Saturday, 27 January 2018

Automation and Real Wages



Economists generally work on the basis that improvements in technology lead to higher real wages.  Conventional production functions (when combined with other assumptions) invariably produce this result.  Looking at the Cobb-Douglas production function, as the most common example, an increase in total factor productivity raises labour's marginal product.  With the normal assumptions about competition, this leads to higher real wages.

Although I have no problem with using such things at times, I am wary of simple aggregate production functions.  It seems clear to me that technological developments can lead to reduced real wage levels, even without considering how such developments might impact on monopoly power.

Although I'm sure others have produced models that illustrate this, I'm not aware of any and, in any event, I like to experiment with things myself, so I have constructed a little model of production in which technological innovation leads to a fall in real wages.

There is a single good produced by a combination of labour and capital.  There are two possible production techniques, each of which requires a fixed quantity of labour and a fixed quantity of capital to produce a fixed quantity of the good.  These quantities are set out in the table below:


Output
Labour Input
Capital Input
Technique A
12
1
1
Technique B
24
1
4


Total labour and total capital are fixed.  There is perfect competition and no demand deficiency.  This has two implications:

1. The use of each techniques will normally be such that will ensure full employment of both labour and capital.  However, if the ratio of available labour to available capital is too extreme, then all production will use a single technique (whichever maximises use of the abundant factor) and some of that abundant factor will be unutilised.  This possibility is ignored here - it is assumed that available labour and capital always lead to some combination of possible techniques.

2. The real wage will be at a level that ensures the return on capital is the same for each technique.  If it were not, suppliers of capital would switch technique which would lead to shortages or surpluses in the labour market.  With the numbers here, the real wage works out as 8 units of output per unit of labour.

Marginal productivity is not well defined for each individual technique.  However, given the above conclusion about how the techniques are combined, there is a marginal productivity of labour at an aggregate level.  It is relatively easy to show that this is equal to the real wage.

If we take factor supplies of 12 units of labour and 24 units of capital, we get the following output matrix:


Labour Used
Capital Used
Production
Technique A
8
8
96
Technique B
4
16
96
Total
12
24
192


We now want to consider the innovation of a new technique involving a more intensive ratio of capital to labour.  For this to be beneficial overall, it must yield a higher return on capital given the prevailing real wage.  The details of this new technique are:


Output
Labour Input
Capital Input
Technique C
30
1
5


Technique C dominates technique B at all levels of the real wage, so the latter is completely abandoned.  Since the relative factor input ratios have changed, there is also a change in the amount of resources devoted to technique A.  The new levels of production are shown below:


Labour Used
Capital Used
Production
Technique A
9
9
108
Technique C
3
15
90
Total
12
24
198

The change in techniques also impacts on the real wage, which must settle at a new level to continue to equate return on capital across techniques.  In fact this involves a fall in the real wage to 7.5.   

The interesting thing here is that although labour productivity (labour's average product) has increased, its marginal product has decreased.  Real wages have therefore fallen, even though output per head has risen.  The corresponding change is that the return on capital has increased.

My belief is that, on the whole, technological progress results in higher overall real wages.  However, I do not think we can assume that all new innovations will do this.

6 comments:

  1. Thanks for the post on an important subject. Your words prompted me to consider further.

    By no means am I done thinking about this but I seem to return to the same point summarized by your comment "Real wages have therefore fallen, even though output per head has risen. The corresponding change is that the return on capital has increased."

    You don't seem to say who owns the capital involved or whether it is fixed capital (like a building) or consumed capital (like coal burned). Not that the omission effects the conclusion but, instead, it leaves open whether micro-economic analysis or macro-economic analysis should be applied.

    I think the quote is a product of micro-economic thinking. Wages of workers are most important in the micro enviroment.

    On the other hand, output per worker has increased as has return on capital. These are both positive for the macro-economic sphere. I think we can safely assume that the increased output and better deployment of capital can be rightfully shared by the members of the macro-economy even if some workers directly affected are narrowly (micro-economically) impacted in a negative way. These workers should get at least partial economic recovery to the extent that they will share the improvement in the macro-economy.

    Summing my thoughts, I seem to find myself bouncing between the micro and macro viewpoints. Interesting.

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    1. It doesn't matter here who owns the capital, given the assumptions about competition. Of course, if the same people are both workers and capitalists then they will be better off overall, even if their income from labour has fallen, because their income from capital has risen.

      I have assumed that there is no consumption of capital, so you could think of it as buildings or durable machinery.

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  2. Nick,

    I always appreciate your posts! I thought replicating your calculation would be a good exercise. But I couldn't get the same numbers after the technology C takes over.

    Base case: I get MPL = 16 and MPC = 8. Do you get the same numbers? Do we assume here that the real wage reflects marginal productivity of labor? If yes, I got, given the factors, compensation levels of 8 (real) production units per unit of labor and 4 production units per unit of capital. Total labor compensation is 12*8 = 96 and total capital compensation is 24*4 = 96. So no problem here.

    After the innovation: I get MPL = 16.5 and MPC = 8.25. I got these by reducing labor from 9 to 8.25 (A tech) and 3 laborers to 2.75 (C tech), which gives total production of 181.5, which is 16.5 units less than 198. So this gives compensation levels of 8.25 production units per unit of labor and 4.125 production units per unit of capital. Here total compensations levels per factors are labor: 8.25*12 = 99 and capital: 4.125*24 = 99. So here I cannot replicate your numbers?

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    Replies
    1. Hi Jussi,

      I'm really pleased you've done the numbers for yourself - it's the best way to see what's going on. And I do make mistakes, so it's good for me to have the check.

      First note that MPL isn't well defined for each technique at the relevant point, because each function is Leontieff.

      So, MPL only exists as an aggregate of the techniques. We can calculate it analytically, but it's easiest with an example, seeing what happens if we reduce labour supply from 12 to 11, keeping capital constant at 24. Optimal use of resources now results in 6.67 units of labour and 6.67 of capital being used in Technique A and 4.33 units of labour and 17.33 units of capital being used in Technique B for total output of 184. MPL of 8. Playing around will show that MPL is constant, as long as both labour doesn't become so scarce or abundant that it becomes impossible to fully employ all resources.

      In the second case, reducing labour units to 11 leads to 7.75 units of labour and 7.75 units of capital beimg used in Technique A and 3.25 units of labour and 16.25 units of capital being used in Technique C for total output of 190.5. MPL of 7.5.

      The key here is that if total labour varies holding total capital constant, the blend of techniques needs to change.

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    2. Ah, I see my mistake now. Indeed I missed how the optimal blend needs to be determined.

      I wrongly assumed that it would be optimal to keep constant ratio of labor units allocated to different technologies while calculating the derivative. Thus I e.g. had 11 units of labor allocated as 7.33 units in technology A and 3.67 in technology B and total production of 176 (instead of optimal 184).

      Thank you for your time!

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