WageScaling <- function(n = 1, K = 1, m = 2, L = 1, a = 1/3, A = 1, p = 1, rnd = 3){
NewWage <- (1 - a) * A * ((n * K)/(m * L))^a
RealWage <- NewWage/p
cat(paste0("With a labor share of production of ", round(a * 100, rnd), "%, increasing the labor supply by a factor of ", m, " and multiplying the capital supply by ", n, ", wages adjust to ", round(RealWage * 100, rnd), "% of economic output. In other words, they scale to ", round(((((n * K)/(m * L))^a) * 100)/p, rnd), "%.\n"))}To answer the title question, probably not. In the Cobb-Douglas production function (Cobb & Douglas, 1928),
\[Y = AK^{\alpha}L^{1-\alpha}\]
where Y, is total production or GDP, A is total factor productivity, L and K are the total factors of Labor and Capital, respectively, \(\alpha\) is capital’s share of production, and \(\beta\), or \(1-\alpha\) is labor’s share. The real wage is given by
\[\dfrac{\partial Y}{\partial L} = \frac{w}{p}\]
where w and p are the nominal wage and price level. And the real rent is given by
\[\dfrac{\partial Y}{\partial K} = \frac{r}{p}\]
where r is nominal rent. With these quantities in hand, it can be seen that profit is maximized when the marginal product of labor is equal to its marginal cost (i.e., wages) and the marginal product of capital is equal to its marginal cost (i.e., rent). Thus, where m is the multiplier,
\[\dfrac{\partial Y}{\partial L} = (1 - \alpha)A(\frac{K}{L})^{\alpha}\]
and
\[\dfrac{\partial Y}{\partial K} = \alpha A(\frac{L}{K})^{1 - \alpha}\]
With those equations, a constant price level, a constant capital share of, say, \(\frac{1}{3}\), and a constant TFP, we can estimate the effect of rapidly expanding the labor supply on wages. This is because wages scale by the proportion of added labor, m, when you have a fixed capital stock (or scaling by n). This goes by the amount added where 2L in the original wage equation is a doubling of the labor supply.
\[W_{\text{new}} = (1 - \alpha)A(\frac{nK}{mL})\]
This is equivalent to testing the effect of adding all women at once to the workforce, assuming equal productivity with men, no effect on men’s own productivity, and that men do not use women in their household who work as substitutes for their own labor.
WageScaling(m = .5) #Halving## With a labor share of production of 33.333%, increasing the labor supply by a factor of 0.5 and multiplying the capital supply by 1, wages adjust to 83.995% of economic output. In other words, they scale to 125.992%.WageScaling(m = 1) #No change## With a labor share of production of 33.333%, increasing the labor supply by a factor of 1 and multiplying the capital supply by 1, wages adjust to 66.667% of economic output. In other words, they scale to 100%.WageScaling() #Doubling## With a labor share of production of 33.333%, increasing the labor supply by a factor of 2 and multiplying the capital supply by 1, wages adjust to 52.913% of economic output. In other words, they scale to 79.37%.WageScaling(m = 3) #Tripling## With a labor share of production of 33.333%, increasing the labor supply by a factor of 3 and multiplying the capital supply by 1, wages adjust to 46.224% of economic output. In other words, they scale to 69.336%.From the above examples, it’s apparent that halving the labor supply immediately would increase wages for the remainder. But that assumes the economy remains the same size, and that is unrealistic. If we hold everything constant, GDP declines equal to the decline in the wage share in the doubling scenario, and in the other direction, it grows when we add labor. In other words, with a halving of the labor force GDP becomes roughly 80% as large.
K = 1; L = 1; a = 2/3; m = .5; M = 2
Y_Before = K^(a)*L^(1-a)
Y_After = K^(a)*(m * L)^(1-a)
Y_Double = K^(a)*(M * L)^(1-a)
Y_Before; Y_After; Y_Double #Total production before versus after removing half the workforce and workforce doubling## [1] 1## [1] 0.7937005## [1] 1.259921Y_After * 1.259921 #Wages in a neutrally shrunken economy## [1] 1But this only works with overly generous assumptions. If half of labor was removed, people would have to engage in less efficient labor, and consumption would also fall. Wages would be a larger share of a smaller economy and under a generous set of assumptions, wages wouldn’t change for the people who remained at work. An alternative in the scenario where the removed half is women is that they would thus go to work in the home, where they might have a comparative advantage over men. But it’s extremely doubtful that labor in the home actually contributes as much to the economy as a job and of course, all women are not married or parents, or whatever else would be needed to make this scenario work.
This scenario where wages decrease by about 20% by doubling the labor supply is unrealistic for more reasons than that it leaves TFP unaffected. In all likelihood, it would greatly increase productivity, while also causing rapid K/L convergence, but more pointedly, women did not enter the labor force in one moment. It is more likely that women never depressed wages, because they never entered the workforce rapidly enough to lower wages. In the real world, women entering the workforce increased wages (Weinstein, 2017).
The results I have described here are replicable with the Solow-Swan model and they likewise apply to immigrants but typically with caveats about productivity. That is, a one-time population shock has an effect that, when traced, becomes pretty benign. Wages fall in the short-term because there’s less capital per worker, but wages return to the level path very quickly if you have constant aggregate returns to scale. If you have improved productivity, as you likely would, you return to the level path faster and, realistically, we’re all wealthier for it.
Cobb, C. W., & Douglas, P. H. (1928). A Theory of Production. The American Economic Review, 18(1), 139–165.
Weinstein, A. L. (2017). Working women in the city and urban wage growth in the United States. Journal of Regional Science, 57(4), 591–610. https://doi.org/10.1111/jors.12336