И.З.И

In the previous chapter we have analyzed the behaviour of a producer who indeed had little choice. Her only degree of flexibility was to produce a little bit more or a little bit less by hiring a little bit more or a little bit less labour. Luckily enough, to be a producer, to start her own business is more exciting than that! That chapter will enlarge the analysis of producer's choice. “In the long run” the producer is much less passive and can take many more initiative.

The first thing the producer can do is to modify or change completely her product. She can also add new products to her catalogue of goods and services offered to consumers. A second thing that can be done is to keep the same product but improve the production process, that is to say, she can modify the technology, or, if you prefer, change her production function. We will postpone the study of those two types of decision until chapter 10, when we will be discussing the nature of the firm and the challenge the owner of the firm have to face in a world of competition.

For now we will concentrate on another type of long run decision, one which still assumes technology to be fixed . But, even when the technology is given, the producer has to decide how much to invest in the production. How much capital does she want to have? How big she wishes the firm to be? Those are the questions addressed in this chapter.

The decision to invest, or not to invest, is a difficult one. What makes it more difficult than other decisions is precisely that it engages the decision maker for a long period of time; and if the economic environment changes in an unexpected way, then that investment might well be lost.

Indeed most investment decisions have a higher degree of irreversibility than other choices. Or to be more precise, the irreversibility of that choice has lasting consequences. If I choose the wrong food for tonight, my supper will be not as good as it could have been, but next day I will be more careful. There are no long lasting consequences. But, if I go to the medical school and then find out that I like literature much more than medicine, then a large part of my investment will be useless and I will have lost a lot of time. 1 In the same way, if I decide to double the production capacity of my company and the demand for my product falls sharply, then I will be left with machines, and stocks of intermediary goods which are likely to be of very low value; no one being willing to buy them from me. Hence, investment decisions are complex because they have lasting consequences and the future is uncertain.

Also, like any decision, investing always comes at a cost. In order to invest I must give up on something. Economists sometime say that to invest is to make a detour of production (roundabout). The traditional example is the fisherman on the desert island. The fisherman can catch fishes with his hands, but then his productivity is very low. So he can decide to invest in a net. But while he makes the net, he can no longer catch fishes. This is the cost of his decision. As always, an opportunity cost. If he accepts to pay that cost it is because the detour of production consisting in making a net will hopefully enable him to catch more fish in the future. Let us also note that in order to invest he must first constitute a stock of fishes that will allow him to survive during the time he makes the net. Hence, investment must be preceded by some saving , and in order to save I must sacrifice some present consumption.

We have been so far stressing the difficulty and complexity of an investment decision. Let me end up those general remarks on investment by recalling that investment is essential to progress: it unable to increase productivity and therefore to satisfy more needs in the future. Surely, it is a risky decision because it is a long run decision. But we should not be too afraid by that. Remember: to do nothing and take each day as it comes is often riskier than to invest!

We have decided for now to consider the case in which the function of production, Y =F( K , L ), is given to the producer. The difference with the previous chapter is therefore simply that the producer can now choose both the level of labour and the level of capital. What levels are in his best interest, that is, maximize profits?

To answer that question we can rely on the analysis of the previous chapter. As a matter of fact to each level of capital that he can choose is associated a set of marginal and average short term cost curve. It is like if, once he will have chosen the level of capital, he will be back to the problem as set in the previous chapter. This is illustrated on the graph below: choosing a level of K 1 of capital, is like choosing to be on the average and marginal cost curves AC 1 and MC 1 . If a higher level of capital is chosen, K 2 , then the costs of production will be those indicated by the curves AC 2 and MC 2 , and so on.

Now, let us assume that the producer expects a demand for his product such as will be able to sell be able to sell Y E . We see immediately that he should invest in a quantity K 3 of capital, because that minimizes the average cost of production. If he invests in K 4 , the average cost will be higher. Same thing if he invests K 1 , or K 2 .

That graph illustrates well the general remarks made earlier on the investment decision. To see that assume the producer anticipates a demand such that he will have to produce Y A . He therefore chooses to invest in order to obtain a quantity K 2 of capital. If in fact the demand turns out to be higher, at Y E , he will not be able to produce it. Inversely, if he anticipates a production of Y E and the demand is lower, at Y A , he will have over invested. In both cases, the profits will be lower than it would have been, had his expectations be accurate. It is even easy to see that he might lose money because, when price is below the break-even point, one can not always stop production at no cost.

The graph above allows us to discuss what might happen to the firm as it gets bigger. Of course, as usual, the cost curves have been drawn in a very, but not totally!, arbitrary way. The exact shape of the curves will depend on the technology. As we can see, as the production scale increases, the average cost has first a tendency to decrease. This is the area where Y A is located. Then, on the right side of the graph, the average cost starts to increase. Y E is located in that area. Hence, to get bigger is sometimes a good idea as far as costs of production are concerned, but not always.

Economists like to use the concept of economies of scale to study what happens as the firm increases its productive power. Assume that the producer decides to double the size of his production activities through a doubling of the quantities of labour and capital. Then, clearly, three outcomes are possible.

If the production more than doubles, let us say, that is multiply by three, then we will say that the firm benefits from increasing return to scale , or, equivalently, that it realizes economies of scale. Increasing scale of production leads to an increase in productivity and consequently reduces the average cost of production.
If the production just double when labour and capital employed are multiplied by two, we say that production is done at constant return to scale . The average cost remains unchanged.
If the production does not even double when labour and capital are multiplied by two, then we have decreasing return to scale , or, diseconomies of scale .

We hear many stupid things regarding the choice between capital and labour. How often, for instance, do we hear that “the market” induces producers to substitute machine to labour, and that this is shameful because human beings are more important than machines! The most stupid commentators go even as far as to suggest that with fewer machines we would have less unemployment! The purpose of this section is to provide a rigorous analysis of the way producers combine labour and capital.

To conduct this analysis we will assume that the producer wishes to produce a given quantity Y *. Then, regardless of how the price is determined, profits will be maximized by producing the quantity Y * at the lowest possible cost . The cost of production will vary of course with the quantity of labour and capital employed during the production process. We will denote C( K , L ) the total-cost function. The goal of the producer is hence to choose a bundle ( K, L ) that minimizes C( K , L ) while allowing at the time the production of Y*, that is a bundle such that F( K , L )= Y *.

This problem can be studied with the same tools that were used to study the choice of the consumer. So, you can now rip the benefits from your earlier investment! To start with we will gather all the bundles (K, L) according to how much they can produce. The locus of all the bundles (K, L) producing a given level Y, that is such that F(K, L) = Y, will be called an isoquant . 2 On the graph below are represented some isoquants corresponding respectively to levels of production Y 1 , Y 2 , Y 3 , Y 4 . The combination (K 2 , L 2 ) enables the production of Y 2 , as do all the other bundles (K, L) located on the same curve.

Once again, the exact shape of those curves will depend on the available technology. As drawn here, it assumes a technology that allows for some substitution between the two factors of productions. Hence, a quantity Y 2 can be produced with “a lot” of capital and little labour (point A), or with “a lot” of labour and little capital (point B). Sometimes no such substitution is possible. If for instance you run a cab company, then it is pointless to own ten cabs and hire only one cab driver. You can hardly substitute labour for capital in the production of cab service.

To measure the degree of substitutability between labour and capital, economists have designed a tool very similar to the one used to study the degree of substitutability between two goods that can potentially satisfy the needs of a consumer (see chapter 4). It is also a ratio of marginal magnitudes, it is also called a marginal rate of substitution, but, because we are now talking about technology, we call it the marginal rate of technical substitution .

More precisely, the marginal rate of technical substitution is defined as the ratio of the marginal productivity of labour to the marginal productivity of capital. (Recall that the marginal productivity of a factor of production is just how much more I will produce if I put one more unit of that factor in the production process, keeping the other factors constant 3). It tells you how much units of labour you must add if you get rid off one unit of capital and wish to keep production constant (i.e., to remain on the same isoquant).

As was the case when we studied consumer's choice, the MRTS changes with the composition of your bundle (K, L). For each point (K,L) it tells you how steep is the isoquant curve at that given point. To say things differently, the MRTS gives you the slope of the tangent to the curve at that point. Hence, at point A the producer produces a quantity Y 2 with a lot of capital and little labour. Hence the ratio of the two marginal productivities is high: for instance, you will need only half a unit of labour to replace one unit of capital. In point B, it is just the reverse, so that the slope is gentle (for instance, you need two units of labour to replace one unit of capital).

Let us note in passing an important characteristic of the production function which we study here. It is clear from the discussion of the previous paragraph that what explains the shape of those isoquant curves is the assumption that when I use a lot of capital, the marginal productivity of capital is low (point A), but when I use little capital, the marginal productivity of capital is high (point B). And the same thing can be said of the labour factor. Hence we have here implicitly into action a law of decreasing marginal productivity . The extra unit of one factor of production is always less productive than the preceding one. Is this a general law? Well, it can be pointed out that if I receive my tenth unit of labour I will put it, if I am rational, to its most productive use. Consequently, the next unit of labour, the eleventh one, will be used necessarily in a less productive way. Hence there exists a good reason to believe that marginal productivity is decreasing. Remember however that this is given a technology . In real life technologies change and for that reason, the next unit of labour could be more productive than the previous one.

Let us now come back to our problem: the producer wishes to produce a given quantity Y *. We have all the combination labour-capital leading to that level of production. To find which one minimizes costs, it is useful to draw another set of curves, the isocost curves . An isocost curve is the locus of all the bundles (K, L) that cost the same to the producer. If we call w the price of one unit of labour and r the price of one unit of capital, then the isocost curve is the set of all the combination (K, L) such that wL + rK = C , where C is a given amount.

To give an illustration, if the price of labour is w = 10, and the price of capital is r = 20, then all the bundles that cost C = 100 must verify: 10 L + 20 K = 100. On that isocost curve you will find among others the bundles (K=2, L=6) or the bundle (K=0, L=10), or (K=5, L=0), or (K=4, L=2). As you can easily check, all those bundles are located on the same line . The slope of the line is given by the ratio of prices: - w / r . In our example, you can quickly verify that the relation between K and L on that isocost curve is given by: K = -½ L + 5.

Some isocost curves are drawn below assuming w = 10, r = 20, so that the slopes of all the curves is -1/2.

We are just one step away from completing our analysis of the choice of the bundle (K, L) that minimizes costs of production. The last thing we have to do is to plug on that graph the isoquant corresponding to the level of production we wish to reach, Y *.

We have therefore shown that, given the technology, given the prices of the factors of production, the producer will chose the bundle ( K *, L *). All the other bundles, either cannot produce that much (they are located on an isoquant below the Y *-isoquant), or cost more (they are located on an isocost curve located above the one passing through ( K *, L *)).

Finally, we are in position to answer the question asked at the beginning of that section: Is it more advantageous to use capital or labour? And the answer is not so simple. Sorry for that! In fact, a property of the rational choice will enable us to state our answer in a way which I hope will appear clear. Let us remark that, according to our analysis, the rational producer will choose a bundle ( K *, L *) such that at that point, the MRTS is equal to the slope of the isocost curve (the isoquant is tangent to the isocost line). At the optimum, the following relation prevails:

Hence, when choosing with which combination of capital and labour she will produce, the producer looks at two things:

And, let us stress again, that the answer to the second question depends of how much labour and capital that producer is already using since the MRTS changes with your factors endowment.

The lesson to be drawn is that it is non sense to say that the capital has more value than labour, or the reverse. The value of capital, like the value of labour, will vary with the technology, the type of good considered, with how much factors I already have and, of course, with the relative price of those factors.

The costs of labour and capital play an important role in determining producer's choice. But where do they come from? A short answer is that they are given by the labour market and the capital market. This is true and we will come back to it in the next chapters. But let me close that chapter by pointing out to some characteristics of those costs.

Let us start with the cost of capital. Assume you must buy a computer on sale for1000 leva; is 1000 leva the cost of your capital? Usually not! Because most of the time you don't have that much money on your current bank account and you must therefore ask for a loan. If you are a trustworthy person, your banker will be more than happy to lend you that money. Money that you don't have yet, but that you expect to get, precisely thanks to that computer. So, what the banker does for you in fact is to allow you to spend now some money that you will have in the future. And of course you will have to pay for that service. This is the interest . Consequently, the cost of that computer is 1000 leva plus interest, let us say 100 leva. The total cost therefore turns out to be 1100 leva. Now, you can ask the following question: what if I have the 1000 leva and don't have to borrow? Isn't total cost then equal to 1000? The answer is a simple: NO. Why? Because costs are always lost opportunities; with those 1000 leva you could have made another investment. You could have, for instance, put your money at the bank and the banker will have paid you some interest. (May be not 100 leva, but something big enough to convince you not leave the money at the bank; money that he will then lend to someone who needs it immediately.) For that reason, even if you purchase it with your own funds, the “true cost” of that investment is not just the price of the computer, but the price of the computer plus the value of the lost opportunity. A good, rational, manager should take that into account.

Regarding labour costs, it is clear that when a producer studies the possibility to hire more labour he takes into account the full cost of labour. By full cost I mean all the “obligations” that come with hiring a new employee. Of course you will have to pay some wage to the employee (this is why we used the letter w for the price of labour). But you will also have to write down a contract, you will also have may be to take some insurance (for instance, to cover damages in case an accident occurs during work-time). In many countries the company must pay unemployment insurance, health insurance. You will also have to control the quality of the work done and, if the work is not well done, or if you have to reduce the scale of your production, you will have to pay some extra costs to lay-off that worker. All those aspects are taken into account by the employer. On the other hand, it should be recalled that an employee can brings a lot to a company, much more than is “labour force”. She can bring her experience, her dynamism, her good mood, her imagination, her talents. Contrarily to what we have been assuming all along so far, labour is not a homogeneous good. Far from it! And we should therefore expect the “cost of labour” to vary greatly from one place to the other, and from one person to the other.

1 In the language of economists, when you go to school, to university, and the like you invest in your human capital. Human capital is surely one of the most precious forms of capital as far as economic progress is concerned (but also as far as human flourishing is concerned).

2 The term is easy to understand iso means equal, and quant is for quantity. So, on an isoquant you find all the combinations (K,L) producing the same quantity.

3 For the mathematically inclined, when the function of production is Y = F(K, L), the marginal productivity of, let us say, labour, is just the partial derivative of F with respect to L, that is ¶ F/ ¶ L, or if you prefer ¶ Y/ ¶ L. And the derivative is defined as the limit of D F/ D L when D L become arbitrarily small, that is to say, the increased in production obtained after an infinitely small increase in labour.