Having seen a real business cycle in the data you de-trended in PS#2, the next step is to create a cycle in a model world. It turns out that doing so is relatively hard in a model with infinitely lived agents.
There we have to deal with uncertainty, which is fun to do, but it is not that easy as far as the math is concerned. Therefore our model world will have people living for only one period.
In fact, there is just one person each period, but this person has a child that is around in the next period, and so on.
The person, let us call her Jill, cares about consumption ct and the bequest of capital kt+1 she makes to her child, also named Jill. The utility function is:
ln (ct) +Aln(kt+1)
where A > 0 is a parameter. Jill uses the capital she got from her mother to produce consumption ct and investment it , according to the resource constraint:
ct+it+gt=Bkt+et
where B > 0 is a parameter, and et a random shock to the production function. The shock takes different values in different periods. Jill knows et once she is born, so for her it is just a constant. The capital that is left to Jill the daughter is determined by:
kt+1=(1-d)kt+it
where the parameter d, the depreciation rate, is a number between zero and one. This just means that capital tomorrow is what is left over today after depreciation, plus investment.
Compute Jill's decision of consumption and investment as a function of the parameters kt , and et .
If we want to examine the behavior of this model relative to the real world, the next step would be to set the parameters in a way that matches certain features of the real world. Since that is a complicated task, we will give some values to you. B is a scale parameter and does not affect the qualitative behavior of the model.
Therefore we set it to B=0.1. d is the depreciation rate, for which a realistic value is d= .05. A determines the relative size of ct and ktin equilibrium. A rough approximation is A = 4.
In the last step, you will simulate business cycles in the model economy. All you need to know is the capital k1at the beginning of time and the random shocks et. As a starting capital, use k1=3.7. You can generate the random shocks with the random number generator in your spreadsheet.
In Excel, just type "=RAND()", and you will get a uniformly distributed random variable between zero and one. Generate 50 such random numbers, and use your formulas for ctand it and the equation for capital in the next period, kt+1=(1-d)kt+it , to simulate the economy.
Assume g=0 for five periods then g=0.4 for 10 periods and then back to g=0 after that. Plot consumption, investment, government spending, total output and just private sector output each on a spate graph. Calculate the percent change from the initial value (x(1)-x(0))/x(0) for each of the variables.
Are there spending multipliers?
What should the interest rate be equal to?
Plot the interest rate over the 50 periods. Is there crowding out?