4/14/2016. Intermediate Microeconomics W3211. Lecture 18: Equilibrium with Firms 2. Today. The Story So Far. Quantity Taxes.

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1 1 Intermediate Microeconomics W3211 Lecture 18: Equilibrium with Firms 2 Introduction Columbia University, Sring 2016 Mark Dean: mark.dean@columbia.edu 2 The Story So Far. 3 Today 4 Last lecture we talked about equilibrium with firms and consumers Distinguished between artial equilibrium and general equilibrium Showed how to find a artial equilibrium Showed how suly and grahs could be used for olicy analysis Do a artial equilibrium analysis of tax olicy Show how we can solve for a (simle) general equilibrium with firms Quantity Taxes 6 One articularly imortant tye of olicy we might want to examine is the effect of taxes A quantity tax levied at a rate of is a tax of aid on each unit traded. Partial Equilibrium The Effect of Quantity Taxes 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. 5 1

2 Quantity Taxes 7 Quantity Taxes 8 Questions we might want to ask What is the effect of a quantity tax on a market s equilibrium? How are rices affected? How is the quantity traded affected? Who ays the tax? How are gains-to-trade altered? A tax rate t makes the rice aid by buyers, b, higher by t from the rice received by sellers, s. If there is a excise tax If there is a sales tax In both cases Quantity Taxes 9 Quantity Taxes 10 Even with a tax the market must clear. I.e. quantity ed by buyers at rice b must equal quantity sulied by sellers at rice s. b s t and D ( b) S ( s) describe the market s equilibrium. Notice that these two conditions aly no matter if the tax is levied on sellers or on buyers. D ( b) S ( s) Quantity Taxes 11 Quantity Taxes 12 b s t and D ( b) S ( s) describe the market s equilibrium. Notice that these two conditions aly no matter if the tax is levied on sellers or on buyers. b s t and D ( b) S ( s) describe the market s equilibrium. Notice that these two conditions aly no matter if the tax is levied on sellers or on buyers. This is an Imortant Point! This is an Imortant Point! The equilibrium conditions are not affected by who faces the tax 2

3 Quantity Taxes & Equilibrium suly No tax 13 Quantity Taxes & Equilibrium suly 14 An excise tax raises the market suly curve by D(), S() D(), S() Quantity Taxes & Equilibrium b suly 15 An excise tax raises the market suly curve by, raises the buyers rice and lowers the quantity traded. Quantity Taxes & Equilibrium b s suly 16 An excise tax raises the market suly curve by, raises the buyers rice and lowers the quantity traded. D(), S() D(), S() And sellers receive only s = b -t. Quantity Taxes & Equilibrium suly No tax 17 Quantity Taxes & Equilibrium suly 18 An sales tax lowers the market curve by D(), S() D(), S() 3

4 Quantity Taxes & Equilibrium s suly 19 An sales tax lowers the market curve by, lowers the sellers rice and reduces the quantity traded. Quantity Taxes & Equilibrium b s suly 20 An sales tax lowers the market curve by, lowers the sellers rice and reduces the quantity traded. D(), S() D(), S() And buyers ay b = s + t. Quantity Taxes & Equilibrium b s suly A sales tax levied at rate has the same effects on the market s equilibrium as does an excise tax levied at rate. D(), S() 21 Quantity Taxes & Equilibrium Who ays the tax of er unit traded? The imortant thing is not who hysically hands over the tax, but who is affected by the tax i.e. for the buyer, the difference between the buyers rice and the equilibrium rice with no tax for the seller, the difference between the sellers rice and the equilibrium rice with no tax The division of the between buyers and sellers is the incidence of the tax. 22 Quantity Taxes & Equilibrium b s suly 23 Quantity Taxes & Equilibrium b s suly Tax aid by buyers 24 D(), S() D(), S() 4

5 Quantity Taxes & Equilibrium b s suly Tax aid by sellers 25 Quantity Taxes & Equilibrium b s suly Tax aid by buyers Tax aid by sellers 26 D(), S() D(), S() The incidence of a quantity tax deends uon the own-rice elasticities of and suly. Remember, the elasticity of a good is roortional change in quantity over roortional change in rice We can also define the suly elasticity in the same way b s suly D(), S() b s suly 29 Change to buyers rice is b -. Change to quantity ed is q. Around = the own-rice elasticity of is aroximately q q D b 30 D(), S() q 5

6 Around = the own-rice elasticity of is aroximately q q D b q b D q. 31 b s suly 32 D(), S() b s suly Change to sellers rice is s -. Change to quantity ed is q. 33 Around = the own-rice elasticity of suly is aroximately q q S s 34 q D(), S() Around = the own-rice elasticity of suly is aroximately q q S s q s S q. 35 We will define the tax incidence as the ratio of the rice change for the buyer to the rice change for the seller Note, because rices rise for buyers and fall for sellers, this way of doing things will ensure that the number is ositive A high number means that the incidence falls on the buyer A low number means the incidence falls on the seller 36 6

7 b Tax incidence =. s q b D q. q s S q. So b S. s D 37 b Tax incidence is S. s D The fraction of a quantity tax aid by buyers rises as suly becomes more own-rice elastic or as becomes less own-rice elastic. 38 b s suly As market becomes less ownrice elastic, tax incidence shifts more to the buyers. D(), S() 39 b s suly As market becomes less ownrice elastic, tax incidence shifts more to the buyers. D(), S() 40 s = b suly 41 As market becomes less ownrice elastic, tax incidence shifts more to the buyers. b s = suly 42 As market becomes less ownrice elastic, tax incidence shifts more to the buyers. = D(), S() = D(), S() When D = 0, buyers ay the entire tax, even if it is levied on the sellers. 7

8 This makes sense Think of a good that is very rice inelastic E.g. cigarettes Peole who are addicted will buy cigarettes whatever the rice If there is a tax on cigarettes what haens? Peole buy the same amount of cigarettes and just ay more 43 b Tax incidence is S. s D Similarly, the fraction of a quantity tax aid by sellers rises as suly becomes less own-rice elastic or as becomes more own-rice elastic. 44 This is one of the reasons that tax of cigarettes is so high A quantity tax imosed on a cometitive market reduces the quantity traded and so reduces gains-to-trade (i.e. the sum of Consumers and Producers Surluses). The lost total surlus is the tax s deadweight loss, or excess burden. 45 Deadweight Loss and suly No tax 46 D(), S() Deadweight Loss and suly No tax 47 Deadweight Loss and b s suly The tax reduces both and 48 D(), S() D(), S() 8

9 b s Tax suly The tax reduces both and, transfers surlus to government 49 b s Tax suly The tax reduces both and, transfers surlus to government 50 D(), S() D(), S() b s Tax suly The tax reduces both and, transfers surlus to government 51 b s Tax suly The tax reduces both and, transfers surlus to government, and lowers total surlus. 52 D(), S() D(), S() suly 53 suly 54 b s Tax Deadweight loss b s Deadweight loss D(), S() D(), S() 9

10 b s suly 55 Deadweight loss falls as market becomes less ownrice elastic. b s suly 56 Deadweight loss falls as market becomes less ownrice elastic. D(), S() D(), S() b s = suly 57 Deadweight loss falls as market becomes less ownrice elastic. Deadweight loss due to a quantity tax rises as either market or market suly becomes more own-rice elastic. If either D = 0 or S = 0 then the deadweight loss is zero. This is another reason why there is such a high tax on cigarettes! 58 = D(), S() When D = 0, the tax causes no deadweight loss. General Equilibrium with Firms 60 As we discussed last lecture, artial equilibrium is art of the story General Equilibrium with Firms When we look at markets one at a time, we miss out on the fact that different markets effect each other In order to rectify this error, we need to switch to general equilibrium Here we solve for all the markets in an economy simultaneously 59 10

11 General Equilibrium with Firms 61 The Robinson Crusoe Economy 62 Why might we be interested in general equilibrium? In order to demonstrate this I am going to tell you a story Well, when there were only consumers, we roved the first and second fundamental theorem of welfare economics (Under certain conditions) Any equilibrium is areto efficient and any areto efficient allocation could be suorted as an equilibrium Does this result survive if we have firms? The story of Robinson Crusoe As you robably know, Robinson Crusoe was stranded on a desert island Had to subsist only on coconuts until he was rescued The answer is generally yes With the same caveats we had with the consumer re externalities, rice taking, equity etc The Robinson Crusoe Economy 63 The Robinson Crusoe Economy 64 As is less well known (excet to economists), he went a bit bonkers while on the desert island In articular he develoed a articularly handy form of multile ersonality disorder Secifically, he slit himself into two economic units RC, the consumer RC Inc., the firm RC Inc acts as the firm in the economy Hires labor from RC at the wage rate w Uses the labor to roduce coconuts, which are sold at the market rice (normalized to 1) Does so to maximize rofits RC acts as the consumer in the market Buys coconuts and sells labor to maximize utility Decides how much to work at wage w Uses that money to buy coconuts at the market rice 1 Also owns the firm, so collects any rofits the firm makes Consumer and Firm Problems 65 Why Are We Doing This? 66 RC s Inc s roblem 1. CHOOSE coconuts and labor 2. IN ORDER TO MAXIMIZE rofits, 3. SUBJECT TO technology constraint RC s consumer roblem 1. CHOOSE coconuts and labor 2. IN ORDER TO MAXIMIZE utility, This (admittedly a little artificial) set u allows us to study general equilibrium with firms and consumers How many markets are there in this economy? Two for coconuts for labor We can figure out equilibrium in both markets at the same time Key question: Is equilibrium areto efficient? 3. SUBJECT TO the budget constraint 11

12 Pareto Efficiency 67 Pareto Efficiency 68 What is areto efficiency here? How can we solve the lanner s roblem? Well, there is only one erson (Robinson) Draw some ictures! So we want to make him as well off as ossible To do this we can solve the lanner s roblem 1. CHOOSE coconuts and labor 2. IN ORDER TO MAXIMIZE utility, 3. SUBJECT TO technology constraint Notice this is what we would get if we merged RC and RC Inc Robinson Crusoe Inc s Production Function 69 Robinson Crusoe s Preferences 70 coconuts coconuts Higher Utility labor labor Pareto Otimality for Robinson 71 Pareto Efficiency 72 coconuts c 1 Pareto efficiency occurs at the oint at which the sloe of the roduction function is equal to the sloe of the indifference curve What does this mean? l 1 labor Sloe of roduction function Marginal Product of Labor Rate at which one additional unit of labor creates one additional coconut Sloe of the indifference cure Marginal Rate of Substitution The number of coconuts that the consumer has to be given to make them indifferent about sulying one more unit of labor 12

13 Pareto Efficiency 73 Equilibrium 74 This makes sense! If the MRS is lower that the marginal roduct of labor, can make RC better off by working harder and having more coconuts If MRS is higher than the marginal roduct of labor, can make RC better off by working less hard and having less coconuts We want to know whether equilibrium in this economy is areto efficient In order to figure this out, we need to know what an equilibrium is! At the otimum the two have to be equal Equilibrium 75 Equilibrium 76 An Equilibrium with consumer and firms is Let s draw some more ictures A set of rices w A consumtion decision for RC (i.e. an and ) First, let s think about the Firm s otimization roblem A roduction decision for RC Inc (i.e. an and ) We have already solved this tye of roblem once Such that The allocation is feasible and and are otimal for the consumer given rices w and are otimal for the firm given rices w We draw iso-rofit lines So Highest ossible iso rofit line will occur at the oint of tangency with the roduction function Cometitive Behavior of RCI 77 Equilibrium 78 F(L) Next, let s think about the consumer's otimization roblem C f (w) (w) Iso-rofit line with sloe w They will try to get on the highest ossible indifference curve, subject to the budget constraint What is the budget constraint? l f (w) L 13

14 Cometitive Behavior of RC 79 Equilibrium 80 F(L) C c (w) (w) Budget Constraint with sloe w and intercet So now, for a given w, we can figure out the otimal behavior of the firm, and their resulting rofits And the otimal behavior of the consumer, given that they receive such rofits What does an equilibrium look like? l c (w) L Not an Equilibrium 81 An Equilibrium 82 F(L) Sloe=w F(L) Sloe=w C c (w) C f (w) C c (w )=C f (w ) (w) l f (w) l c (w) L l f (w )=l c (w ) L Equilibrium 83 Equilibrium 84 At the equilibrium The marginal roduct of the firm will equal the wage rate The marginal rate of substitution of the consumer will equal the wage rate This means that marginal roduct of the firm equals the marginal rate of substitution for the consumer i.e. this is the solution to the lanner s roblem! The cometitive equilibrium is Pareto Efficient! The FFWE carries over to this setting! What is going on? Well, it is the same as in the Edgeworth Box cases For otimality we need the MRS of the individual to equal the MP of the firm In a cometitive equilibrium the consumer will set the MRS equal to the rice ratio, and the firm will set the MP equal to the rice ratio Because consumers and firms face the same rices, this will equalize MRS and MP Magic! But again subject to the usual caveats. 14

15 Summary 86 Today we have dealt with 1. Partial Equilibrium with taxes 2. General equilibrium with firms Summary 85 15

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