## Welfare cost of business cycles - Lucas' formula ## It is straight forward to measure \sigma from available data. Using United States|US data from between 1947 and 2001 Lucas obtained \sigma=.032. It is a little harder to obtain an empirical estimate of \theta; although it should be theoretically possible, a lot of controversies in economics revolve around the precise and appropriate measurement of this parameter. However it is doubtful that \theta is particularly high (most estimates are no higher than 4).
## Welfare cost of business cycles - Lucas' formula ## As an illustrative example consider the case of log utility (see below) in which case \theta=1. In this case the welfare cost of fluctuations is
## Welfare cost of business cycles - Lucas' formula ## In other words eliminating 'all' the fluctuations from a person's consumption path (i.e. eliminating the business cycle entirely) is worth only 1/20 of 1 percent of average annual consumption. For example, an individual who consumes $50,000 worth of goods a year on average would be willing to pay only $25 to eliminate consumption fluctuations.
## Welfare cost of business cycles - Lucas' formula ## The implication is that, if the calculation is correct and appropriate, the ups and downs of the business cycles, the recessions and the booms, hardly matter for individual and possibly social welfare. It is the long run trend of economic growth that is crucial.
## Welfare cost of business cycles - Lucas' formula
## Welfare cost of business cycles - Lucas' formula ## or 1/5 of 1 percent. An individual with average consumption of $50,000 would be willing to pay $100 to eliminate fluctuations. This is still a very small amount compared to the implications of long run growth on income.
## Welfare cost of business cycles - Lucas' formula ## One way to get an upper bound on the degree of risk aversion is to use the Ramsey model of intertemporal savings and consumption. In that case, equilibrium real interest rate is given by
## Welfare cost of business cycles - Lucas' formula ## where r is the real (after tax) rate of return on capital (the real interest rate), \rho is the subjective rate of time preference (which measures impatience) and g is the annual growth rate of consumption
## Welfare cost of business cycles - Lucas' formula
## Welfare cost of business cycles - Mathematical representation and formula ## Lucas sets up an infinitely lived representative agent model where total lifetime utility(U is given by the present discounted value (with \beta representing the Discount factor#Discount factor|discount factor) of per period utilities (u(.)) which in turn depend on consumption in each period (c_t)
## Welfare cost of business cycles - Mathematical representation and formula ## where A is initial consumption and g is the growth rate of consumption (as it turns out neither of these parameters turns out to matter for costs of fluctuations in the baseline model so they can be normalized to 1 and 0 respectively).
## Welfare cost of business cycles - Mathematical representation and formula
## Welfare cost of business cycles - Mathematical representation and formula ## where \sigma is the standard deviation of the natural log of consumption and \epsilon is a random shock which is assumed to be Log-normal distribution|log-normally distributed so that the mean of ln (\epsilon_t) is zero, which in turn implies that the expected value of e^ \epsilon_t is 1 (i.e
## Welfare cost of business cycles - Mathematical representation and formula
## Welfare cost of business cycles - Mathematical representation and formula ## For the case of isoelastic utility, given by
## Welfare cost of business cycles - Mathematical representation and formula ## we can obtain an (Approximation|approximate) closed form solution which has already been given above
## Welfare cost of business cycles - Mathematical representation and formula ## A special case of the above formula occurs if utility is logarithmic, u(c_t)=ln(c_t) which corresponds to the case of \theta=1, which means that the above simplifies to \lambda=.5 \sigma^2. In other words, with log utility the cost of fluctuations is equal to one half the variance of the natural logarithm of consumption.
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