Malthusian Simulation

The model in the simulation is as follows:

Per capita income at time t is



where A is land/technology, L is amount of labor and alpha<1. A grows at the rate g (possibly 0) so



Growth of population is births minus deaths or



where f is the crude birth rate (CBR) and m/m+y is the crude death rate. So here we're assuming that the fertility rate is independent of income (which isn't too much at odds with empirical evidence) and the mortality rate is a declining function of income. If per capita income is zero, everybody dies and if per capita income income is very very large then very very few people die. Each period.

That's pretty much the model and if we got numbers for the parameters we can simulate it. Some parameters, like the fertility rate we can get more or less from the data. Right at the start we can pick a value for alpha, or labor's share in output. Others we can pin down by using observed data and assuming that at particular points in time the economy is at its long run steady state. To do that we need to solve for it:

The change in per capita income from one period to next is given by






If technological progress is slow enough relative to rate at which the population adjustment takes place (whole another post - but the condition will be evident in a second) then in steady state per capita income will be constant even with on going technological progress. So dy=0, or



or winding that puppy backwards



where L(0) is the initial level of population (assuming we start out in steady state) and L(t) is our final level of population (assuming we end in steady state). So from that we can calculate/calibrate the annual growth rate of technological progress as



Then we calculate the steady state level of per capita income from the population growth equation:



solving for y (and dropping the time sub script since this is steady state per capita income) we have



where



from which you can also see the necessary condition for the steady state to exist (if g is too high then you will get ever increasing income) and with a bit of thought can also figure out the approximate rate of convergence to the steady state.

From this, if we know the steady state income at some point in time (basically you want to pick to points in time where income is the same and which are far enough apart so that any kind of transitory shocks, like, um, the Black Death, have time to sort themselves out) we can calibrate m, the mortality rate parameter. Alternatively if we observe a mortality level in what we think is a steady state year than we can calibrate the income. In practice the former turns out to be easier. In particular if we only care about the magnitude of changes rather the magnitude of levels (as in the previous post) we can just "normalize" the steady state level of income to 1 and see how much of a rise we can get in % terms with various shocks.

Then all that's left is to calibrate the initial level of technology which is easy if we have steady state income:



And that's pretty much it.

The nature of this calibration excercise and the model is that if we shock the model (by, say, killing off a third of population of England in 1350) and then leave it alone then it will be pretty good in matching the population data.



But this is pretty much by construction so it doesn't really constitute a "test" of the model. The problem, as noted in the post below, is that it is pretty much impossible to match the 100% increase in per capita incomes (or wages, but that might be even worse) that occurred between 1350 and 1480. Here's what it looks like with just an initial shock of wiping out a third of the population:



The fact that in the model the shock has an immediate impact on incomes while this doesn't happen until 130 years later historically is not that important. You know, sticky wages, market frictions and all that, means that we shouldn't expect an immediate adjustment. The important aspect is that at its peak income rises only 14.5% above its steady state level, rather than the 100% seen in the data.

Like I said below, you can try to mess with it in order to get those higher incomes;
1. Increase mortality rate or decrease the fertility rate. But then you'll screw up that nice match up of population data in the above graph, winding up with way too few people in 1570 and on.
2. Increase the technological growth rate. But then you'll screw up that nice match up of population data in the above graph, winding up with way too many people in 1570 and on.

Of course you can try a combination of these to try to match it up. Or play with the other parameters. That's why here is the Excel file used for the simple simulations (Thanks to Gabriel).