In a previous post, I wrote:
Supporters of a higher minimum wage invoke efficiency wage arguments without explaining why that should be expected to dampen disemployment effects rather than amplify them. While opponents of higher minimum wages describe faster productivity growth as a cost, without, as far as I can tell, being against it in any other context. This is a case where abstraction — some equations, even one of those awful supply and demand graphs — might actually be helpful.
Well, for some reason, I sat down and did the math. And the results are interesting.
I need to stress at the outset: This is a “model,” in the sense that it is a set of mathematical equations that are intended to give a simplified description of a phenomenon in the real world. But it is not a model in the sense that you may be used to in economics. I am not making any behavioral assumptions here of any kind. I am simply trying to formalize the logical relationships that everyone in this conversation is discussing. For example, someone says “higher wages will increase labor productivity,” and someone else says, “no they won’t.” Then we can write
y = a w,
where y is the change in percentage output per worker, w is the percentage change in the wage, and a is a parameter. Writing it this way, and reframing the questions themes likely value of a, is not taking a position in the argument or adding anything to the statements that were made. It is just rephrasing them in a way that, first, allows for more precise statement of the disagreement (maybe both people actually agree that a is probably around 0.1, but one of them considers that a meaningful effect and the other thinks it is trivial) and, second, makes it possible to systematically explore how these claims are logically related to others. As soon as you have more than one or two quantitative relationships to fit together, it is easier to work with algebra than with words.
So let’s do it.
As in that other post, we will use lower case variables here as rates of change. e is the percentage change in employment, w is the percentage change in the wage, y is the percentage change in output per worker, q is the percentage change in the quantity of output, and p is the percentage change in the price of output. q and p apply to whatever firm, industry or region wages are increasing for. w, e and y are for whatever group of workers we are interested in — it doesn’t matter for this purpose whether we think of a minimum wage increase as a big percentage increase for a small group of workers (those directly affected) or a small percentage increase for a bigger group. a1 through a4 are parameters. I am writing as if the relationships are all linear, to keep things simpler. But I’m pretty sure everything would turn out the same with a more general functional form.
First, we know that employment depends on the quantity of goods produced and the amount of labor used for each one. So we can write 
(1) e = q – y
This is an accounting identity.
(2) q = – a1 p + a2 (e + w – p)
a1 here is the price elasticity of demand faced by the unit where wages have increased. a2 reflects the demand created by the wage increase; it is equivalent to the fraction of wages from the unit (firm, industry, region, etc.) in which wages are changing, that is spent on the output of that unit. We know a1 > 0, and a2 >= 0. For an individual business, or a relatively narrow group of businesses or workers (such as in the case of a minimum wage increase), a2 will be very low — McDonald’s employees are probably McDonald’s customers, yes, but the percent of their total income they spend there can’t be above single digits. Conversely, as we talk about a broader and broader wage increase, a1 will approach zero as there is less and less margin for substitution away from higher-cost producers. Probably there aren’t many cases where we need both parameters.
(3) p = a3 (w – y)
a3 is the fraction of changes in unit labor costs that are passed on to prices. We know that 0 <= a3 <= 1, with a3 = 1 describing a constant markup.
(4) y = a4 w
a4 is the fraction of (exogenous) wage increases that are absorbed in higher labor productivity. Assuming rational (profit-maximizing) behavior by businesses, 0 < a4 < 1 for wage increases imposed from outside. (For voluntary wage increases like the one depicted in the Doonesbury cartoon, presumably the employer thinks a4 > 1.) It doesn’t matter whether we imagine the substantive reality described by Equation 4 as an efficiency wage/motivation/turnover story, or a mechanization/robots story.  The question we are interested in is, what are the implications of a high value for a4 for the disemployment effects of an increase in wages?
First we combine these equations and express e as a function of w:
(5) e = w [ a3 (-a1 – a2) (1 – a4) + a2 – a4] / (1- a2)
This isn’t very transparent. For a simpler case, assume that the markup is fixed and demand effects are not important. So a3 = 1 (fixed markup) and a2 = 0 (no demand effects.). Then we have:
(6) e = w (-a1 + a1 a4 – a4)
It’s clearer here that the wages-productivity link enters twice. Which makes sense if you think about it. Higher productivity may translate into lower prices, which in turn may translate into higher sales. That’s the middle term, which also depends on the price elasticity of output. But on the other hand, higher productivity means lower employment per good produced. That’s the second term, –a4 by itself.
Which dominates? The answer is straightforward: For it to be the case that when higher wages lead to higher productivity, that ameliorates any employment loss from higher wages, it must be the case that the price elasticity of demand for goods is greater than one. But in this case, as you can see from equation (6), the employment elasticity with respect to the wage must be at least negative one — a one percent increase in wages must lead to at least a one percent fall in employment. Because in the best case, if wage increases are entirely offset by productivity improvements, then a4 = 1. In which case, we have e = w(-a1 + a1 – 1) = -w. (Recall that all variables here are in percent changes.)
So the conclusion is: if we assume a fixed markup and ignore the demand created by higher wages, it is impossible that efficiency-wage or other productivity stories can explain an absence of disemployment from minimum wage increases. If there are large disemployment effects, efficiency-wage type stories can explain why they are not even larger. But they cannot explain why we don’t see disemployment at all, if that is the case. This might seem like a strong conclusion but it really is unavoidable — and it’s actually quite logical when you think about it. Productivity gains only ameliorate the disemployment effects by moderating the reduction in sales that would otherwise result from higher prices. But they also reduce the number of workers required at any given level sales. The first effect can only dominate when the loss of sales is large, i.e. where there is already a substantial disemployment effect.
Now what if we reintroduce variable profit margins and demand from workers?
Take the derivative of (5) with respect to w (which just gives you the rest of the right hand side) and then with respect to a4, and that tells you how the (dis)employment effect of wage increase varies with the extent to which that increase results in higher productivity. This is:
(7) d(de/dw)/d a4 = [a3(a1 + a2) – 1] / (1 – a2)
For productivity gains to ameliorate rather than exacerbate the disemployment effect, this must be greater than one. Since a2 will always be less than one —affected workers can’t spend more than all of their incremental wages at affected employers — we can ignore the denominator. So we need:
(8) a3(a1 + a2) > 1
Notice that this includes the simple case we considered before, with fixed margins and no important demand from affected workers. Then a3 =1 and a2 = 0 so condition(8) is just that demand for relevant output have price elasticity of at least unity.
In the more general case, we can see that, if demand from the affected workers is important, high productivity is more likely to ameliorate disemployment effects. We have to think carefully, in this case, about how large a2 might plausibly be. Even if we are imaging an across-the-board wage increase in a closed economy, not all of the incremental wages will be spent concurrently produced goods. And in any real-world case, there will also be increased imports, and reduced investment and consumption out of profits. If we are considering one business or a limited sector, a2 will certainly be close to zero. For an economy as a whole,it might be bigger — but it also might be negative. (This is the whole debate about wage-led versus profit-led demand.) Meanwhile, recall that a3 is the share of higher costs that is passed on to prices. (So 1 – a3 gives the fall in profit margins.) Notice that if a3 is small, condition 8 will not be satisfied. In other words, if higher wages come out of profits, then any resulting productivity increases will mean more disemployment, not less.
Again, this initially comes out of the algebra, but if you think about it it makes sense. If costs can’t be passed on to prices, then higher productivity just goes straight to the bottom line. And if costs are not passed on to prices, then lower costs can’t lead to higher sales. All they mean is fewer workers needed for the same output. You might think — I did think, until I worked through this — that “higher wages will mean higher productivity” and “higher wages just come out of profits” reinforce each other as two reasons not to worry about higher wages costing jobs. But actually they are contradictory. Higher productivity only mitigates job losses if higher wages do not come out of profits.
Again, I want to emphasize that this is not a model in the usual economics sense of starting some standard description of individual behavior and then tweaking the assumptions so as to produce the desired results.  Rather, I am simply stating the claims everyone in the debate agrees on, and then asking what their logical implications are. In particular, no one disputes that higher productivity both offsets the effect of higher wages on costs, and reduces the number of workers required to produce a given output. But that doesn’t mean they consistently take both effects into account when they consider the effects of a wage increase.
Higher productivity alone cannot explain why disemployment effects so small. By itself, higher productivity will only ameliorates job loss if a one percent wage increase causes a fall in employment of more than one percent; otherwise, productivity gains will make the job loss worse. If demand from the affected workers is important, then productivity gains can help even if the elasticity is smaller than negative one, but probably not too much smaller. And to the extent cost increases come out of profits rather than being passed through to prices, productivity gains will definitely make job loss worse. These seem like strong conclusions, but they follow logically from premises that are disputed by no one — that higher productivity means that the same output an be produced by few workers, that higher costs may be passed on wholly or impart to higher prices, and that higher prices may result in lower sales.
One specific lesson I take from this is that the observed lack of disemployment from minimum wages must be the result of higher prices and/or lower profit margins, not efficiency wage type effects. In fact, the lack of observed job loss suggests rather strongly that productivity does not rise with higher wages. An interesting question is why people are so attracted to stories where it does.
UPDATE: I should add — I’m not labor economist, I have only a superficial knowledge of this literature; I’m sure someone else has done exactly this exercise and reached the same conclusions. I don’t claim I’m adding anything anything new. I just did this because I was puzzled about myself, and wanted to know under what conditions higher productivity would moderate job losses from higher wages, and under what conditions it would make them worse. And now I do.
 Again, using the fact that percentage changes are approximately equal to log differences.
 I wanted to explicitly justify this statement but the post is too long already.
 In my experience, it’s basically impossible to talk about this kind of thing with a professional economist without them immediately wanting to reframe it as problems optimization under constraints, even if that’s irrelevant to the question at hand. In this case, for instance, if we want to know what the effects of the wage increase are on sales, ,it’s sufficient to know how much the higher wage is passed on to prices, and how strongly sales fall with higher prices. Both these parameters might well be the result of maximization of profits (or some other objective) but you don’t need to know anything about that to answer to question. With respect to employment, what matters is whether the whole cost increase is passed on topics, or some it, or none; the effect will be the exactly same whether or not that fraction is the result of an optimization process.