1 Competitive Equilibrium

Transcription

1 1 Competitive Equilibrium Each household and each firm in the economy act independently from each other, seeking their own interest, and taking as given the fact that other agents will also seek their best. In the previous section we have described the behavior of each agent in the economy, here we show how all individual actions aggregate into the behavior of the whole economy. For this purpose, we use the notion of equilibrium, meaning that in each market aggregate demand equals aggregate supply, so the corresponding equilibrium price clears the market. We make a simplifying assumptions. We assume that all households are the same. Identical consumers behave in identical ways, so it s enough to analyze the behavior of one consumer, the representative consumer oftheeconomy. Wemakethesame assumption for firms. We assume all firms have the same CRS technology, and just study the behavior of the representative firm. Representative household (who buys good and sells labor) and firm (who sells goods and buys labor) play the role of a stand-in for all consumers and firms in the economy. The third actor is the government thatbuysgoodsandtaxesagentstofinance such purchases. Definition of CE We need to distinguish the exogenous variables determined outside the model, and treated as parameters into the model from the endogenous variables determined in equilibrium. In our case, the exogenous variables are ª and the endogenous variables are the quantities { }, the tax policy,thewage We are now ready to define a competitive equilibrium for the one-period economy we have described: Definition 1 A competitive equilibrium of this economy is a set of quantities { }, a government policy,aprice such that 1. Given { }, the consumers choose { } optimally to solve ( ) 2. Given ª the firms choose { } optimally to solve ( ) and distribute dividends = ( ) ( ) 3. The market for labor clears, i.e. =1,and = ( ) is the market clearing wage 4. The goods market clears: + = 1

2 5. The government sets the tax policy such that the budget constraint is balanced in equilibrium: = Therefore a competitive equilibrium is the combination of optimizing behavior for firms and workers and a set of market clearing conditions. How do we solve for the CE? We have 7 endogenous variables which are determined in equilibrium { } Let s count equations: 1. Budget constraint: (1 )+ = 2. Household optimality: = 3. Firm s optimality: = 4. Distributed dividends equal profits: = = 5. Output determination: = 6. Labor market clearing: 1 = 7. Goods market clearing: = + 8. Balanced government budget: = It seems we have eight equations, so one more than we have unknowns. But we can show that one of the two market clearing conditions (6-7) is redundant. By redundant we mean that it is not an independent equation: it can be obtained as a combination of some other equilibrium conditions. This result is called Walras Law. WalrasLaw states that: in an economy with markets, if 1 markets are in equilibrium, also the market is in equilibrium. To prove that, for example, equilibrium condition (7) is redundant, we start from the budget constraint (1 )+ = (1 )+ = + = + = + where the first line is just the budget constraint; the second line uses the equilibrium dividends, the third line uses the market clearing condition in the labor market; the fourth line uses the government balanced budget condition. The last line is exactly (7) 2

3 Graphical Approach We now show that the competitive equilibrium lends itself naturally to a graphical analysis in the ( ) space. First, we can rewrite the goods market clearing condition as = = 1 which can be thought of as a function = ( ) describing the production possibility frontier (PPF) of the economy. This function describes the trade-off dictated by the production technology between leisure and consumption: the economy can enjoy more leisure at the cost of lower time devoted to work, lower production and therefore lower consumption good. All the points within the PPF (including those on the frontier) are said to be feasible in the sense that they can be reached given the available production technology. The slope of the production possibility frontier is given by the marginal product of labor. By applying the chain rule: 0 ( ) = 1 ( 1) = 1 Thisslopeisalsocalledmarginal rate of transformation between consumption and leisure: it is the rate at which leisure can be transformed into consumption through work, given the existing technology. The slope is negative, but is the curve ( ) concave or convex? A further application of the chain rule yields: 00 ( ) = 1 ( 1) = 1 0 by decreasing marginal returns to labor. So the production possibility frontier is concave. From the optimization conditions of the households, we have that Using the market clearing wage ( ) ( ) ( ) ( ) = into the equation above, we conclude that the equilibrium allocation can be characterized as the point ( ) where ( ) ( ) ( ) = = ( ) 3

4 Thus, at the equilibrium point ( ) the rate at which households are willing to substitute between ( ) equals the real wage which equals, in turn, the marginal product oflabor, i.e. therateatwhichfirms can convert leisure (and work) into consumption through the production technology. Graphically, ( ) is the point where the convex indifference curve is tangent to the production possibility frontier. Finally, note that the budget constraint cuts exactly at the point ( ) separating the indifference curve from the production possibility frontier: to understand, recall that the slope of the budget constraint is and that the equilibrium allocations have to be affordable (i.e., they must be on the budget constraint). This graphical representiation teaches that the competitive equilibrium (CE) allocationhastobefeasible(ontheppf), affordable (on the budget constraint), optimal for the firm (where the PPF is tangent to the budget constraint) and optimal for the household (on the highest indifference curve, thus tangent to the budget constraint). Comparative Statics Let s do some comparative statics on the equilibrium allocations ( ) using this graphical approach. We study the effect of a change in and a change in on the equilibrium allocations. TFP change: What happens when there is an increase in total factor productivity? Think of this as the recent improvement in information and communication technologies that improved the productivity in the economy over the last 30 years. The term enters the production possibility frontier (PPF), so we need to understand how the latter shifts. Again, let s use the total differential on the equation = ( 1 ). Differentiate with respect to both and keeping fixed to see how the curve shifts in the ( ) space: = 1 = 1 0 so the curve shifts outward. But is the shift equal at every point? To understand, differentiate one more time the expression above with respect to This differential measures how large is the shift at different levels of : 2 = 1 0 which means that the shift is small for high and large for low The reason is that the harder the household work, the more she can benefit from a higher level of productivity. If no labor is supplied to the market, the larger productivity does not translate to any output improvement. 4

5 Remark 1 If we assume that labor is essential to production, then 0 =0and the PPF will tilt around the point =1since = 0 =0and the PPF does not shift at =1 The result of this increase in productivity is that consumption increases unambiguously: the economy is more productive and the level of consumption will go up. The change in leisure is ambiguous and depends on income vs substitution effect, driven by the preferences: a higher productivity level makes it a good time to work (substitution effect), but the fact that wages increase creates an income effect that leads workers to decide to reduce their hours worked, since now they can earn the same by working fewer hours. Data on all developed countries for the last years suggest that growth is balanced, i.e., average hours worked by households have remained constant, roughly, so income and substitution effects offseteachotherinthelongrun. Itmustbethis way, if productivity is what drives long run growth: if the substitution (income) effect dominated, for example, hours worked would grow (fall) continuously over time, but at some point would hit the time endowment (zero): this outcome makes no sense, and it is not what the data say. Finally, note that an increase in induces a rise in welfare ( ) even if the income effect dominates the substitution effect and leisure falls. The reason is the following. If the household kept leisure (and thus hours worked) constant, consumption would go up because is higher and would go up. Therefore, if the households optimally chooses to work more, he does so because her utility level would increase even more. Change in : Consider now an increase in government expenditure The PPF shifts down and both leisure and consumption decrease. The explanation is as follows: suppose leisure does not change, then production remains constant. But because grows, this means that has to fall proportionately: government expenditure crowds out consumption through higher taxes. Now, recall from the graph of the budget constraint that a pure negative income effect (rise in or fall in ) decreases leisure because leisure is a normal good, i.e., a good whose consumption increases when (nonlabor) income rises. So hours worked will rise and this increases production, which means that consumption will fall less than the rise in i.e. crowding out is less than one for one. For this reason, output goes up since = + and we just showed that Does household welfare fall after a rise in government expenditures? Clearly, it depends on how the increase in the public good is valued by the agents: if it is 5

6 strongly valued through ( ) then welfare i.e. the total utility ( )+ ( ) could become larger, notwithstanding the decrease in both ( ) Finally, we note that in the data, business cycles fluctuations are characterized by a co-movement of output and private consumption (when one rises, the other rises too). This outcome is consistent with fluctuations in productivity. Instead, fluctuations in government consumption move output and privare consumption in opposite directions. Based on this observations, we conclude that productivity shocks are a more likely force behind aggregate fluctuations than government expenditures shocks. 2 Social Efficiency of Equilibrium Allocations The competitive allocation is the aggregate outcome the economy will settle upon, through the self-interested behavior of individuals and the natural functioning of markets. One important question we should ask, in relation to the competitive allocation, is whether this allocation is the best possible allocations among the feasible ones, or if there are better allocations than this one. In principle, all the allocations on the PPF are feasible: what is that guarantees us that the point picked up by the competitive equilibrium is the so called first best, i.e. is the one guaranteeing the highest utility to the agents? To answer this question, we introduce a new character, the benevolent dictator. Consider a fictitious benevolent social planner who maximizes households welfare ( ) by choosing consumption and leisure on their behalf, subject to the aggregate feasibility constraint of the economy that tells us which combinations of ( ) are attainable and which are not, i.e. the usual constraint given by the PPF. + 1 We can write the Social Planner Problem (SP) as (SP) max ( ) { } + 1 The solution to this problem tells us what is the best feasible consumption and leisure bundle that can be achieved in this economy, i.e. the solution to this problem is the socially efficient allocation. We want to compare the socially optimal allocations to the competitive equilibrium allocations. 6

7 Writing the Lagrangian of the (SP) problem and denoting with the multiplier on the feasibility constraint, we conclude that the solution of (SP) is = = ³1 ˆ where we use hats to denote socially optimal allocations to distinguish them from the star competitive equilibrium allocations. Dividing through those two equations, we arrive at the familiar condition: = ³1 ˆ which implies that the = exactly as in the competitive equilibrium. Since the equilibrium allocations ( ) also satisfy ³ the same condition, it means that they are equal to the socially optimal allocations ˆ ˆ 2.1 Pareto Optimality The notion of social efficiency that we have just derived, as a solution to the (SP) problem, is equivalent to the notion of Pareto Optimality. Therefore, we can use the terms social efficiency and Pareto optimality interchangeably. Definition 2 A feasible allocation is Pareto Optimal if there is no other feasible allocation that can improve the welfare of at least one agent in the economy without reducing the welfare of some other agent. The concept of Pareto optimality takes its name from the 19th century Italian economist Wilfredo Pareto. In economies with one representative agents, like the static model we studied, a Pareto optimal allocation is the feasible allocation that makes the agent as well off as possible, so it is the solution to the (SP) problem described above. It is useful to look at an example of a planner s problem with more than one agent. Suppose there are two types of agents, 1 and 2 The Planner s problem is max 1 1 )+(1 ) ( 2 2 ) { } (SP2)

8 where istherelativeweighttheplannerputsonagent1 relative to agent 2. The solution to this problem is indexed by For each value of we have a different solution and each one is a PO. When 0 5, the PO allocation is such that agent 1 consumes more than agent 2 Try to characterize the solution of this problem on your own, by using the Lagrangean. 2.2 Welfare Theorems Technically the equivalence between competitive equilibria and social efficient allocationsisstatedbythewelfare Theorems. The First Welfare Theorem states that every CE is PO. The Second Welfare Theorem states that every PO allocation (i.e., every solution to a planner s problem) can be decentralized as a CE allocations (i.e., obtained as an equilibrium of the market economy) through the right initial transfers to the agent in the economy. These transfers are going to be proportional to the welfare weights: an agent who has a lot of weight in the planner s solution will consume a lot in the PO allocation. How do you get that agent to consume a lot in the CE? By starting off that agent with a high level of wealth relative to the others. The Welfare Theorems hold under some technical conditions. Often these conditions do not hold. When these conditions fail, the CE allocations feature a wedge between the MRS and the MRT. Recall that = is always the solution to the Planner s problem, but in equilibrium we can have that the = = is violated because the firstequalitydoesnothold(wehaveawedgeonthehoousehold side), or because the second equality does not hold (we have a wedge on the firm side). Examples where socially efficient allocations differ from competitive equilibrium and the Welfare Theorem fails are economies with: Distortionary taxes, like a proportional labor income tax in presence of an endogenous labor supply decision, or a proportional profit taxonfirms. Externalities not internalized by individuals and markets. These are cases where the consumption choice of an individual will affect the utility of another. The individual does not take this external effect (or, externality) into account, but the social planner does. There are negative externalities (e.g., pollution, smoking) and positive externalities (e.g., keeping your backyard clean and cute-looking is a pleasure for your neighbors too). Non-competitive markets, e.g., monopolies or oligopolies where firms are not price-takers, they can affect prices. Prices (and profits) are too high and quantities too low compared to the efficient benchmark. 8

9 Finally, note that this equivalence result between competitive equilibrium and social planner allocation is important in another way: we can compute an equilibrium just by solving the planner s problem, which as you might have noticed is simpler: for example, it has no prices. This is key: the planner dictates allocations of consumption and leisure to the agents, he doesn t have to worry about markets and prices. There are no market and no prices in the fictional world of the social planner. 2.3 Efficiency and Fairness Note that Pareto optimality has to do with economic efficiency, not with fairness. For example, an allocation where all resources are assigned to only one agent in the economy and zero is given to everyone else is Pareto-efficient, but not fair according to the common definition of fairness which is related to economic equality. There are other ways to rank allocations based on fairness. For example, suppose that you have types in the economy, each one with different productive abilities. Then one can use several criteria: Rawlsian criterion: maximizes the utility of the least able agent. Here, the implicit assumption is that ability of an individual is the outcome of luck (e.g., being born from smart or parents or rich parents who can afford a good school for their kids). Because luck is outside the control of an agent, it is fair to compensate those who are the most unlucky. Elitist criterion: maximizes the utility of the most able agent. Here the assumption is that ability is the outcome of effort not luck (e.g., having studied very hard since very early ages which led to a high level of human capital and hence ability). Because effortisundertheagentcontrol, itneedstoberewarded. Benthamian (or utilitarian) criterion: maximizes the weighted-sum of the utilities where every agent receives the same weight. It is a middle-ground between the first two where the social welfare function is agnostic and puts the same weight on every agent. 2.4 Socialism vs market economies The result that competitive equilibrium allocations are socially efficient, in absence of wedges, has some normative implications for competitive markets: Adam Smith said that an unrestricted competitive market economy will behave as if there was an invisible hand guiding and coordinating the actions of self-interested individuals 9

10 towards a state of affairs that is beneficial for all. What he meant is that the market equilibrium is the first best, as we showed. But we also showed that a planner can do as well as the market. So why did the socialist planning economies fail so miserable? They should have behaved exactly as market economies, according to our derivation above. The reason lies in the strong informational requirements of the Planner s solution: the Planner needs to have a lot of information about preferences of every household in the economy, and production technology of every firm. In the representative agent economy, this requirement is minimal, but in a real economy with millions of different families and thousands of firms, it is a huge requirement. In the decentralized competitive equilibrium, nobody needs to have all this information: it is enough for every firm and individual to know their own preferences and technology, then the market will take care of aggregation towards the achievement of the common goods through prices. Prices signal scarcity. For example, suppose that there is a rise in the demand for apples in the economy. In the planning economy, the planner would need to be aware of this change in preferences and would then command an increase in the production of apples. The market economy reaches the same objective through prices. The rise in demand leads to a rise in prices. Firms realize that there is a profit opportunity and enter the market, so the production of apples increases. There is another reason why socialist economies failed. The planner needs to be benevolent, but we know that many dictators are corrupt and divert resources towards their own welfare rather than towards the welfare of their citizens. 10