One story of the 1930’s, which I discussed recently, is that agricultural workers were displaced by tractors and other forms of mechanization. I find this an interesting story, and I went on to I post Who Will Write This Paper? to suggest that it could be formalized.

I then worried that maybe a formal model would disprove my thinking, so I decided to do a little modeling myself, enough to convince me that I am not crazy. Since it involves, math, it goes below the fold.

[UPDATE: Bruce Bartlett did a survey of the long history of economic analysis of technology and unemployment. A quarter century later, that survey is still useful. At that time, Bartlett could point to a high share of labor income in GDP as an indicator that technological change was broadly beneficial. More recently, that share has come down somewhat, and in addition one can argue that the dispersion of labor income has increased due to technological change. For example, consider the Goldin-Katz observation that the gap has gotten larger between average pay for college-educated workers and those without a college education. My guess is that within the college-educated category (and perhaps within the other as well), dispersion has also increased.]The intuition that I have is this. We have an industry (agriculture) that faces somewhat inelastic demand. A technology comes along that raises the average product of labor but lowers the marginal product of labor. Hence wages and employment fall in that sector.

I want to compare wages and employment in a pre-innovation and post-innovation setting. In the pre-innovation setting, I assume a linear relationship between output and employment.

1. Q = L

In the post-innovation setting, I assume a diminishing-returns relationship between output and employment.

1′. Q = dL1/2

where d is a constant greater than one, so that for relevant values for the amount of labor, output is higher under 1′ than under 1.

What is strange here, and perhaps a swindle, is that this is an industry production function. It is not the production function of a representative firm. A representative firm with diminishing returns to scale won’t help me if the industry as a whole can scale by adding firms. Somehow, I want to imagine lots of competitive agricultural firms, but because land is a fixed factor, there are diminishing returns to labor for the industry as a whole. Maybe the best way to come to terms with this is to assume lots of entrepreneurs bidding to farm on the marginal plot of land. Thus, prices and wages are set competitively, and for a given level of industry output, everyone faces the same marginal cost.

Next, assume that labor supply is an upward-sloping linear function of the wage.

2. L = aW

where a is a constant

Next, let demand be given by a downward-sloping linear function

3. Q = b – cP

At this point, if you do not want to bother with the calculus and algebra, feel free to skip to the numerical example.

Assume that the industry is competitive, then the price is equal to the wage times the marginal product of labor, Q’.

4. P = WQ’ = (L/a)Q’

where the second equality was obtained by solving the labor supply function (2) for the wage rate in terms of L and substituting.

Next, we need to find the marginal product of labor, by taking the derivative of the production function. For the pre-innovation case, this is

5. Q’ = 1.

For the post-innovation case, this is

5′. Q’ = (d/2)L-1/2

Substituting (5) into (4) into (3) and setting equal to (1) gives, for each case respectively:

6. b – cL/a = L; L = ab/(a+c)

6′. b – cdL1/2/2a = dL1/2; L = (2ab/[2ad+cd])2)

Numerical Example:

Suppose that we let a = 1, b = 12, c = 3 and d = 4. Then, in the pre-innovation case, we have:

L = 3, Q = 3, W = 3

In the post-innovation case, we have

L = 1.44, Q = 4.8, W = 1.44

So, we have the desired result, namely that after the innovation, there is more output, but less labor input and lower wages.

This is a partial-equilibrium story. The (urban) consumers of our farm products are wealthier than they were before, and I have not shown what they do with this extra wealth. In the long run, I assume that they spend some of it on goods that can employ the agricultural workers. But in the short run, there may be nothing that the agricultural workers can produce that interests the urban consumers, so the urban consumers, too, may increase their leisure. At least, that is the story I aimed to tell in Who Will Write This Paper? And, of course, I have to translate “leisure” into involuntary unemployment, as I pointed out in Who Will Write This Paper, No. 2?. If you look at some of the references suggested by Tyler and Alex in their comments on that post, they talk about “indivisibilities” of labor, which is pretty much what I am talking about when I say fixed cost of labor. Basically, on your farm you would much rather have one worker for 10 hours than 10 workers for one hour each.

(As an aside, I suspect that in a slump, this fixed cost of labor improves the bargaining power of firms, lowering wages more than would be the case without such fixed costs. But I could have that wrong. It needs another modeling exercise.)

Note that if for some reason wages are sticky, then relative to the solution presented above the wage rate will be higher and employment will be lower.