Economics 352: Intermediate Microeconomics Notes and Sample uestions Chapter Ten: The artial Euilibrium Competitive Model This chapter will investigate perfect competition in the short run and in the long run. The difference between the short run and long run here will be that new firms can enter the market in the short run and, as a result, economic profits for firms will be driven to zero in the long run, regardless of what they are in the short run. Market emand Market demand is the sum of the individual demand curves of each person. In a graph, this looks like: Three graphs, side by side, showing that the horizontal sum of two individual demand curves is the market demand curve. Total market demand, t, is the horizontal sum of the demand functions of person a ( a ) and person b ( b ). Also, note that the lower case letters are used for individual uantities while the upper case is used for the market uantity. In terms of demand functions expressed mathematically, if each person has individual demand function for good x of x i = x i (p x, p y, I i ) where x i is the uantity demanded by person i and I i is the income of person i, then the total market demand where there are n people is: X = n i= 1 x i ( p, p, I ) x y i 1
Changes in the price of x will cause movement along the demand curve and are referred to as a change in the uantity demanded. Changes in anything else result in a shift in the demand curve and are a change in demand. Events that could result in a change in market demand would include: -a change in the price of another good -a change in incomes -a change in weather -new information about health effects of the good -a tax or subsidy on the good You should work through Example 10.1 to see how this all plays out with euations. This can also be generalized to multiple goods and multiple people, as shown on page 282 of the text, but I don t believe that this sort of generalization will be exploited further in the chapter. Some Notation x i is the uantity of good x demanded by person i is the total uantity demanded in the market is the market price of the good in uestion is the prices of other goods (or for you math fans, a vector of the prices of other goods) I is the incomes of all the people in the market (again, a vector of incomes) What s up with the vectors? and I are one symbol that stand for multiple values. So, if we re talking about good 1, then is the set of all the other prices in the market, so it would be [p 2, p 3,, p n ]. Similarly, I is the set of all of the incomes of people that are potential customers in the market, so I is really [I 1, I 2,, I m ]. Elasticities of Market emand These are definitional and given in terms of partial derivatives: rice Elasticity of Market emand Cross rice Elasticity of Market emand (, ', I) (,',I) ' ' 2
Income Elasticity of Market emand (,',I) I I Short Run and Long Run iscussion of supply responses in a market really depends on the time frame involved. As more time is allowed, the magnitude of suppliers response to a price change can increase. The very short run is defined as the time period over which the uantity supplied is fixed, so no supplier can alter the uantity that she will offer for sale. The market supply curve is vertical in the very short run. The short run is defined as the time period in which suppliers who are currently in the market can change the uantity that they supply. However, new firms cannot enter the market and existing firms cannot leave the market. In the long run it s all in play. Existing firms can leave the market and new firms can enter. The important thing about this will be that if economic profits are positive in the short run, new firms will be attracted to the market and supply will expand in the long run, driving prices down and driving profits toward zero. How long the very short run, short run and long run might be depends on the industry in uestion. It might take a couple of weeks or a month to get a new espresso cart up and running, but it could take a decade to bring a new automobile plant on line. Assumptions of erfect Competition The following are the assumptions about perfectly competitive markets. They actually describe only a few markets, generally financial markets and some commodity markets. However, the analysis is perfect competition is important, even though it doesn t perfectly describe all that many real-world markets, because lots of markets are sufficiently similar to perfectly competitive markets that the conclusions we get from analysis of perfect competition can be applied. Also, perfect competition is an ideal against which other types of markets can be compared. 1. A large number of firms 2. A homogeneous product every firm produces exactly the same thing 3. Each firm maximizes its own profits 4. Each firm takes the market price as given 3
5. All prices are known by all market participants 6. There are no secrets about the production of the product 7. There are no transactions costs 8. There are no barriers to the entry of new firms or exit of existing firms The short run market supply curve is the horizontal sum of the individual supply curves of the existing firms in the market. This is just like the market demand curve assembled above. The total supply curve is the sum of the uantities supplied by each individual firm at each price. Three graphs, side by side, showing that the market supply curve is the horizontal sum of individual firms supply curves. In mathematical terms, if each firm has the supply function i (,v,w) then the market supply is given by: S (, v, w) = ( n i= 1 i, v, w ) To see how this all works out, you should work through Example 10.2 in the textbook, knowing that the real start of this is back in Example 8.1 with cost minimization of the Cobb-ouglas production function: = f (k,l) = k α l β In Example 8.5 we get the cost function SC (v, w,, k 1 α β β 1) = vk1 + w k1 And in Example 9.3 we get the short run supply function for one firm 4
= w β β 1 β k α 1 β 1 β 1 β Short Run Euilibrium in A erfectly Competitive Market In a perfectly competitive market, the short run euilibrium price will be the price for which the market uantity supplied is eual to the market uantity demanded. This should come as no surprise. In math terms, this is that the euilibrium price * satisfies: (*,,I) = S (*,v,w) In terms of a simple diagram of a market with 1000 firms and an euilibrium price of $8, this could be: Two diagrams, side by side, showing supply and demand in the market generating a price that an individual firm takes as given in determining its profit maximizing uantity to produce. The profit maximizing uantity is the uantity for which marginal cost is eual to the market price. That is, the euilibrium price is determined in the market to be $8, and the resulting market uantity is 10,000 units. For an individual firm, the price of $8 causes it to produce 10 units. The 1000 identical firms together produce the 10,000 units in the market. If market demand increases, the market price will rise and, in the short run, all of the existing firms will increase production. 5
If market demand falls, the market price will fall and, in the short run, all of the existing firms will decrease production. Each of these events will be represented in the market diagram by a shift in the demand curve, but by a movement along the existing supply curve. There will be a change in demand, but a change in the uantity supplied. You should work through Example 10.3. Mathematical Model of Market Euilibrium This is shown on page 293 of the text. I ll try to work through it a bit more carefully here. We start with the demand and supply functions, except that the uantities to be supplied and demanded are expressed as a function of the price and just one other factor. This other factor (which we won t name right now) is called α in the case of the demand function and is called β in the case of the supply function. S = S ( α ) =, (, β ) If we totally differentiate each of these we get: d = d + dα α = d + dα α d S S S = d + dβ β = S d + S dβ β In euilibrium we have = S and, totally differentiating this we get: d =d S And combining these we get: dα = S d + α d + S dβ β 6
Now, if β doesn t change (that is, if β is held constant) we have dβ=0 and this euation becomes: d + dα d α = dα S α = S d So, this describes the effect of an increase in α on the euilibrium price,. The denominator, S is positive (you should know why this is positive), so if the effect of an increase in α on demand is positive, then an increase in α will increase the price. Similarly, if the effect of an increase in α on demand is negative, then an increase in α will lower the price. α might be something like income, and if the good is a normal good then an increase in income will increase demand for the good and its euilibrium price will rise. You should work through Example 10.4 and be sure it makes sense to you. Long Run Competitive Euilibrium In the long run, if economic profits in an industry are positive, new firms will be attracted to the industry and supply will increase. The increase in supply means that prices will fall and, as a result, profits will fall, too. This will continue to happen until profits in the industry are driven to zero, giving us the long run competitive euilibrium condition that profits are eual to zero or, alternatively, that =AC. If we combine this with the firm s profit maximizing condition that =MC, we get =MC=AC and profits eual zero. Similarly, if economic profits are negative, firms will exit the industry (shut down, or maybe start making something else, it doesn t matter), causing the supply to diminish and the price to rise. This will continue to happen until the price rises enough to bring economic profits up to zero, at which point firms will stop exiting. The standard, simple graph of long run euilibrium is: 7
Two graphs, side by side, showing long run euilibrium in which the market price is eual to the minimum average cost. The individual firm produces a uantity such that price is eual to minimum average cost, which is eual to marginal cost. Example Imagine that a perfectly competitive industry has firms whose cost functions are given by: C() = 5000 + 5 + 2 2 and that there are currently 1000 firms in the industry. emand in the industry is given by: () = 1000 (/1000) Which may be rewritten as: = 1,000,000 1000 Solve for the short run euilibrium price and for the long run euilibrium price. iscuss how the market will move from the short run euilibrium to the long run euilibrium. The short run supply curve for a typical firm will be given by the firm s marginal cost function, which is: 8
MC() = MC () = p s p 5 = 4 dc d = 5 + 4 That is, the marginal cost function is MC()=5+4 and, at a profit maximizing uantity the marginal cost euals the price, so we set marginal cost eual to the price and solve for the uantity to get the function for the uantity supplied. Now, there are 1000 firms, so the market supply function is: s = 1000 s p 5 = 1000 = 250(p 5) = 250p 1250 4 We can combine this with the market demand function to get 1,000,000 1000 = 250 1250 1,001,250 = 1250 * = 801 = = S S = 199,000 So, in the short run the uantity sold in the market is 199,000, so each of the 1000 firms in the market produces 199 units. Now, in the long run, =MC=AC. It also happens that MC=AC at minimum average cost, so there are a number of ways to solve for minimum average cost. You could either find the average cost function and find its minimum (take the derivative and set it eual to zero) or you could set the marginal cost function eual to the average cost function and solve for. Let s do both. 9
AC() = 5000 AC() = + 5 + 2 dac 5000 = + 2 = 0 2 d 2 = 2500 = 50 C() 5000 + 5 + 2 = 5000 AC(50) = + 5 + 2(50) = 205 50 2 5000 AC() = + 5 + 2 = 5 + 4 = MC() 5000 = 2 5000 = 2 = 50 2 MC(50) = 5 + 4(50) = 205 So, the long run euilibrium has a price of 205 and each firm is making 50 units. Total market demand at a price of 205 is: 1,000,000 1000(205) = 795,000 If each firm is producing 50 units, this means that in the long run there will be 795,000/50 = 15,900 firms. Increasing, ecreasing and Constant Cost Industries There s a uestion about how well new firms in an industry can compete with previously existing or incumbent firms. If new firms tend to have higher average costs, either because they lack some advantage that previously existing firms enjoy or because they just don t know the business as well, then in the long run as the uantity supplied increases, average costs in the market will increase. Such an industry is known as an increasing cost industry, because as the market uantity increases in the long run, the price or average cost will rise. In this case, the long run supply curve will be upward sloping. The long run elasticity of supply will be positive. If prices or average costs don t rise with the entry of new firms in the long run, then long run supply will be infinitely elastic. That is, the long run supply curve will simply be a horizontal line at the level of a typical firm s minimum average cost. This is called a constant cost industry. The long run elasticity of supply will be infinite. If prices or average costs fall as more and more firms enter the industry, perhaps due to positive externalities in production, then the long run supply curve will be downward sloping and this will be a decreasing cost industry. The long run elasticity of supply will be negative. 10
The Effects of Shocks to the Market on the Structure of the Market Market structure, in the case of perfect competition, basically means how many firms there are and how much each firm produces. If demand increases, the euilibrium price will rise in the short run, but it will fall back to the minimum average cost in the long run. In a constant cost industry, this will result in the entry of new firms and, in the long run, each firm will produce just as many units as they did before the demand increase. If producer costs rise, the effect on the number of firms and on each individual firm s output will be ambiguous. If the cost increase is an increase in the firm s fixes costs, this will shift the average cost function up but will leave the marginal cost function unchanged. It will likely be the case that there will be fewer firms in the market, but that the remaining firms will produce more units than they did previously. If the cost increase takes the form of a large increase in marginal cost, then the optimal (average cost minimizing) uantity might fall enough that each firm will produce fewer units and the number of firms could actually rise. roducer Surplus in the Long Run In a constant cost industry, producer surplus or profit will go to zero in the long run as new companies come in and drive the price down to minimum average cost. The applicable principle here is that if firms in an industry without significant barriers to entry are making extranormal profits, these profits won t last long because new firms will enter the industry and take those profits away. ut somewhat differently, if firms in industries without barriers to entry have highly priced stocks, the value of those stocks will probably fall over time as a result of new firms entering. In an increasing cost industry, incumbent firms will have some cost advantage over new entrants and the profits of incumbent firms can remain higher than normal (that is, they can enjoy positive economic profits on a sustained basis), but these extra profits will really be returns to the advantage that they possess. For example, a farmer who owns particularly fertile land will be able to produce crops more cheaply than a farmer working less fertile land. The extra profit earned as a result of the superior fertility of the land might show up as extra profit for the farmer, but it is really a rent accruing to the very fertile land that he owns and works. In fact, the opportunity cost of his working the land is the rent that he could charge to someone else to work the land. 11
ractice roblems 1. Recreate the analysis on page 293, except that you should assume that supply shifts and that demand is held constant. Explain how changes in β affect euilibrium price and why this makes sense. 2. For the short run and long run competitive example presented in these notes, fill in the appropriate numbers in the following diagrams. Short run: Long Run 12
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