General Equilibrium - Exchange Economy PDF

13 General Equilibrium - Exchange Economy For the rest of the class, we will be using the tools we have developed to explore the concept of general equilibrium. As discussed in the last notes, general equilibrium refers to the situation where all markets are in equilibrium simultaneously. Be...

13 General Equilibrium - Exchange Economy For the rest of the class, we will be using the tools we have developed to explore the concept of general equilibrium. As discussed in the last notes, general equilibrium refers to the situation where all markets are in equilibrium simultaneously. Because general equilibrium models can become very complicated very quickly, we will build up a general equilibrium model from the simplest possible case. Concepts Covered The Exchange Economy Walrasian General Equilibrium Pareto Efficiency and The Welfare Theorems 13.1 The Exchange Economy Setup We will start with a very simple economy. We make the following assumptions about the economy 1. 2 Consumers: {A, B} 2. 2 Goods: {x, y} 3. Each consumer receives some initial amount of each good, called their endowment. We will label the endowments for each consumer with an e. These endowments are the only supply of each good. Label the total endowment E so that eA B x + ex = Ex eA B y + ey = Ey 4. Each consumer can buy or sell x and y at prices px and py With the assumptions above, we can write each consumer’s budget constraint. Since there are no direct sources of income in the model, consumers can only get income by selling their endowments. It is easiest to think about each consumer’s income as the total value they would receive if they sold their entire endowment I A = px e A A x + py ey I B = px eB B x + py e y 1  Consumers then spend their income on the two goods. We will use xA to represent consumer A’s demand for good x so that we can write the budget constraints are p x xA + p y y A = I A p x xB + p y y B = I B From here we could set up a standard utility maximization problem to solve for consumer demand as a function of income and prices. Notice that given prices, this problem is no different than the standard utility maximization problem we have introduced earlier in the class. The key difference is that changes in prices will now also change consumers’ incomes. 13.2 Walrasian General Equilibrium Market Clearing An equilibrium in the economy will require two pieces: optimization and market clearing. Es- sentially, market clearing means that supply and demand are equal. Although we had market clearing in our partial equilibrium model, that setup only equated supply and demand in one market. In a general equilibrium framework, we need the markets to clear for all goods at the same time. In this case, supply is fixed by the endowments Ex and Ey. Demand will be given by the consumers’ choices for x and y. Therefore, we can say an allocation satisfies market clearing as long as xA + xB = Ex y A + y B = Ey Luckily, there is a theorem known as Walras’s Law that guarantees that if all but one markets clear, the final market will always clear as well. Since we only have two markets here, Walras’s law means if one market clears, the other will as well. Equilibrium We are now ready to define an equilibrium in this market. An equilibrium requires us to find 1. An allocation of goods across each consumer: {xA , y A , xB , y B } 2. Prices for each good: {px , py } Subject to the following conditions 1. The allocation maximizes all consumers’ utility given their budget constraints 2. All markets clear In practice, we can solve for an equilibrium price and allocations by first solving for Marshallian demands and then using the market clearing conditions to find a price such that supply equals demand. 2 Example Let’s use an example to help clear up the ideas introduced above. Assume endowments are given by eA A x = 25, ey = 75 eB B x = 75, ey = 25 And utility functions for each consumer are both U (x, y) = x1/2 y 1/2 We know that the Marshallian demands for this utility function are given by 1 I 1I x∗m = ∗ ym = 2 px 2 py We can replace incomes for each consumer by plugging in the value of their endowments I A = 25px + 75py I B = 75px + 25py Therefore, we have 1 25px + 75py 1 75px + 25py xA = xB = 2 px 2 px 1 25px + 75py 1 75px + 25py yA = yB = 2 py 2 py The last step is to find prices that clear the market. Using the market clearing condition for good x gives us xA + xB = e A B x + ex = 100 1 25px + 75py 1 75px + 25py + = 100 2 px 2 px   1 py 100 + 100 = 100 2 px py =1 px Which tells us that prices of the two goods should be equal to clear the market. Plugging this price back into the Marshallian demands gives us the equilibrium allocations xA = y A = xB = y B = 50. There’s a couple of things to notice about this result. First, we can only solve for the relative prices between the two goods. It doesn’t really matter to the consumer whether the price of both goods is 1 or 100. What matters is that they are equal to each other. This result comes because consumers’ income is also related to the prices of each good. For example, if the price of both goods doubled, the value of a consumer’s endowment would double as well. So everything would be twice as expensive but consumers would all have twice as much to spend. Therefore, their optimal decision wouldn’t change. Usually, for simplicity, we set the price of one good to 1 and call this good the numeraire. Then all prices will be relative to this numeraire. For example, we could set px = 1, then we would get py = 1. 3 13.3 Pareto Efficiency and the Welfare Theorems Pareto Efficiency Let’s compare our new allocation to the original endowments by calculating the utilities. At the original endowment, both consumers had a utility of U (x, y) = (25)1/2 (75)1/2 ≈ 43 At the new allocation, both consumers get a utility U (x, y) = (50)1/2 (50)1/2 = 50 So both consumers are better off at this new equilibrium than they were at the original allocation. This result is not a coincidence. It wasn’t specific to the utility function we chose or the specific endowments. To define the property more generally, we need a definition first. We will say an allocation is Pareto efficient if there is no other feasible allocation where at least one person is better off and no person is worse off. Let’s look at the result we got in the example above. The original endowment was clearly not Pareto efficient since we found an allocation that was better for both consumers. But what about our Walrasian equilibrium allocation? Can we come up with any feasible allocation where one consumer is better off but no other consumer is worse off? The answer is no. The allocation we found was a Pareto efficient outcome. In fact, this is a general result (that we will not prove) called the First Welfare Theorem (see the formal definition below). An easy way to check whether an allocation is Pareto efficient is to look at each consumer’s MRS at the allocation. If the two consumers have different values for their MRS, then there are most likely gains from trade since one consumer values one good more than the other. However, if the two consumers have the same MRS, then is no way to make one consumer better off without making the other worse off. The only exception to this idea is in the case of corner solutions. With corner solutions, consumers could have different MRS but there are still no gains from trade because one consumer doesn’t have anything left to trade. The First Welfare Theorem The first welfare theorem says that every Walrasian equilibrium is Pareto efficient. In other words, for any utility function and initial endowments, if we find an allocation and prices where consumers are maximizing their utilities and markets clear, that allocation will always be efficient. The Second Welfare Theorem The second welfare theorem says that any Pareto efficient outcome can be achieved through a Walrasian equilibrium given some initial set of endowments (each set of endowments will produce a different equilibrium). 4 Discussion The first welfare theorem represents a powerful result in economics. Essentially, it implies that if consumers are free to make trades as they please, the economy will always end up at a place where there are no unrealized gains from trade. All mutually beneficial transactions are carried out. However, we need to be careful when applying this result. While it implies that a competitive Walrasian equilibrium will always be Pareto efficient, that doesn’t mean that the allocation is necessarily the “best” allocation. For example, a trivial Pareto efficient allocation is one where consumer A has the entire endowment of both goods and consumer B has nothing. Clearly there is no allocation that can make both consumers better off. If we want to improve consumer B 0 s situation, we have to take something from consumer A. However, even though this allocation is Pareto efficient, there might be other Pareto efficient allocations that seem more equitable (like the 50-50 case we found above). Another concern is that we actually need some conditions to make the First Welfare Theorem hold. Specifically, we need to assume that there is perfect competition. If market imperfections, like incomplete information, market power, or any friction that prevents trades from happening costlessly, the first welfare theorem will not necessarily hold. Finally, a more philosophical reason to be concerned about the concept of Walrasian equi- librium is that it is unclear about the adjustment process to get from disequilibrium prices to an equilibrium. In other words, imagine we started at arbitrary prices. How would the economy figure out the equilibrium prices that clear the market? To think about this problem, Walras used the thought experiment of an auctioneer who would call out prices for every good in every market and market participants would all write down how much they wanted to buy or sell at that price. The auctioneer would then compare supply and demand in each market and set the price at the point where all markets clear. Of course, in reality, there is no auctioneer. But Walras assumed that the economy would find the correct prices through a similar kind of trial and error, which he called tâtonnement (French for trial and error or “groping” - firms feel around until they find the price that clears the market). However, the model itself assumes that consumers and firms will reach this outcome without really describing how it works in reality. 5