Chapter 13: Firm Supply and Producer Surplus

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1 Chapter 13: Firm Supply and Producer Surplus 13.1: Introduction In the last two chapters we have prepared the way for this chapter in which we find the optimal output of a price-taking (competitive) firm. We have all the apparatus that we need in chapter 11 we worked out the cheapest way to produce any given output, and we used this in chapter 12 to work out the (lowest) cost of producing any output. We can now work out the profit from any output and hence find the profit-maximising output. In this chapter we assume that the firm takes the output price as given. This is termed by economists competitive behaviour, and the assumption is usually justified by arguing that the firm is so small relative to the market that it has no control over the price it simply has to take it as given. Later we shall consider the opposite extreme when the firm is the only firm in the market and can thus choose the price and we shall also consider cases in between. But for the moment we assume that the firm takes the output price as given. 13.2: Profit Maximisation We denote the output of the firm by y. Let us denote the price by p. This price p is given and the firm can not change it. What we are considering in this chapter is the optimal choice of y by which I mean the choice of y which maximises the profit of the firm. We will build up a figure in which the choice variable y is on the horizontal axis, and where revenue, costs and profits are on the vertical axis so that we can see easily where profits are maximised. Let us start with the firm s revenue, which is simply given by py. Recall that p is constant it is then clear that revenue is simply a straight line through the origin with slope p. In the figure that follows I have assumed a price of 30 so the slope of the revenue function is 30. If we graph cost against output we know that the form depends upon the returns to scale in the technology. Let us start here with decreasing returns to scale later we shall look at constant and increasing returns. We know that if the returns to scale are decreasing then the cost function is (increasing, obviously) and convex costs rise more than proportionately with output. We thus get the following figure.

2 The straight line is the revenue function and the convex curve the cost function. Also drawn in this figure is the difference between revenue and costs namely profits. The concave curve at the bottom is profits. Note that profits are zero at an output of zero (this must be the long run as costs are zero at an output of zero), they are then positive up to an output of around 76, after which they become negative. It is fairly easy to see from this figure that there is a unique level of output at which profits are maximised. Indeed it should be easy to see that this must always be the case unless the cost curve is steeper at an output of zero than the revenue curve (in which case the best thing to do is to produce nothing). This profit-maximising output is indicated in the next figure.

3 Can we write down a condition to help us find the optimal output? If we look at the figure above it is clear that profits are maximised when the vertical distance between the revenue curve and the cost curve is maximised. The condition for this is simply that the slope of the revenue curve must equal the slope of the cost curve. The first of these is simply the price p, and the second, by definition, the marginal cost at the optimal output. The condition then is simply p = marginal cost This is the profit-maximising condition for the price-taking firm. We have found the profit-maximising condition and output using the space 1 above with the output on the horizontal axis and total revenue total cost and total profit on the vertical axis. This space is simple to use and easy to understand. However, it is sometimes useful to do the analysis in a difference space one again with output on the horizontal axis but now with marginal revenue and marginal cost on the vertical axis. You will recall how to go from total cost to marginal cost we simply find the slope of the total cost function. The same is true for the marginal revenue function to find it from the total revenue function we simply find the slope of the total revenue function. But we already know the slope it is constant and equal to the price of the good p. So we have that, for this competitive firm, its marginal revenue is simply equal to the fixed price of the good: every extra unit the firm sells it receives an extra revenue equal to the price. If we find the corresponding marginal revenue and marginal cost curves for the figure above, we get the following. Note that marginal cost (the slope of the total cost curve) is everywhere increasing 2. 1 The term space was defined in Chapter 1. It merely defines the variables on the axes. 2 The fact that it is increasing at a decreasing rate that is marginal cost is concave comes from the fact (which is not exactly obvious) that the third derivative of the total cost function is negative.

4 In the analysis so far a price p of 30 has been assumed, so that the marginal revenue curve is horizontal at 30. (A price of 30 was also assumed when drawing figure 13.3 above.) We have indicated in this figure the profit-maximising output where price equals marginal cost notice that it is around 33, exactly as before. Obviously it is the same point all that differs is the graphical representation. Note from the figure above that if the price changes then so does the optimal output. We shall explore the implications of this later. 13.3: Loss Minimisation We should be a little careful in applying the profit-maximising condition it does not guarantee a positive profit, but merely that profit is maximised. This is very easy to see if we take the example above and add to the firm s costs some new fixed cost perhaps a government tax which is sufficiently large to make the profits everywhere negative. What happens? To the total revenue and hence marginal revenue curves nothing. To the total cost curve the new fixed costs shifts the entire cost curve up by a constant amount. To the marginal cost curve nothing because shifting a curve vertically upwards by a constant amount does not change the slope at any point. So the addition of the fixed costs, while making profits negative everywhere, does not change the figure 13.5 it looks exactly the same. Is the identified point still a profit-maximising point? Well, yes in the sense that it minimises the losses which now exist everywhere because of the fixed costs. But there are losses at that point. It may pay the firm to give up production altogether. 13.4: Increasing Returns to Scale We also have to be a little careful when applying the condition that price should equal marginal cost, for we might have identified a point where the profits are minimised rather than maximised. Consider what happens if we have increasing returns to scale. In this case, the cost function becomes concave and the total curves look as follows:

5 Also inserted in this figure are the profits of the firm and is identified an output (around 18) for which the price (the slope of the revenue curve) is equal to the marginal cost (the slope of the cost curve). If we move to marginal space we can see this more clearly. In this figure the downward sloping curve is the marginal cost curve, the line is the constant marginal revenue horizontal at the price (here assumed to be 1.8). If we compare figures 13.9 and we see that the condition (price = marginal cost) has identified a point where profit is minimised. Where is the profit maximising output? In this instance, if we always have increasing returns to scale, at infinity. Let us compare figure 13.9 and figure In both of these the point identified is where price = marginal cost but they differ in that 13.5 identifies a profit maximisation point while figure 13.9

6 identifies a profit minimisation point. You will see the difference: in figure 13.5 the marginal cost curve intersects the price from below; in figure 13.9 the marginal cost curve intersects the price from above. We should ensure that we have checked this second condition. So with increasing returns to scale the optimal output of the firm is infinite. This seems a bit unrealistic and alerts us to the fact that there must therefore be an incompatibility between increasing returns to scale and competition (price-taking behaviour). In practice, the possibility of increasing returns to scale everywhere is highly remote, but, even so, this result indicates that if an industry has increasing returns to scale over most realistic levels of output, then the industry can not really be competitive (price-taking). We might expect a monopoly (one big firm exploiting the returns to scale) taking over the industry. Indeed that is what we see in practice in industries that are such natural monopolies. We will have more to say about this in Chapters 28 and : Constant Returns to Scale An interesting intermediate case is when the technology displays constant returns to scale. In this case the total cost curve is linear as is the total revenue curve. What then is best for the firm to do depends upon where these two curves are in relation to each other. We shall just draw one case you should be able to envisage the others yourself. The case illustrated is the case when the cost curve is everywhere below the total revenue curve that is, when the constant marginal cost is smaller than the price. We have: The top line is total revenue; the second total cost; and the third total profit the vertical difference between the first two. Where is the profit-maximising point? If the returns to scale stay constant it is clear that profits continue to increase so the profit-maximising point is at infinity. The opposite is the case if the total cost curve is everywhere above the total revenue curve the optimal output is zero as any output causes a loss and the greater the output the greater the loss. When the total cost and the total revenue coincide then it really does not matter what the firm does its profits are always zero. So the constant returns to scale case appears rather weird when combined with price-taking. We have (where c denotes the constant marginal cost under constant returns): Optimal output = infinity if p > c Optimal output = anything if p = c

7 Optimal output = zero if p < c But there is an interesting implication: since output can not be infinite in practice, the only way that constant returns to scale can co-exist with competitive (price-taking) behaviour is to have an industry in which the price is equal to the marginal cost (and the firms sharing the aggregate demand between them). Perhaps this is truly what is meant by a competitive industry. 13.6: The Supply Curve of the Firm The supply curve tells us, for each output price, how much the firm would be willing to supply at that price. We know that the optimality condition is that the firm produces the quantity of output for which price = marginal cost. In the following figure we examine the long run supply curve of the firm so we draw in the long run marginal cost curve. We take the example of the technology with which we started this chapter. For any given price of the output the firm chooses that level of output for which that price is equal to the marginal cost. One case is pictured where the price is 10. For this price the optimal output is about 2.5. If the price were to rise then the optimal output would also rise along the marginal cost curve. Indeed we can see that the marginal cost curve is the curve relating output to price and hence is the supply curve of the firm. (We should, of course, check that the profits are indeed positive at every point on the supply curve. As it happens this is true for the technology that we have assumed.) Note that the variable along the horizontal axis is the quantity of output and the variables on the vertical axis are the price and marginal cost. The same is the case in the following figure.) In the short run the principle is the same. We equate the price with the short run marginal cost. But we also ought to check whether the firm might not be better off simply closing down the firm. So we need to check that the profits (while possibly negative) are greater than the fixed costs in the figure below (one of the short runs that we have considered previously) this is always the case so the short run supply curve of the firm coincides with its short run marginal cost curve.

8 In this figure the U-shaped curve is the short run average cost. The solid upward sloping curve is the short run marginal cost curve and is also the supply curve of the firm, and the dashed upward sloping curve the short run average variable cost (excluding the fixed cost). 13.7: Producer Surplus We can prove a nice result about the producer surplus. You may like to anticipate it. Consider, for some price (here 30) the optimal output decision. Here it is an output of about 33. Where in the above figure is the profit or surplus of the firm? The profit or surplus is the difference between revenue and costs. If the firm sells 33 units at a price of 30 its revenue is the product of the price (30) and the quantity (33) it is precisely equal to the area of the rectangle bounded by the price and the quantity sold. The total cost of producing 33 units is given by the area below the marginal cost curve between 0 and 33. The difference between these two areas is the profit. It should be immediately obvious that this difference is the area between the price and the marginal cost curve. But note that we know that the marginal cost curve is the supply curve of the firm. So we get the nice result (which should be familiar) that: The surplus of the firm is the area between the price received and the supply curve.

9 This enables us to find immediately the effect of a policy change which affects the price that the firm can get. For example if the price goes up from 30 to 40, then the firm is better off by the area between 30 and 40 and the supply curve. Note that the same result is true in the short run though the absolute surplus omits the fixed cost. 13.8: Summary We found in this chapter the optimal profit-maximising output for the price-taking firm. The price-taking (competitive) firm produces where the price equals the marginal cost, at a point where marginal costs are increasing. The implications of this condition are important: The supply curve of a competitive firm is upward sloping. Most of this chapter assumed decreasing returns to scale. Indeed the combination of price-taking behaviour and constant or increasing returns was somewhat odd. With increasing returns to scale everywhere, the optimal output for a price-taking firm is infinite. With constant returns to scale the optimal output is either infinite, or indeterminate or zero. Finally we confirmed an important result that we have shown in another context. The surplus of the firm is the area between the supply curve and the price received.

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