Explaining the shift for an increase in capital

A classmate asks the following question, which I could not figure out on-the-spot (paraphrasing):

If the capital stock increases by 10%, how do we know that the horizontal magnitude of that shift will be a 10% increase in labor demanded at each quantity (as opposed to a vertical increase of 10% as with a price increase)?

Great question and challenge accepted! But buckle your mathematical safety belts, it’s going to be a bumpy ride…

First, and as we have mentioned before, in Chapters 3 and 4 the book (and most trade models) assume constant returns to scale (CRS). Actually it does so in Chapter 2 as well, but the CRS assumption is trivial there given the even stricter assumption of linear a production function. I’ll use the example of a Cobb-Douglass production to “illustrate.”

Q = L^aK^{1 -a}.

By definition, along any isoquant, output is a constant level of revenue, i.e. Q = Q^*. This also means that along any isoquant, the change in Q is zero, i.e. dQ = 0:

dQ = aL^a-1K^{1 -a}dL + (1 - a)L^aK ^-adK= 0.

Solving for dK/dL, we have:

dL/dK = -(a/1 – a)(K/L).

In other words, the slope of the isoquant is the same for any given capital-labor ratio. The isoquants will look something like this (using a = 0.5 and Q = 1, 4, and 16):

Also, notice that the tangent lines in the graph represent costs of a given number of units of output, i.e. the equation of those lines represent wL + rK = C. Since they are tangent to the curves they also represent cost-minimizing combinations of L and K at a given Q. Additionally, the slopes are equal to the wage-rental ratio.

So, for a given market wage and a given market rental rate (return on capital), at any profit maximizing allocation of labor and capital (which will have to also be cost minimizing for that particular quantity of output), the proportion of capital to labor will be constant (as seen by the straight-line ray from the origin through the points of tangency).

Whew! Is our anonymous student glad to have asked about this yet?!? Anyway, returning to the “bucket,” and assuming CRS, and DK/K = 10%, then if the wage doesn’t change, then to maximize profits, labor in the manufacturing sector will have to increase by 10% (i.e. the quantity of labor demand will be 10% higher at each wage). This means that the shift will be a horizontal shift to the right by 10%.