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Perusing the properties of the production function
After you understand that firms are trying to choose the amount of labour and capital they want to hire in order to maximise their profits (see the preceding section), you need to understand how the production function f(K, L) works.
Firms’ demand for labour and capital (and ultimately wages and the returns to capital) depends on the behaviour of the production function.
To work out how much capital and labour firms will hire, you first need to understand how the production function works:
The production function is increasing in labour means that if a firm hires an additional worker, it’s able to produce more output. The amount that output increases by if an additional worker is hired is called the marginal product of labour (MPL).
The production function is increasing in capital means that an extra unit of capital is going to increase the firm’s output. The amount that output increases by if a unit of capital is added is called the marginal product of capital (MPK).
Mathematically, we can write:
Clearly, MPL and MPK should always be positive: having more people or more/better machines can never reduce output. At this point precocious students usually put up their hands and ask, ‘Yeah, but what if you have so many people in the factory that they literally can’t move?’ To which we reply, ‘You can always ask people not to turn up to work if you really have too many!’ Economists call this the assumption of free disposal.
The next question is: what happens to MPL as L rises? Here, a thought experiment helps. Imagine that a firm has 5 machines but no workers to work them. With no workers, it’s unable to produce anything, that is, f(5,0) = 0. Now imagine adding a single worker (while keeping capital fixed at 5 machines). The first worker is likely to be able to add a lot to output, or in other words, his MPL is high. The next worker also adds to output, but probably a bit less than the first worker – his MPL is high but a bit lower. Adding a third worker increases output further – but probably less than the increase due to the first and second workers.
At some point, adding more workers isn’t going to add much to output at all – imagine having 100 workers and still only 5 machines: you don’t need an extra worker, you need more machines!
The idea that the MPL should fall as the quantity of labour increases (holding capital fixed) is called the diminishing marginal product of labour. A similar argument can be made for what happens when a firm increases its capital while holding its labour fixed. The first unit of capital adds a lot to output (high MPK), but subsequent units add less to output. In other words, you can also have diminishing marginal product of capital.
Figure 4-3 shows what happens to output as the quantity of labour varies (holding capital fixed). Looking closely, you can see that when there isn’t much labour, adding an extra unit increases output by a lot. But when a lot of labour already exists, increasing it further doesn’t increase output by much. This is precisely what diminishing MPL means. If we were to draw the equivalent diagram for capital, it would look much the same: output would increase a lot for the first few units of capital and then less for each additional unit of capital (holding labour fixed).
© John Wiley & Sons
Figure 4-3: Diminishing marginal product of labour; if amount of labour and capital is then
resulting output will be .
Working out the demand for labour and capital
Okay, almost there. You know how firms work and how the production function behaves. Now we tie it all together to derive firms’ demand for labour and capital. This determines wages and the rental price of capital, which determines what share of GDP individuals receive.
Imagine that you’re the manager of a firm thinking about whether to hire an additional worker. You know that it will cost you £w, but what will you gain from the new worker? Well, by hiring him, your firm would produce more output. How much more output? Exactly the marginal product of labour from the preceding section! If you sell that extra output, you’d receive £p per unit of additional output, and so the additional revenue would equal p × MPL.
So if p × MPL > w, hiring the extra worker makes sense, whereas if p × MPL
w, it doesn’t make sense. In other words, you should only hire if the revenue gained from an extra worker is greater than the wage. Equally, the absolute most a firm would be willing to pay a worker for his services is p × MPL (the marginal revenue product of labour or MRPL for short). The firm would like to pay less, but this is the most that it would pay. For example, if employing Ahmed is going to mean that you can produce 5 more units of output that you can sell for £1,000 each, the most you’d be willing to pay Ahmed for his services (his MRPL) is £5,000.
Knowing that the most a firm would be willing to pay for a unit of labour is equal to the MRPL has certain implications for the demand for labour. After all, the first worker has a relatively high MRPL, the second worker a lower MRPL and the third worker lower still. Why? Because of the diminishing marginal product of labour! We can show this on a graph.
Figure 4-4 shows that when the wage rate that a firm has to pay is high, it doesn’t hire many workers because only the first few workers have an MRPL higher than the wage w. As the wage falls, hiring more workers becomes profitable for the firm; even though their marginal productivity is less than the previous workers, hiring them still makes sense.
© John Wiley & Sons
Figure 4-4: The demand for labour: At different wage rates w0’, w*, w1 firms wish to hire amount of labour LD0, L*, LD1 respectively.
The first few workers are more productive than later workers not because they’re intrinsically more skilled or talented, but because of the diminishing marginal product of labour.
The thing is, however, that the firm probably can’t just choose the wage that it pays its workers. It has to pay the going wage in the labour market – the wage that all the other employers are paying. Calling this wage w*, we can now say exactly how many workers the firm will hire:
The firm will continue to hire workers until the MRPL is exactly equal to the wage rate.
All the workers employed are paid w* and they all (now) have MRPL equal to w* whether they were hired first or not. The reason is: if any individual employee were to now leave, the firm would experience a loss of revenue exactly equal to that person’s MRPL, which is equal to w*.
Of course, the real world contains many different types of firms and different types of workers. Some people are good with their hands, others with numbers; some people are very productive, and others are less so. Whatever the case may be, the logic is still the same: economists expect that people’s wages reflect their marginal productivity, so each person is paid a wage equal to his marginal revenue product.
If you want to know how to earn more money, you need to increase your (marginal) contribution, either by producing (or helping to produce) more or by producing something that’s highly valued in the marketplace and can therefore demand a high price. Easier said than done!
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