Welfare cost of business cycles
Encyclopedia
In macroeconomics
, the welfare cost of business cycles refers to the decrease in social welfare, if any, caused by business cycle
fluctuations.
Nobel economist
Robert Lucas
proposed measuring the cost of business cycles as the percentage increase in consumption
that would be necessary to make a representative consumer
indifferent between a smooth, non-fluctuating, consumption trend and one that is subject to business cycles.
Under the assumptions that business cycles represent random shocks around a trend growth path, Robert Lucas
argued that the cost of business cycles is extremely small and as a result the focus of both academic economists and policy makers on economic stabilization policy rather than on long term growth
has been misplaced. Lucas himself, after calculating this cost back in 1987, reoriented his own macroeconomic research program away from the study of short run fluctuations.
However, Lucas' conclusion is controversial. In particular, Keynesian economists typically argue that business cycles should not be understood as fluctuations above and below a trend. Instead, they argue that booms are times when the economy is near its potential output trend, and that recessions are times when the economy is substantially below trend, so that there is a large output gap
. Under this viewpoint, the welfare cost of business cycles is larger, because an economy with cycles not only suffers more variable consumption, but also lower consumption on average.
and the same initial level of consumption – and as a result same level of consumption per period on average
– but with different levels of volatility
, then, according to economic theory, the less volatile consumption path will be preferred to the more volatile one. This is due to Risk aversion
on part of individual agents. One way to calculate how costly this greater volatility is in terms of individual
(or, under some restrictive conditions, social) welfare
is to ask what percentage of her annual average consumption would an individual be willing to sacrifice
in order to eliminate this volatility entirely. Another way to express this is by asking how much an individual with a smooth consumption path would have to be compensated in terms of average consumption in order to accept the volatile path instead of the one without the volatility. The resulting amount of compensation, expressed as a percentage of average annual consumption, is the cost of the fluctuations calculated by Lucas. It is a function of people's degree of risk aversion and of the magnitude of the fluctuations which are to be eliminated, as measured by the standard deviation
of the natural log of consumption.
where
is the cost of fluctuations (the % of average annual consumption that a person would be willing to pay to eliminate all fluctuations in her consumption), 
is the standard deviation of the natural log of consumption and 
measures the degree of risk aversion.
It is straight forward to measure
from available data. Using US
data from between 1947 and 2001 Lucas obtained
. It is a little harder to obtain an empirical estimate of 
; 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 
is particularly high (most estimates are no higher than 4).
As an illustrative example consider the case of log utility (see below) in which case
. In this case the welfare cost of fluctuations is
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.
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.
If
is at the upper range of estimates found in literature, around 4, then
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.
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
where
is the real (after tax) rate of return on capital (the real interest rate), 
is the subjective rate of time preference
(which measures impatience) and
is the annual growth rate of consumption. 
is generally estimated to be around 5% (.05) and the annual growth rate of consumption is about 2% (.02). Then the upper bound on the cost of fluctuations occurs when 
is at its highest, which in this case occurs if 
. This implies that the highest possible degree of risk aversion is
which in turn, combined with estimates given above yields a cost of fluctuations as
which is still extremely small (13% of 1%).
model where total lifetime utility(
is given by the present discounted value (with 
representing the discount factor) of per period utilities (
) which in turn depend on consumption in each period (
)
In the case of the a certain consumption path, consumption in each period is given by
where
is initial consumption and 
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).
In the case of a volatile, uncertain consumption path, consumption in each period is given by
where
is the standard deviation of the natural log of consumption and 
is a random shock which is assumed to be log-normally distributed so that the mean of 
is zero, which in turn implies that the expected value
of
is 1 (i.e. on average, volatile consumption is same as certain consumption). In this case 
is the "compensation parameter" which measures the percentage by which average consumption has to be increased for the consumer to be indifferent between the certain path of consumption and the volatile one. 
is the cost of fluctuations.
We find this cost of fluctuations by setting
and solving for
For the case of Constant Relative Risk Aversion utility, given by
we can obtain an (approximate
) closed form solution which has already been given above
A special case of the above formula occurs if utility is logarithmic,
which corresponds to the case of 
, which means that the above simplifies to 
. In other words, with log utility the cost of fluctuations is equal to one half the variance of the natural logarithm of consumption.
(hence 
and in fact, possibly to Lucas' entire approach is the so-called equity premium puzzle
, first observed by Rajnish Mehra
and Edward Prescott. The analysis above implies that since macroeconomic risk is unimportant, the premium associated with systematic risk
, that is, risk in returns to an asset that is correlated with aggregate consumption should be small (less than 0.5 percentage points for the values of risk aversion considered above). In fact the premium has averaged around six percentage points.
In a survey of the implications of the equity premium, Simon Grant
and John Quiggin
note that 'A high cost of risk means that
recessions are extremely destructive'.
Macroeconomics
Macroeconomics is a branch of economics dealing with the performance, structure, behavior, and decision-making of the whole economy. This includes a national, regional, or global economy...
, the welfare cost of business cycles refers to the decrease in social welfare, if any, caused by business cycle
Business cycle
The term business cycle refers to economy-wide fluctuations in production or economic activity over several months or years...
fluctuations.
Nobel economist
Economist
An economist is a professional in the social science discipline of economics. The individual may also study, develop, and apply theories and concepts from economics and write about economic policy...
Robert Lucas
Robert Lucas, Jr.
Robert Emerson Lucas, Jr. is an American economist at the University of Chicago. He received the Nobel Prize in Economics in 1995 and is consistently indexed among the top 10 economists in the Research Papers in Economics rankings. He is married to economist Nancy Stokey.He received his B.A. in...
proposed measuring the cost of business cycles as the percentage increase in consumption
Consumption (economics)
Consumption is a common concept in economics, and gives rise to derived concepts such as consumer debt. Generally, consumption is defined in part by comparison to production. But the precise definition can vary because different schools of economists define production quite differently...
that would be necessary to make a representative consumer
Representative agent
Economists use the term representative agent to refer to the typical decision-maker of a certain type ....
indifferent between a smooth, non-fluctuating, consumption trend and one that is subject to business cycles.
Under the assumptions that business cycles represent random shocks around a trend growth path, Robert Lucas
Robert Lucas, Jr.
Robert Emerson Lucas, Jr. is an American economist at the University of Chicago. He received the Nobel Prize in Economics in 1995 and is consistently indexed among the top 10 economists in the Research Papers in Economics rankings. He is married to economist Nancy Stokey.He received his B.A. in...
argued that the cost of business cycles is extremely small and as a result the focus of both academic economists and policy makers on economic stabilization policy rather than on long term growth
Economic growth
In economics, economic growth is defined as the increasing capacity of the economy to satisfy the wants of goods and services of the members of society. Economic growth is enabled by increases in productivity, which lowers the inputs for a given amount of output. Lowered costs increase demand...
has been misplaced. Lucas himself, after calculating this cost back in 1987, reoriented his own macroeconomic research program away from the study of short run fluctuations.
However, Lucas' conclusion is controversial. In particular, Keynesian economists typically argue that business cycles should not be understood as fluctuations above and below a trend. Instead, they argue that booms are times when the economy is near its potential output trend, and that recessions are times when the economy is substantially below trend, so that there is a large output gap
Output gap
The GDP gap or the output gap is the difference between potential GDP and actual GDP or actual output. The calculation for the output gap is Y*–Y where Y* is actual output and Y is potential output...
. Under this viewpoint, the welfare cost of business cycles is larger, because an economy with cycles not only suffers more variable consumption, but also lower consumption on average.
Basic intuition
If we consider two consumption paths, each with the same trendTrend estimation
Trend estimation is a statistical technique to aid interpretation of data. When a series of measurements of a process are treated as a time series, trend estimation can be used to make and justify statements about tendencies in the data...
and the same initial level of consumption – and as a result same level of consumption per period on average
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
– but with different levels of volatility
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
, then, according to economic theory, the less volatile consumption path will be preferred to the more volatile one. This is due to Risk aversion
Risk aversion
Risk aversion is a concept in psychology, economics, and finance, based on the behavior of humans while exposed to uncertainty....
on part of individual agents. One way to calculate how costly this greater volatility is in terms of individual
Economic surplus
In mainstream economics, economic surplus refers to two related quantities. Consumer surplus or consumers' surplus is the monetary gain obtained by consumers because they are able to purchase a product for a price that is less than the highest price that they would be willing to pay...
(or, under some restrictive conditions, social) welfare
Social welfare function
In economics, a social welfare function is a real-valued function that ranks conceivable social states from lowest to highest. Inputs of the function include any variables considered to affect the economic welfare of a society...
is to ask what percentage of her annual average consumption would an individual be willing to sacrifice
Willingness to pay
In economics, the willingness to pay is the maximum amount a person would be willing to pay, sacrifice or exchange in order to receive a good or to avoid something undesired, such as pollution...
in order to eliminate this volatility entirely. Another way to express this is by asking how much an individual with a smooth consumption path would have to be compensated in terms of average consumption in order to accept the volatile path instead of the one without the volatility. The resulting amount of compensation, expressed as a percentage of average annual consumption, is the cost of the fluctuations calculated by Lucas. It is a function of people's degree of risk aversion and of the magnitude of the fluctuations which are to be eliminated, as measured by the standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
of the natural log of consumption.
Lucas' formula
Robert Lucas' baseline formula for the welfare cost of business cycles is given by (see mathematical derivation below):where
is the cost of fluctuations (the % of average annual consumption that a person would be willing to pay to eliminate all fluctuations in her consumption), is the standard deviation of the natural log of consumption and measures the degree of risk aversion.It is straight forward to measure
from available data. Using USUnited States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
data from between 1947 and 2001 Lucas obtained
. It is a little harder to obtain an empirical estimate of ; 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 is particularly high (most estimates are no higher than 4).As an illustrative example consider the case of log utility (see below) in which case
. In this case the welfare cost of fluctuations isIn 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.
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.
If
is at the upper range of estimates found in literature, around 4, thenor 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.
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
where
is the real (after tax) rate of return on capital (the real interest rate), is the subjective rate of time preferenceTime preference
In economics, time preference pertains to how large a premium a consumer places on enjoyment nearer in time over more remote enjoyment....
(which measures impatience) and
is the annual growth rate of consumption. is generally estimated to be around 5% (.05) and the annual growth rate of consumption is about 2% (.02). Then the upper bound on the cost of fluctuations occurs when is at its highest, which in this case occurs if . This implies that the highest possible degree of risk aversion iswhich in turn, combined with estimates given above yields a cost of fluctuations as
which is still extremely small (13% of 1%).
Mathematical representation and formula
Lucas sets up an infinitely lived representative agentRepresentative agent
Economists use the term representative agent to refer to the typical decision-maker of a certain type ....
model where total lifetime utility(
is given by the present discounted value (with representing the discount factor) of per period utilities () which in turn depend on consumption in each period ()In the case of the a certain consumption path, consumption in each period is given by
where
is initial consumption and 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).In the case of a volatile, uncertain consumption path, consumption in each period is given by
where
is the standard deviation of the natural log of consumption and is a random shock which is assumed to be log-normally distributed so that the mean of is zero, which in turn implies that the expected valueExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of
is 1 (i.e. on average, volatile consumption is same as certain consumption). In this case is the "compensation parameter" which measures the percentage by which average consumption has to be increased for the consumer to be indifferent between the certain path of consumption and the volatile one. is the cost of fluctuations.We find this cost of fluctuations by setting
and solving for
For the case of Constant Relative Risk Aversion utility, given by
we can obtain an (approximate
Approximation
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...
) closed form solution which has already been given above
A special case of the above formula occurs if utility is logarithmic,
which corresponds to the case of , which means that the above simplifies to . In other words, with log utility the cost of fluctuations is equal to one half the variance of the natural logarithm of consumption.Risk Aversion and the Equity Premium Puzzle
However, a major problem related to the above way of estimating (hence and in fact, possibly to Lucas' entire approach is the so-called equity premium puzzleEquity premium puzzle
The equity premium puzzle is a term coined in 1985 by economists Rajnish Mehra and Edward C. Prescott. It is based on the observation that in order to reconcile the much higher returns of stocks compared to government bonds in the United States, individuals must have implausibly high risk aversion...
, first observed by Rajnish Mehra
Rajnish Mehra
Rajnish Mehra is an Indian American economist. He currently holds the E. N. Basha Chair at Arizona State University and is a research associate of the NBER...
and Edward Prescott. The analysis above implies that since macroeconomic risk is unimportant, the premium associated with systematic risk
Systematic risk
In finance, systematic risk, sometimes called market risk, aggregate risk, or undiversifiable risk, is the risk associated with aggregate market returns....
, that is, risk in returns to an asset that is correlated with aggregate consumption should be small (less than 0.5 percentage points for the values of risk aversion considered above). In fact the premium has averaged around six percentage points.
In a survey of the implications of the equity premium, Simon Grant
Simon Grant
Simon Grant in Falmouth, Cornwall is a British television presenter and actor.-Biography:Before becoming a CBBC presenter, he studied acting at Middlesex University....
and John Quiggin
John Quiggin
John Quiggin is an Australian economist and professor at the University of Queensland. Quiggin studied at the Australian National University, obtaining bachelor's degrees in Arts and Economics in 1978 and 1980 respectively, and completing a master's degree in Economics in 1984. Quiggin was awarded...
note that 'A high cost of risk means that
recessions are extremely destructive'.