# More on misery: how consistent are alternative indices? A comment.

A series of comments in this journal (cited below) addresses the
appropriateness of the "misery index," attributed to the late
economist Arthur Okun, which is popularly reported as a general
indicator of the state of health of the macro-economy. The index is the
sum of the absolute value of the inflation rate(1) and the aggregate
rate of unemployment in the economy:

|M.sub.1~ = /i/ + U. (1)

Any unemployment above zero adds to the index given by. Golden, Orescovich and Ostafin (1987) point out that the "best" level of unemployment may not be zero. Since a natural unemployment rate of 4-6% is considered normal and even desirable by many economists, it follows that only rates in excess of the normal level should augment the index. A reformulation of (1) that recognizes natural unemployment is

M|prime~ = /i/ + /U - |U.sub.n~/, (2)

where |U.sub.n~ is the natural rate of unemployment.

Since no index of aggregate well-being can be meaningful in other than an ordinal sense, it is worth noting that the number attached to such an index serves only to rank levels of welfare in different time periods, but not to determine the amount or degree of such welfare differences. In view of this, |M.sub.1~ and M|prime~ are equivalent for U |is greater than~ |U.sub.n~, over which range the one is merely a linear transformation of the other by an additive constant. For U |is less than~ |U.sub.n~, |M.sub.1~ and M|prime~ are not equivalent: as unemployment declines below |U.sub.n~, M|prime~ increases, indicating greater misery. Support for this might be based on the view that excessively low unemployment implies non-optimally short job search periods, which will waste resources over the longer run due to inappropriate use. The contrary view would emphasize the desirability of lower unemployment in the near term as an alternative to the resource waste associated with higher unemployment. Thus, the issue is an empirical one of the relative short- versus long-term costs and benefits of below normal unemployment. The approach taken here will be, where relevant, to keep the alternative indices invariant for changes in the unemployment rate less than the natural rate. The modification of (2) that does this is

|Mathematical Expression Omitted~

Both |M.sub.1~ and |M.sub.2~ implicitly assume a constant, unitary rate of substitution between the inflation and unemployment rates, i.e., the index is unchanged when a change in one rate is matched by an oppositely-signed change of equal magnitude in the other. This has drawn comment from Zaleski (1990), who states that a constant rate of substitution is "not true," and recommends a "new misery index,"

|M.sub.3~ = |i.sup.2~ + |U.sup.2~, (4)

which is the equation for a circle centered at the origin. This form incorporates the plausible assumption(2) that the rate of substitution between the two "goods," low inflation and low unemployment, declines. Accordingly, when unemployment exceeds inflation a given percentage reduction in unemployment requires a larger percentage increase in inflation to keep the index unchanged, and conversely.

In a reply to Zaleski, it is noted by Golden, et al (1990) that the formulation given by (4) again fails to account for natural unemployment. Their objection can be accommodated by modifying |M.sub.3~ to center the indifference curves at i = 0 and U = |U.sub.n~. Incorporating comments above relating to unemployment below the natural rate gives

|Mathematical Expression Omitted~

There is evidence from election returns (See Smyth and Dua, 1989) that people view high rates of unemployment as more undesirable than high inflation rates. A variant offered here which incorporates this feature is

|Mathematical Expression Omitted~

where K is a constant greater than unity. This gives indifference curves which are elliptical and symmetrical about i = 0 and U = |U.sub.n~, with foci on U = |U.sub.n~. The absolute values of the curves' i-axis intercepts are equal to K times the excess of their U-axis intercepts over |U.sub.n~.

Figure 1 depicts indifference maps associated with the formulations in (1) and (3) through (6). Indices |M.sub.3~, |M.sub.4~ and |M.sub.5~--but not |M.sub.2~--can potentially yield rankings different from M1 at any value of U. Also, whereas |M.sub.1~ and |M.sub.3~ treat an unemployment reduction below Un as welfare-increasing, |M.sub.2~, |M.sub.4~ and |M.sub.5~ treat it neutrally.

Here, as a test for consistency of the various possible measures, the formulations in (3) through (6) were each compared with (1). It is commonly observed that the natural rate of unemployment seems to have been rising in recent decades, due among other things to the changing age and sex composition of the labor force. Indeed, this appears to be the case, with the unemployment rate during successive high-employment booms ratcheting upwards. For operational purposes, the value of |U.sub.n~ was allowed to trend upward, and was estimated visually from a graphical display of the unemployment rate over the period of the study. This resulted in an estimated value that is constant at 3 percent from 1930 to 1952, and then increases linearly to a value of 5 percent in 1989. A value of 2 was assigned to K in |M.sub.5~, giving the indifference curves i-axis intercepts twice as large as their U = |U.sub.n~ intercepts. This is consistent with the estimates of Smyth and Dua based on election data.(3)

As noted, each of the alternative indices was compared with the standard misery index |M.sub.1~. To this end, pairwise comparisons were made between individual years in the study period for each of the alternative indices. The relative index values of the two years being compared were then compared with the relative values assigned by |M.sub.1~. For example, if |M.sub.5~ gives a lower value for 1981 than 1936 and |M.sub.1~ does not, a difference in rankings is noted. For each index, there are 1770 pairwise annual comparisons within the set of 60 years. Table 1 gives the number and percentage of cases in which such comparisons differed from the ordering assigned by |M.sub.1~.

Much of the interest in the misery index focuses on year-to-year changes in its value. Of the total number of pairwise comparisons in Table 1, a subset of 59 are for sequential years in the period. For this set, Table 2 gives the number and percentage of differences in ordering for each of the alternative indexes as compared to |M.sub.1~. These are cases where the index in question indicates either an increase or decrease from the prior year and |M.sub.1~ indicates the reverse or no change.

The results rather strongly suggest that as a practical matter there is not a compelling reason to use an index other than the standard Okun measure. The relatively simple standard misery index gives the same ranking as the most "sophisticated," M5, more than 90 percent of the time. For annual changes, the performance is less consistent, but not greatly so. There is no apparent reason why two indices should be less consistent for sequential years than for all years, so one might expect that Table 1 values, taken from a larger sample, would better reflect the long term future sequential performances than Table 2.

Obviously, the effort that one cares to put into constructing a misery index is related to the importance that is attached to exactitude of the measurements and comparisons to be made. The simplicity of the standard index, its ease of interpretation by the lay public, and its use by the media as a somewhat crude index only, all argue for its usefulness over alternatives. Its empirical consistency with the alternatives reinforces this position.

Notes

1. The basis of this is the view that the social loss from inflation or deflation is due to higher variability in the rate as it deviates from zero (Mankiev, p. 169.) Correctly planning for the future becomes more difficult, and mistakes and ancillary inefficiencies in resource use therefore become greater. All the indices discussed here are treated accordingly.

2. "Plausible" rather than "true" is used here in light of the sketchy evidence, which comes from election return data. The diminishing marginal rate of substitution of microeconomic theory applies to a single consuming unit, rather than society at large, and even then is invariably regarded as an assumption.

3. These authors fitted a parabola having a trade-off similar to the elliptical form used here.

References

Golden, J. M., Orescovich, Robert, and Ostafin, David. "Optimality on the Short-Run Phillips Curve: A "Misery Index" Criterion, A Note," The American Economist, 31 (Fall 1987), 72.

-----. "Optimality on the Short-Run Phillips Curve: A "Misery Index" Criterion, A Reply," The American Economist, 34 (Fall 1990), 92.

Mankiw, N. Gregory. Macroeconomics, Worth Publishers, 1992.

Smyth, D. J., and Dua, P. "The Public's Indifference Map between Inflation and Unemployment: Empirical Evidence from the Nixon, Ford, Carter and Reagan Presidencies," Public Choice, 60 (January 1989), 71-85.

Zaleski, Peter A. "On the Optimal Level of Macroeconomic Misery: A Comment." The American Economist, 34 (Fall 1990), 90-91.

Associate Professor of Economics, School of Business Administration, Gonzaga University. Helpful suggestions were made by John Beck and Randall Bennett.

|M.sub.1~ = /i/ + U. (1)

Any unemployment above zero adds to the index given by. Golden, Orescovich and Ostafin (1987) point out that the "best" level of unemployment may not be zero. Since a natural unemployment rate of 4-6% is considered normal and even desirable by many economists, it follows that only rates in excess of the normal level should augment the index. A reformulation of (1) that recognizes natural unemployment is

M|prime~ = /i/ + /U - |U.sub.n~/, (2)

where |U.sub.n~ is the natural rate of unemployment.

Since no index of aggregate well-being can be meaningful in other than an ordinal sense, it is worth noting that the number attached to such an index serves only to rank levels of welfare in different time periods, but not to determine the amount or degree of such welfare differences. In view of this, |M.sub.1~ and M|prime~ are equivalent for U |is greater than~ |U.sub.n~, over which range the one is merely a linear transformation of the other by an additive constant. For U |is less than~ |U.sub.n~, |M.sub.1~ and M|prime~ are not equivalent: as unemployment declines below |U.sub.n~, M|prime~ increases, indicating greater misery. Support for this might be based on the view that excessively low unemployment implies non-optimally short job search periods, which will waste resources over the longer run due to inappropriate use. The contrary view would emphasize the desirability of lower unemployment in the near term as an alternative to the resource waste associated with higher unemployment. Thus, the issue is an empirical one of the relative short- versus long-term costs and benefits of below normal unemployment. The approach taken here will be, where relevant, to keep the alternative indices invariant for changes in the unemployment rate less than the natural rate. The modification of (2) that does this is

|Mathematical Expression Omitted~

Both |M.sub.1~ and |M.sub.2~ implicitly assume a constant, unitary rate of substitution between the inflation and unemployment rates, i.e., the index is unchanged when a change in one rate is matched by an oppositely-signed change of equal magnitude in the other. This has drawn comment from Zaleski (1990), who states that a constant rate of substitution is "not true," and recommends a "new misery index,"

|M.sub.3~ = |i.sup.2~ + |U.sup.2~, (4)

which is the equation for a circle centered at the origin. This form incorporates the plausible assumption(2) that the rate of substitution between the two "goods," low inflation and low unemployment, declines. Accordingly, when unemployment exceeds inflation a given percentage reduction in unemployment requires a larger percentage increase in inflation to keep the index unchanged, and conversely.

In a reply to Zaleski, it is noted by Golden, et al (1990) that the formulation given by (4) again fails to account for natural unemployment. Their objection can be accommodated by modifying |M.sub.3~ to center the indifference curves at i = 0 and U = |U.sub.n~. Incorporating comments above relating to unemployment below the natural rate gives

|Mathematical Expression Omitted~

There is evidence from election returns (See Smyth and Dua, 1989) that people view high rates of unemployment as more undesirable than high inflation rates. A variant offered here which incorporates this feature is

|Mathematical Expression Omitted~

where K is a constant greater than unity. This gives indifference curves which are elliptical and symmetrical about i = 0 and U = |U.sub.n~, with foci on U = |U.sub.n~. The absolute values of the curves' i-axis intercepts are equal to K times the excess of their U-axis intercepts over |U.sub.n~.

Figure 1 depicts indifference maps associated with the formulations in (1) and (3) through (6). Indices |M.sub.3~, |M.sub.4~ and |M.sub.5~--but not |M.sub.2~--can potentially yield rankings different from M1 at any value of U. Also, whereas |M.sub.1~ and |M.sub.3~ treat an unemployment reduction below Un as welfare-increasing, |M.sub.2~, |M.sub.4~ and |M.sub.5~ treat it neutrally.

Here, as a test for consistency of the various possible measures, the formulations in (3) through (6) were each compared with (1). It is commonly observed that the natural rate of unemployment seems to have been rising in recent decades, due among other things to the changing age and sex composition of the labor force. Indeed, this appears to be the case, with the unemployment rate during successive high-employment booms ratcheting upwards. For operational purposes, the value of |U.sub.n~ was allowed to trend upward, and was estimated visually from a graphical display of the unemployment rate over the period of the study. This resulted in an estimated value that is constant at 3 percent from 1930 to 1952, and then increases linearly to a value of 5 percent in 1989. A value of 2 was assigned to K in |M.sub.5~, giving the indifference curves i-axis intercepts twice as large as their U = |U.sub.n~ intercepts. This is consistent with the estimates of Smyth and Dua based on election data.(3)

As noted, each of the alternative indices was compared with the standard misery index |M.sub.1~. To this end, pairwise comparisons were made between individual years in the study period for each of the alternative indices. The relative index values of the two years being compared were then compared with the relative values assigned by |M.sub.1~. For example, if |M.sub.5~ gives a lower value for 1981 than 1936 and |M.sub.1~ does not, a difference in rankings is noted. For each index, there are 1770 pairwise annual comparisons within the set of 60 years. Table 1 gives the number and percentage of cases in which such comparisons differed from the ordering assigned by |M.sub.1~.

TABLE 1 Number and percentage of differences in annual pairwise rankings of alternative misery indexes as compared with standard index, /i/ + U Number of Ranking Percentage of Total Index Differences Annual Pairs |M.sub.2~ 59 3.33% |M.sub.3~ 81 4.58% |M.sub.4~ 124 7.01% |M.sub.5~ 155 8.76%

Much of the interest in the misery index focuses on year-to-year changes in its value. Of the total number of pairwise comparisons in Table 1, a subset of 59 are for sequential years in the period. For this set, Table 2 gives the number and percentage of differences in ordering for each of the alternative indexes as compared to |M.sub.1~. These are cases where the index in question indicates either an increase or decrease from the prior year and |M.sub.1~ indicates the reverse or no change.

TABLE 2 Number and percentage of differences in sequential pairwise rankings of alternative misery indexes as compared with standard index, /i/ + U Number of Ranking Percentage of Total Index Differences Sequential Pairs |M.sub.2~ 0 0.00% |M.sub.3~ 6 10.16% |M.sub.4~ 8 13.55% |M.sub.5~ 7 11.86%

The results rather strongly suggest that as a practical matter there is not a compelling reason to use an index other than the standard Okun measure. The relatively simple standard misery index gives the same ranking as the most "sophisticated," M5, more than 90 percent of the time. For annual changes, the performance is less consistent, but not greatly so. There is no apparent reason why two indices should be less consistent for sequential years than for all years, so one might expect that Table 1 values, taken from a larger sample, would better reflect the long term future sequential performances than Table 2.

Obviously, the effort that one cares to put into constructing a misery index is related to the importance that is attached to exactitude of the measurements and comparisons to be made. The simplicity of the standard index, its ease of interpretation by the lay public, and its use by the media as a somewhat crude index only, all argue for its usefulness over alternatives. Its empirical consistency with the alternatives reinforces this position.

Notes

1. The basis of this is the view that the social loss from inflation or deflation is due to higher variability in the rate as it deviates from zero (Mankiev, p. 169.) Correctly planning for the future becomes more difficult, and mistakes and ancillary inefficiencies in resource use therefore become greater. All the indices discussed here are treated accordingly.

2. "Plausible" rather than "true" is used here in light of the sketchy evidence, which comes from election return data. The diminishing marginal rate of substitution of microeconomic theory applies to a single consuming unit, rather than society at large, and even then is invariably regarded as an assumption.

3. These authors fitted a parabola having a trade-off similar to the elliptical form used here.

References

Golden, J. M., Orescovich, Robert, and Ostafin, David. "Optimality on the Short-Run Phillips Curve: A "Misery Index" Criterion, A Note," The American Economist, 31 (Fall 1987), 72.

-----. "Optimality on the Short-Run Phillips Curve: A "Misery Index" Criterion, A Reply," The American Economist, 34 (Fall 1990), 92.

Mankiw, N. Gregory. Macroeconomics, Worth Publishers, 1992.

Smyth, D. J., and Dua, P. "The Public's Indifference Map between Inflation and Unemployment: Empirical Evidence from the Nixon, Ford, Carter and Reagan Presidencies," Public Choice, 60 (January 1989), 71-85.

Zaleski, Peter A. "On the Optimal Level of Macroeconomic Misery: A Comment." The American Economist, 34 (Fall 1990), 90-91.

Associate Professor of Economics, School of Business Administration, Gonzaga University. Helpful suggestions were made by John Beck and Randall Bennett.

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Author: | Wiseman, Clark |
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Publication: | American Economist |

Date: | Sep 22, 1992 |

Words: | 1672 |

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