# A Bare Bones Model of Immigration

Think of a Neoclassical model with inelastic labor supply (including inelastic retirement timing). I am going to focus on the effects that occur in the period of time before capital has had much chance to adjust. To keep the numbers simple, let’s imagine there are 100 million workers in the economy. Production is Cobb-Douglas (see my post “The Shape of Production: Charles Cobb’s and Paul Douglas’s Boon to Economics”), with the following shares:

• 25% share for capital;
• 50% share for skilled going to half the population
• 25% share for unskilled labor going to half the working population.

What are the economic effects of allowing 1 million new workers: enough additional immigration to increase the population by 1%?

Case 1: All of the New Immigration is Unskilled. If all of the new immigration is unskilled, the amount of unskilled labor increases by 2% (from 50 million to 51 million). This increases output by .25 * 2% = .5 %. Unskilled labor gets ¼ of this bigger pie. With a .5% bigger pie and 2% more people to share it among, unskilled workers get a 1.5% reduction in their wage.  The skilled workers get the same share of a bigger pie, with no dilution, so they each get a .5 % increase in their wage. Capital also gets .5% more. This helps people in saving for retirement.

Case 2: All of the New Immigration is Skilled. Now the pie is .5 * 2% = 1% bigger (from 50 million to 51 million). With a 1% bigger pie and 2% more people to share it among, each skilled worker now gets 1% less. Unskilled workers get the same share of a bigger pie, with no dilution, so they each get a 1% increase in their wage. Capital also gets 1% more, which helps people in saving for retirement.

Case 3: 60% of the New Immigration is Skilled, 40% Unskilled. In this case, the amount of skilled labor increases by 1.2% (from 50,000,000 to 50,600,000), while the amount of unskilled labor increases by .8% (from 50,000,000 to 50,400,000). This increases output by (.5 * 1.2%) + (.25 * .8%) = .6% + .2% = .8%. Capital gets .8% more, which helps people in saving for retirement. Since there are .8% more unskilled workers to divide their share of the pie among, the unskilled wage is unaffected. The effect on the skilled wage is .8% - 1.2% = -.4% in this simple model.

Everything left out of this bare bones model suggests that in a richer model, output will ultimately grow more. If the skilled workers are willing to bet that the combination of higher returns to capital and the effects of capital accumulation, extra technological progress, the benefits of increasing returns to scale and diversity, and the share of the national debt and of unfunded government liabilities that immigrants will shoulder would make up for the -.4% reduction in wages they see in the bare bones model, the unskilled workers have no reason to object (except perhaps for cultural reasons), since even in the bare bones model, their wage is unaffected, and all other factors then push in the direction of doing better economically.

I suspect that something like the bare bones model of immigration lurks in the back of many brains, and is encoded as saying that immigration is good for capitalists, but not for workers. Therefore, to successfully argue their case, advocates of immigration must confront this bare bones model head on, explain what is missing, and convincingly argue what the quantitative effects of those missing pieces are.