quantity tax depends upon the own-price elasticities of demand and supply.

Tax Incidence and Own-Price Elasticities Market demand Market supply Change to buyers’ price is pb - p*. Change to quantity demanded is Dq. pb $t p* ps qt q* D(p), S(p) Dq

Tax Incidence and Own-Price Elasticities Around p = p* the own-price elasticity of demand is approximately

Tax Incidence and Own-Price Elasticities Market demand Market supply Change to sellers’ price is ps - p*. Change to quantity demanded is Dq. pb $t p* ps qt q* D(p), S(p) Dq

Tax Incidence and Own-Price Elasticities Around p = p* the own-price elasticity of supply is approximately

Tax Incidence and Own-Price Elasticities Market demand p Market supply Tax paid by buyers pb pb pb p* Tax paid by sellers ps qt q* D(p), S(p) Tax incidence =

Tax Incidence and Own-Price Elasticities So

Tax Incidence and Own-Price Elasticities Tax incidence is The fraction of a $t quantity tax paid by buyers rises as supply becomes more own-price elastic or as demand becomes less own-price elastic.

Tax Incidence and Own-Price Elasticities Market demand Market supply p As market demand becomes less own- price elastic, tax incidence shifts more to the buyers. pb $t p* ps qt q* D(p), S(p)

Tax Incidence and Own-Price Elasticities Market demand Market supply p As market demand becomes less own- price elastic, tax incidence shifts more to the buyers. pb $t p* ps qt q* D(p), S(p)

Tax Incidence and Own-Price Elasticities Market demand Market supply p As market demand becomes less own- price elastic, tax incidence shifts more to the buyers. pb $t ps= p* qt = q* D(p), S(p) When eD = 0, buyers pay the entire tax, even though it is levied on the sellers.

Tax Incidence and Own-Price Elasticities Tax incidence is Similarly, the fraction of a $t quantity tax paid by sellers rises as supply becomes less own-price elastic or as demand becomes more own-price elastic.

Tax Incidence and Own-Price Elasticities

Deadweight Loss and Own-Price Elasticities A quantity tax imposed on a competitive market reduces the quantity traded and so reduces gains-to-trade (i.e. the sum of Consumers’ and Producers’ Surpluses). The lost total surplus is the tax’s deadweight loss, or excess burden.

Deadweight Loss and Own-Price Elasticities Market demand Market supply CS p* No tax q* D(p), S(p) Market demand: measures the amount of good that will be demanded at each price  Consumers’ surplus is the area between demand curve and market price.

Deadweight Loss and Own-Price Elasticities Market demand Market supply No tax p* PS q* D(p), S(p) Market supply: measures the amount of good that will be supplied at each price  Producers’ surplus is the area between market price and supply curve.

Deadweight Loss and Own-Price Elasticities Market demand Market supply CS p* No tax PS q* D(p), S(p)

Deadweight Loss and Own-Price Elasticities Market demand Market supply pb CS $t Tax reduces both CS and PS. p* ps PS qt q* D(p), S(p)

Deadweight Loss and Own-Price Elasticities Market demand Market supply p Tax reduces both CS and PS. Tax transfers surplus to government (i.e. the government gains revenue from tax). pb CS $t p* Tax ps PS qt q* D(p), S(p)

Deadweight Loss and Own-Price Elasticities Market demand Market supply p Tax reduces both CS and PS. Tax transfers surplus to government (i.e. the government gains revenue from tax). pb CS $t p* Tax ps PS qt q* D(p), S(p) Finally, tax causes excess burden and lowers total surplus.

Deadweight Loss and Own-Price Elasticities Market demand p Market supply pb $t p* ps Deadweight loss qt q* D(p), S(p) Deadweight loss (excess burden) of the tax: the lost value to the consumers and the producers due to the reduction in the sales of the good. (You can’t tax what isn’t there.)

Equilibrium QUESTION 1. Pareto efficiency Meaning? Is the situation where 𝒒 𝒕 units is sold a Pareto efficient outcome?

Equilibrium Answer 1. Pareto efficiency “is a state of allocation of resources from which it is impossible to reallocate so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off.”