The skill bias of technological change and the evolution of the skill premium in the US since 1970

Notes on the algorithm

As there is very little previous numerical work involving irreversibility constraints [one notable exception being Coleman (1997)], the algorithm is developed from scratch. There are two parts to it: a value function iteration for given production efficiencies and an update mechanism for the guess of the production efficiencies.

Value function iteration with irreversibility constraint, given production efficiencies

At its core is a finite horizon value function iteration that incorporates an irreversibility constraint. I am interested in a period of T=35 years, corresponding to the period from 1970 to 2005. The terminal value is the steady state value towards which the economy converges, given that all values grow at their BGP rates from the final period of interest onwards. Between the last period of interest and the terminal value are m periods at which the production efficiencies grow at their BGP rates and relative labor supply stays constant. m is chosen large enough so that the choice of terminal value does not affect the results in the period of interest, a method known as tatonnement. An assumption implicit in this method is that the model converges to a steady state [see Judd (1998)], hence it is important to ascertain theoretically that a steady state indeed exists and that the model converges to it, to be able to trust the results. My model converges to a balanced growth path with both production efficiency growth rates uniquely determined by the overall growth rate of the economy, and the capital shares β and γ.

Each value function iteration takes as given the (T+m×2)-matrix of skilled and unskilled production efficiencies. Using a vector of n values for unskilled capital, I derive the corresponding values for k_s for which rates of return to capital are equalized each period. I use these values for skilled and unskilled capital to derive the optimal choice of next period’s capital, given each k_s–k_u combination possible in the current period.

As the irreversibility constraint might be binding somewhere along the optimal path, running the value function iteration under the binding constraint is also necessary. For this, a separate grid is introduced for next period’s values of unskilled capital under the binding constraint. Then the value function iteration is used to find the optimal choice of next period’s skilled capital, given that the irreversibility constraint on unskilled capital is binding.

Given the initial values for k_s and k_u, it is straightforward to pick the optimal paths for both types of capital for 35 periods. If the unconstrained optimal choice violates the irreversibility constraint, the optimal value will be that from the constrained choices.

Updating the guess for production efficiency

With the optimal paths for the two types of capital and the values of production efficiency, I calculate skilled and unskilled wages as implied by the model. The next step calculates the difference between these wages and the actual wages as given in the data. If the sum of squared errors implied by this difference is above the convergence criterion specified, the efficiency matrix is updated. The new matrix will be a weighted average of the old matrix used to derive the latest optimal paths for capital and the values for skilled and unskilled production efficiency that would yield the wages from the data, given the paths for capital. The weights on the old and new efficiencies, λ and 1–λ, are chosen such that convergence is as smooth as possible, as giving too much weight to the new solution can lead to overshooting the true solution and slow down convergence (Judd (1998)). Some values of λ may also send the algorithm into an infinite loop that repeats the same few guesses over and over. If this happens, λ is decreased or increased to escape the loop.

With the new efficiency matrix, the value function iteration described above starts again until the sum of squared errors of the model’s wages compared to actual wages is below a critical value. Note that as there is no uncertainty whatsoever in the model, it is possible in principle to arrive at the exact solution, provided the grid on capital is infinitely fine. This would, however, come at a large cost in computing time.

Effects of changing the initial guess of production efficiency, years to terminal value, or steady state growth rates

As mentioned in Section 1.4.2, the initial guess for production efficiency is chosen to be one in the first period and grow at a certain constant rate afterwards. I also run the program for a variety of other growth rates, equal for both production efficiencies and different, constant and changing. The first period production efficiency however stays the same throughout.

Changing the initial growth rate for production efficiency does not affect results. This is true whether the same growth rate is used for all periods or the growth rate varies for blocks of time. As an example, changing the growth rate for the initial guess from 0.025 to 0.08 for five periods yields the same final production efficiency growth rates as the uniform growth rate for the initial guess.

The choice of m, years until terminal value, depends on the number of grid points chosen. The finer the grid, the larger m needs to be for the results to be unaffected by the exact choice of terminal value. Increasing the number of years between the last period of interest and the terminal value beyond the point where results are constant will only lead to increased computing times. Of course, it is important to check that the number of years is large enough in the first place.

The parameter g is the steady state growth rate to which the economy converges eventually. It does not appear in the period of interest, but is needed to provide a terminal value for the value function iteration. It also pins down, together with β and γ, the growth rates of skilled and unskilled production efficiency after T. I assume that in the long run the economy will converge to its past long run growth rate of 2.5%.

As this is only an assumption, it is very desirable that the results do not depend on the exact choice of this value. Decreasing the steady state growth rate by one percentage point, increases both production efficiency growth rates in the last ten periods by less than one percent (not percentage point), while earlier periods are not affected. Since most of the larger differences between the production efficiency growth rates occur before 1990, the main results do not depend on the choice of steady state growth rate.

Table A1 shows the parameter values used for the algorithm. λ gives the weights of the new and old values of production efficiencies for the updated guess and the next value function iteration. Finally, the convergence criterion determines how small the sum of squared differences between the model’s wages and wage data must be for the program to stop updating.

Derivation of capital shares

Tables A2 and A3 show the skill classification of industries. Each industry is classifies as skilled if the average of its 1970 and its 2005 share of hours worked by high skilled labor is at least 20%, and classified as unskilled otherwise.