Consider the following macroeconomic model for real business cycles taken from Dejong and Dave (2011) Chapter 3: Suppose that
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Description
1. Real business cycles:
Consider the following macroeconomic model for real business cycles taken from Dejong and Dave (2011) Chapter 3: Suppose that
we have a representative household whose goal is to maximize their utility from consumption and leisure over time:
where E0 is the expected value at time 0, β is a parameter reflecting the household’s discount rate (time preferences), ct is the household’s consumption at time t, and lt is the household’s leisure time at time t.
For this problem to work, we have to introduce four constraints. The first constraint is that the household
produces goods using a Cobb-Douglas production function. That is,
where yt represents the GDP at time t, zt represents and exogenous (i.e., cannot be optimized or changed)
technology stock at time t, kt represents the gross capital stock at time t, and nt represents the amount of
labor supplied at time t. The second constraint is that total time spent working and at leisure should add to
some fixed constant. Without loss of generality we can define the total to be one (1):
