After contemplating for a couple of days the embarrassment of showing up at the conference with no paper, it suddenly struck us that this same procedure used to solve for steady states might work in solving for the economy’s exact transition path. In the case of the solving for the transition path one would guess the time-path of the amount of capital relative to labor demanded by firms, use this path to calculate a time-path of factor payments, and then use this time-path of factor payments to determine a time-path of the supply of capital relative to labor. If one found a time-path of demands of capital relative to labor that equaled the time-path of the supply of these factors, one would have calculated a dynamic equilibrium.
At this point, we had two questions left to worry about: How should we handle the fact that our economy had no terminal period? And, would our procedure of solving for the transition path converge? To deal with the terminal period issue, we decided to assume that the economy reached a steady state at a date that was sufficiently far off in the future that the economy would, indeed, have plenty of time to reach that steady state. Thus, in forming our initial guess of the time-path of the demand for capital relative to labor and in updating this guess, we assumed that the value of this ratio was constant from year 150 onward; i.e., we guessed the same value of this ratio for years after 150 as for year 150.
To deal with the second question — Would our procedure converge? — we simply had to try it. In those days (the Winter of 1980s), computer technology had improved, so one no longer had to punch out computer cards, but could instead use a terminal that looked like a large typewriter and had no screen. Your results came back by way of the terminal typing them out on paper. Even better, one could run this from home via a big modem into which one stuck the terminal’s phone. Alan and I spent a couple of days writing a prototype program in my Los Angeles apartment. We hadn’t debugged it by the time Alan had to head back East. So for the next couple of days I sat in front of this machine trying to remove the final bugs and see if we could get the right sequence of numbers to come out.
What we were printing out was our successive time-paths of the capital-labor ratio and the difference between the values of the current time-path and the previous one. To me it seemed that we were looking at time-lapsed photographs of a wiggly snake. Each successive picture of the snake (of the time-path of the capital-labor ratio) showed different wiggles, with the question being would the successive snakes start flattening out — start lining up on top of each other. After fixing what I thought had to be our final programming bug, I sat back to watch the snake and sure enough its position started to flatten out and, what’s more, become more flat after each iteration.
When the snake had flattened at all points (at all time periods) to a degree – at was well within our convergence criteria and I saw that convergence was improving across the snake’s entire length in each iteration, I removed the phone from the modem and called Alan. He wasn’t home, but I left a message with his wife (there were no answering machines back then) that “The snake has flattened.” This was an historic moment for us because we realized two things: first, we’d have a paper for the conference and second, we’d probably get tenure.