Microeconomics principles and analysis phần 2 ppt

44 CHAPTER 2. THE FIRM Clearly the analysis in terms of the pro…t function and net outputs has an attractive elegance. However it is not for the sake of elegance that we have introduced it on top of the more pedestrian output-as-a-function-of-input ap- proach. We will …nd that this approach has special advantages when we come to model the economic system as a whole. 2.6 Summary The elementary microeconomic model of the …rm can be constructed rigorously and informatively with rather few ingredients. Perhaps the hardest part is to decide what the appropriate assumptions are that should be imposed on the production function that determines the …rm’s technological constraints. The fundamental economic problem of the competitive …rm can be usefully broken down into two subproblems: that of minimising the cost of inputs for a given output and that of …nding the pro…t-maximising output, given that input combinations have already been optimally selected for e ach output level. Each of these subproblems gives rise to some intuitively appealing rules of thumb such as “MRTS = input price ratio”for the …rst subproblem and “price = marginal cost”for the second subproblem. Changing the model by introducing side constraints enables us to derive a modi…ed solution function (the short-run cost function) and a collection of modi…ed response functions. We get the common-sense result that the more of these side constraints there are, the less ‡exible is the …rm’s respon se to changes in signals from the market. The elementary model of the …rm can usefully be generalised by what amounts to little more than a relabelling trick. Outputs and inputs are replaced by the concept of net output. This trick is an important step for the future development of the production model in chapters 6 and onwards. 2.7 Reading notes On the mathematical modelling of production see Fuss and McFadden (1980). The classic references that introduced the cost function and the pro…t function are Hotelling (1932) and Shephard (1953). See also Samuelson (1983) chapters III and IV. 2.8 Exercises 2.1 Suppose that a unit of output q can be produced by any of the following combinations of inputs z 1 =  0:2 0:5  ; z 2 =  0:3 0:2  ; z 3 =  0:5 0:1  1. Construct the isoquant for q = 1. 2.8. EXERCISES 45 2. Assuming constant returns to scale, construct the isoquant for q = 2. 3. If the technique z 4 = [0:25; 0:5] were also available would it be included in the isoquant for q = 1? 2.2 A …rm uses two inputs in the production of a single good. The input requirements per unit of output for a number of alternative techniques are given by the following table: Process 1 2 3 4 5 6 Input 1 9 15 7 1 3 4 Input 2 4 2 6 10 9 7 The …rm has exactly 140 units of input 1 and 410 units of input 2 at its disposal. 1. Discuss the concepts of technological and economic e¢ ciency with refer- ence to this example. 2. Describe the optimal production strategy for the …rm. 3. Would the …rm prefer 10 extra units of input 1 or 20 extra units of input 2? 2.3 Consider the following structure of the cost function: C(w; 0) = 0; C q (w; q) = int(q) where int(x) is the smallest integer greater than or equal to x. Sketch to- tal, average and marginal cost curves. 2.4 Draw the isoquants and …nd the cost function corresponding to each of the following production functions: Case A : q = z  1 1 z  2 2 Case B : q =  1 z 1 +  2 z 2 Case C : q =  1 z 2 1 +  2 z 2 2 Case D : q = min  z 1  1 ; z 2  2  : where q is output, z 1 and z 2 are inputs,  1 and  2 are positive constants. [Hint: think about cases D and B …rst; make good use of the diagrams to help you …nd minimum cost.] 1. Explain what the returns to scale are in each of the above cases using the production function and then the cost function. [Hint: check the result on page 25 to verify your answers] 2. Discuss the elasticity of substitution and the conditional demand for inputs in each of the above cases. 46 CHAPTER 2. THE FIRM 2.5 Assume the production function (z) = h  1 z  1 +  2 z  2 i 1  where z i is the quantity of input i and  i  0 , 1 <   1 are parameters. This is an example of the CES (Constant Elasticity of Substitution) production function. 1. Show that the elasticity of substitution is 1 1 . 2. Explain what happens to the form of the production function and the elas- ticity of substitution in each of the following three cases:  ! 1,  ! 0,  ! 1. 2.6 For a homothetic production function show that the cost function must be expressible in the form C (w; q) = a (w) b (q) : 2.7 For the CES function in Exercise 2.5 …nd H 1 (w; q), the conditional de- mand for good 1, for the case where  6= 0; 1. Verify that it is decreasing in w 1 and homogeneous of degree 0 in (w 1 ,w 2 ). 2.8 Consider the production function q =   1 z 1 1 +  2 z 1 2 +  3 z 1 3  1 1. Find the long-run cost function and sketch the long-run and short-run marginal and average cost curves and comment on their form. 2. Suppose input 3 is …xed in the short run. Repeat the analysis for the short-run case. 3. What is the elasticity of supply in the short and the long run? 2.9 A competitive …rm’s output q is determined by q = z  1 1 z  2 2 :::z  m m where z i is its usage of input i and  i > 0 is a parameter i = 1; 2; :::; m. Assume that in the short run only k of the m inputs are variable. 1. Find the long-run average and marginal cost functions for this …rm. Under what conditions will marginal cost rise with output? 2. Find the short-run marginal cost function. 3. Find the …rm’s short-run el asticity of supply. What would happen to this elasticity if k were reduced? 2.8. EXERCISES 47 2.10 A …rm produces goods 1 and 2 using goods 3, ,5 as inputs. The pro- duction of one unit of good i (i = 1; 2) requires at least a ij units of good j, ( j = 3; 4; 5). 1. Assuming constant returns to scale, how much of resource j will be needed to produce q i units of commodity 1? 2. For given values of q 3 ; q 4 ; q 5 sketch the set of technologically feasible out- puts of goods 1 and 2. 2.11 A …rm produces goods 1 and 2 uses labour (good 3) as input subject to the production constraint [q 1 ] 2 + [q 2 ] 2 + Aq 3  0 where q i is net output of good i and A is a positive constant. Draw the trans- formation curve for goods 1 and 2. What would happen to this transformation curve if the constant A had a larger value? 48 CHAPTER 2. THE FIRM Chapter 3 The Firm and the Market . . . the struggle for survival tends to make those organisations pre- vail, which are best …tted to thrive in their environment, but not necessarily those best …tted to bene…t their environment, unless it happens that they are duly rewarded for all the bene…ts which they confer, whether direct or indirect. –Alfred Marshall, Principles of Economics, 8th edition, pages 596,597 3.1 Introduction Chapter 2 considered the economic problem of the …rm in splendid isolation. The …rm received signals (prices of inputs, prices of outputs) from the outside world and responded blindly with perfectly calculated optimal quantities. The demand for inputs and the supply of output pertained only to the beh aviour of this single economic actor. It is now time to extend this to consider more fully the rôle of the …rm in the market. We could perhaps go a stage further and characterise the market as “the industry”, although this arguably sidesteps the issue be caus e the de…nition of the industry presupposes the de…nition of speci…c commodities. To pursue this route we need to examine the joint e¤ect of several …rms respon ding to price signals together. What we shall not be doing at this stage of the argument is to consider the p ossibility of strategic game-theoretic interplay amongst …rms; this needs new analytical tools and so comes after the discussion in chapter 10. We extend our discussion of the …rm by introduc ing three further develop- ments:  We consider the market equilibrium of many independent price-taking …rms producing either an identical product or closely related products.  We look at problems raised by interactions amongst …rms in their produc - tion process. 49 50 CHAPTER 3. THE FIRM AND THE MARKET Figure 3.1: A market with two …rms  We extend the price-taking paradigm to analyse situations where the …rm can control market prices to some extent. What are these? One of th e simplest cases –but in some ways a rather unusual one –is that discussed in section 3.6 where there is but a single …rm in the market. However this special case of monopoly provides a useful general f ramework of analysis into which other forms of “monopolistic competition” can be …tted (see section 3.7). We shall build upon the analysis of the individual competitive …rm’s supply function, as discu sse d on page 30 above, and we will brie‡y examine di¢ culties in the concept of market equilibrium. The crucial assumption that we shall make is that each …rm faces determinate demand conditions: either they take known market prices as given or they face a known demand function such as (3.7). 3.2 The market supply curve How is the overall supply of product to the market related to the story about the supply of the individual …rm sketched in section 2.3.1 of chapter 2? We begin with an overly simpli…ed version of the supply curve. Suppose we have a market with just two potential producers –low-cost …rm 1 and high-cost …rm 2 each of which has zero …xed costs and rising marginal costs. Let us write q f for the amount of the single, homogeneous output produced by …rm f (for the moment f can take just the values 1 or 2). The supply curve for each …rm is equal to the marginal cost curve –see the …rst two panels in Figure 3.1. To construct the supply curve to the market (on the assumption that both …rms continue to act as price takers) pick a p rice on the vertical axis; read o¤ the value of q 1 from the …rst panel, the value of q 2 from the second panel; in the third panel plot q 1 + q 2 at that price; continuing in this way for all other prices you get the market supply curve depicted in the third panel. clearly the aggregation 3.2. THE MARKET SUPPLY CURVE 51 Figure 3.2: Another market with two …rms of individual supply curves involves a kind of “horizontal sum”process. However, there are at least three features of this story that strike one imme- diately as unsatisfactory: (1) the fact that each …rm just carries on as a price taker even though it (presumably) knows that there is just one other …rm in the market; (2) the …xed number of …rms and (3) the fact that each …rm’s supply curve is rather di¤erent from that which we sketched in chapter 2. Point 1 is a big one and going to be dealt with in chapter 10; point 2 comes up later in this chapter (section 3.5). But point 3 is dealt with right away. The problem is that we have assumed away a feature of the supply function that is evident in Figure 2.12. So, instead of the case in Figure 3.1, imagine a case where the two …rms have di¤erent …xed costs and marginal costs that rise everywhere at the same rate. The situation is n ow as in Figure 3.2. Consider what happens as the price of good 1 output rises from 0. Initially only …rm 1 is in the market for prices in the range p 0  p < p 00 (left-hand panel). Once the price hits p 00 …rm 2 enters the market (second panel): the combined behaviour of the two …rms is depicted in the third panel. Notice the following features of Figure 3.2.  Even though each …rm’s supply curve has the same slope, the aggregate supply curve is ‡atter –in our example it is exactly half the slope. (This feature was already present in the earlier case)  There is a discontinuity in aggregate supply as each …rm enters the market. A discontinuous supply curve in the aggregate might seem to be rather prob- lematic – how do you …nd the equilibrium in one market if the demand curve goes through one of the “holes”in the supply curve? This situation is illustrated in Figure 3.3. Here it appears that there is no market equilibrium at all: above price p 00 the market will supply more than consumers demand of the product, below p 00 there will be the reverse problem (at a given price p people want to consume more than is being produced); and exactly at p 00 it is not self-evident 52 CHAPTER 3. THE FIRM AND THE MARKET Figure 3.3: Absence of market equilibrium what will happen; given the way that the demand curve has been drawn you will never get an exact match b etween demand and supply. These simple exercise suggests a number of directions in which the analysis of the …rm in the market might be pursued.  Market size and equilibrium. We shall investigate how the problem of the existence of equilibrium depends on the number of …rms in the market.  Interactions amongst …rms. We have assumed that each …rm’s supply curve is in e¤ect independent of any other …rm’s actions. How would such interactions a¤ect aggregate market behaviour?  The number of …rms. We have supposed that there was some arbitrarily given number of …rms n f in the market – as though there were just n f licences for potential producers. In principle we ought to allow for the possibility that new …rms can set up in business, in which case n f becomes endogenous.  Product Di¤erentiation:We have supposed that for every commodity i = 1; 2; :::; n there is a large number of …rms supplying the market with in- distinguishable units of that commodity. In reality there may be only a few suppliers of any one narrowly-de…ned commodity type although there is still e¤ective competition amongst …rms because of substitution in consumption amongst the product types. Instead of supplying identical 3.3. LARGE NUMBERS AND THE SUPPLY CURVE 53 Figure 3.4: Average supply of two identical …rms packets of tea to the market, …rms may sell packets that are distinguished by brand-name, or they may sell them in locations that distinguish them as being particularly convenient for particular groups of consumers. Let us deal with each of these issues in turn. 3.3 Large numbers and the supply curve Actually, this problem of nonexistence may not be such a problem in practice. To see why consider again the second example of section 3.2 where each …rm had a straight-line marginal cost curve. Take …rm 1 as a standard case and imagine the e¤ect of there being potentially many small …rms just like …rm 1: if there were a huge number of …rms waiting in the wings which would enter the market as p hit p 0 what would the aggregate sup ply curve look like?To answer this question consider …rst of all a market in which there are just two identical …rms. Suppose that each …rm has the supply curve illustrated in either of the …rst two panels of Figure 3.4. Using the notation of section 3.2 the equation of either …rm’s supply curve is given by: 1 q f =  0; if p < p 0 16 + [p  p 0 ]; if p  p 0 (3.1) Clearly for p > p 0 total output is given by q 1 + q 2 = 32 + 2[p  p 0 ] (3.2) and so for p > p 0 average output is given by 1 2 [q 1 + q 2 ] = 16 + [p  p 0 ] (3.3) 1 Write down a cost function consistent with th is supply curve. [...]... market only9 and, more interestingly, the case where the monopolist sells in both markets and (3.13) yields p1 q 1 q 1 + p1 q 1 = p2 q 2 q 2 + p2 q 2 = Cq (w; q): q q or, if 1 and 2 are the demand elasticities in the two markets: p1 1 + 1 1 = p2 1 + 1 2 = Cq (w; q): (3.14) It is clear that pro…ts are higher10 than in the case of the simple monopolist and –from (3.14) –that if 1 < 2 < 1 then p2 > p1 We... e¤ectively divide the market and sells in two separated markets with prices p1 ; p2 determined as follows p1 = p1 q 1 p2 = p2 q 2 : where q 1 and q 2 are the amounts delivered to each market and total output is q = q 1 + q 2 Pro…ts are now: p1 q 1 q 1 + p2 q 2 q 2 C(w; q) (3. 12) To …nd a maximum we need the following pair of expressions p i q i q i + pi q i q Cq (w; q); i = 1; 2 (3.13) The outcome of the... a single homogenous good and F0 and a are positive numbers 1 Find the …rm’ supply relationship between output and price p; explain s p carefully what happens at the minimum-average-cost point p := 2aF0 66 CHAPTER 3 THE FIRM AND THE MARKET 2 In a market of a thousand consumers the demand curve for the commodity is given by p = A bq where q is total quantity demanded and A and b are positive parameters... from the approach of chapter 2 and …nd straightforward, interpretable conditions for …rms’equilibrium behaviour A marginal condition determines the equilibrium output for each …rm and a condition on market demand and average costs determines how many …rms will be present in the market The question of what happens when there is no determinate demand curve is a deep one and will be addressed after we... outputs q2 ; :::; qn Hence, using the symmetry of the equilibrium, show that in equilibrium the optimal output for any …rm is qi = A [n 1] n2 c and that the elasticity of demand for …rm i is n n n + 2 Consider the case = 1 What phenomenon does this represent? Show that the equilibrium number of …rms in the industry is less than or equal q A to C0 3.3 A …rm has the cost function 1 2 F0 + aqi 2 where...54 CHAPTER 3 THE FIRM AND THE MARKET Figure 3.5: Average supply of lots of …rms Obviously for p < p0 total – and hence average – output is zero But what happens exactly at p = p0 ? Clearly either we must have either (q 1 = 0; q 2 = 0) or (q 1 = 0; q 2 = 16), or (q 1 = 16; q 2 = 0) or (q 1 = 16; q 2 = 16) In other words total output could have the value 0, 16 or 32, so of course average output... Plot the average and marginal revenue curves that would then face the monopolist Use these to show: (a) If pmax > p and p the …rm’ output and price remain unchanged at q s 3.10 EXERCISES (b) If pmax < c 67 the …rm’ output will fall below q s (c) Otherwise output will rise above q 3.5 A monopolist has the cost function C(q) = 100 + 6q + 1 2 [q] 2 1 If the demand function is given by q = 24 1 p 4 calculate... presented in Figure 4 .2 In the left-hand version a …xed amount of money y is available to the consumer, who therefore …nds himself constrained to purchase a bundle of goods x such that p1 x1 + p2 x2 y: (4.1) if all income y were spent on good 1 the person would be able to buy a quantity x1 = y=p1 In the right-hand version the person has an endowment of resources R := (R1 ; R2 ), and so his chosen bundle... b a A=p 1 and no equilibrium otherwise 3 Now assume that there is a large number N of …rms, each with the above cost function: …nd the relationship between average supply by the N …rms and price and compare the answer with that of part 1 What happens as N ! 1? 4 Assume that the size of the market is also increased by a factor N but that the demand per thousand consumers remains as in part 2 above Show... the industry, 1; 2; :::; N; ::: and suppose the number of …rms currently in the industry is nf Let q N be the pro…t-maximising output for …rm N in a price-taking equilibrium (in other words the optimal output and inputs given market prices as we considered for the single competitive …rm on page 26 ) Allow nf gradually to increase: 1 ,2, 3 : output price p will fall if the market demand curve is downward .  2 z 2 Case C : q =  1 z 2 1 +  2 z 2 2 Case D : q = min  z 1  1 ; z 2  2  : where q is output, z 1 and z 2 are inputs,  1 and  2 are positive constants. [Hint: think about cases D and. market and sells in two separated markets with prices p 1 ; p 2 determined as follows p 1 = p 1  q 1  p 2 = p 2  q 2  : where q 1 and q 2 are the amounts delivered to each market and total. only 9 and, more interestingly, the case where the monopolist sells in b oth markets and (3.13) yields p 1 q  q 1  q 1 + p 1  q 1  = p 2 q  q 2  q 2 + p 2  q 2  = C q (w; q): or, if  1 and