Edward Renshaw
Professor of Economics
State University of New York at Albany
The notion that there is an inverse relationship between investment and the rate of interest is an old idea that was extensively discussed by Fisher in (1907 and 1930) before it was incorporated into the General Theory by Keynes and made a central part of the IS-LM framework by Hicks. One of the problems with interest rates, from a modeling perspective, is that they seem to explain either too much or too little with regard to what has happened to the U.S. economy.
The epitome of a model that probably explains too much is the prime rate formula invented by the Center for International Business Cycle Research. They has found that the duration of post World War II expansions in economic activity have been about equal to 15 months plus 1.57 times the monthly lag for a secular rise in the prime rate after months containing an NBER trough in business activity. See Table 5.1.
Why this formula worked so well from 1945-90, is mystery. One would hope that it doesn't represent an immutable law of nature and that the current business expansion won't come to an end in January 1997. Even if it were to break down and greatly underestimate the staying power of this business expansion, however, the CIBCR formula could still serve a useful purpose in providing a time tested standard with which to assess the success of the Fed's efforts to prevent an acceleration of the inflation rate without tipping the economy into another recession.
Verifying an inverse relationship between investment and changes in interest rates, though, has turned out to be a frustrating endeavor-- especially with regard to expenditures for new plant and equipment by private sector business enterprises.
In their (1970) attempt to develop "A Monetarist Model for Economic Stabilization"--what is often referred to as the St. Louis model--Andersen and Carlson do have an equation that endeavors to explain the yield on Aa corporate bonds but do not include an investment function in their model.
In (1984) Ray Fair, a professor at Yale University, documented a small scale econometric forecasting model that has been quite successful at predicting year-year changes in real GNP. The Fair model does have an equation for plant and equipment. His list of explanatory variables, however, doesn't contain an interest rate.
In staff paper 44, documenting the 1985 Version of BEA's quarterly econometric model of the U.S. economy, it is lamented that "despite a strenuous effort to include a cost-of-capital expression, only a simple accelerator model could be developed" to explain fixed nonresidential investment.
It has been reported that DRI's 1990 equation for nonauto/non office equipment has no interest rate variable and depends only on very long lags in output and the rental price of equipment (Mosser 1992).
When the error terms for the accelerator model in Table 4.1 of Essay 4 are correlated with the average prime rate and the year-year growth rates for the consumer price index one obtains a slightly positive relationship for the prime rate during the 1960-94 period rather than the inverse relationship that one would expect on the basis of economic theory.
In this model the growth facilitating and cyclical dynamics of the investment process may be swamping and hiding any rational propensity to invest more when interest rates decline. Once it becomes clear that the economy has entered a recession credit rationing and the fear of bankruptcy may prevent many firms from taking advantage of lower interest rates.
When the economy begins to recover from a recession and interest rates begin to rise, business enterprises may rush in to acquire new plant and equipment to avoid bottle necks and a possible loss of sales to their competitors. History would suggest, in any event, that it can take a long time for a gradual increase in the prime rate, which is closely linked to federal funds rate, to slow an upsurge in investment spending to a more sustainable pace.
Economists are very fond of the equilibrium values implied by simultaneous equations such as those representing demand and supply. To identify the slope coefficients for such curves one needs independent shifts in the two curves. When a marginal efficiency of capital equation is combined with a Keynesian multiplier equation to obtain an "IS" curve the identification of a negatively sloping IS curve, with respect to changes in the interest rate, may be obscured to a considerable extent by offsetting shifts in the "LM" curve. The Fed's policy of "leaning against the wind", so-to-speak, may be one of the reasons why interest rates have added little additional explanatory power to equations that endeavor to explain investment behavior.
Another reason economic model builders have not had much luck at verifying an inverse relationship between investment and the behavior of interest rates, perhaps, is that the writers of most economic text books have been remiss about informing their students how interest rates can be used to help evaluate a chain of replacements.
Since the publication of Hotelling's paper on depreciation in 1925, it has been taken for granted by most theorists that the appropriate criterion for choosing between investment projects is maximizing the present value of a firm's net revenue. If output is presumed to be optimal in all future periods, this is equivalent to minimizing the present value of all future costs associated with a chain of capital replacements (Smith 1961, p. 161).
An assumption that is often made by investment analysts in order to simplify the process of estimating an optimum replacement interval is that operating expenses will increase with the age of equipment, on the average, by a constant amount per period. The hypothesis that a machine can be expected to accumulate "operating inferiority" at a constant rate over its service life was first advanced by George Terborgh in a Dynamic Equipment Policy. By operating inferiority, Terborgh was referring to both the opportunity costs associated with technological advance and the more direct costs resulting from physical deterioration with respect to age and use.
If a machine is expected to produce a constant volume of output over its service life and if operating costs attributable to obsolescence and age are assumed to increase at the end of each period by the dollar amount, $, one can easily solve for the optimum life or replacement interval, N--given a particular before (income)-tax cost of capital or discount rate--by dividing $ into the acquisition cost of new equipment, C, minus its expected salvage value, V, and looking up the resulting annuity factor, k, in a gradient annuity table.
(C - V)/$ = k (1)
A simple gradient annuity or declining balance saving series that assumes a constant rate of output can be constructed from scratch by taking a running sum of the discount factors in the columns of a standard present worth of an annuity table. See Table 5.2. The optimum replacement interval for any discount rate is simply the number of years or periods associated with the present value factor, k.
Consider the case of a truck that costs $20,000 and is not expected to have any salvage value. If operating costs amount to $5,000 the first year and are expected to increase $400 per year, one can use equation (1) to determine that: k = $20,000/$400 = 50. If the cost of capital is 15 percent one can then use Table 5.2 to quickly determine that the optimum replacement interval is a little over 13 years. If the cost of capital were eleven percent it would only be about 12 years. For a truck that lasts 13 years the average acquisition cost per year will be: C/N = $20,000/13 = $1,538. If the truck is replaced every 12 years the average cost of new equipment will be $20,000/12 = $1,667. With these numbers one can construct a longer run equilibrium marginal efficiency of capital schedule and determine that the interest rate elasticity of demand for replacement trucks is about - .3 when easier monetary policy lowers the perceived cost of capital from 15 to eleven percent.
The more important conclusion to be derived from this exercise and numerous efforts to establish a negative relationship between investment by business enterprises and the interest rate is that the marginal efficiency of capital function is probably not very elastic, even in the long run, as far as equipment is concerned. This type of investment function, in any event, helps to confirm the Keynesian notion of a relatively steep marginal efficiency of capital function which in turn will produce a steep IS curve when the investment function is substituted into the multiplier equation.
Andersen, L. and K. Carlson (1970). "A Monetarist Model for Economic Stabilization," Federal Reserve Bank of St. Louis, Review, April.
Fair, Ray (1984). Specification, Estimation, and Analysis of Macroeconometric Models(Cambridge, MA: Harvard University Press).
Fisher, I. (1907). The Rate of Interest(New York: Macmillan).
----, (1930). The Theory of Interest(New York: Macmillan).
Hicks, John R (1937). "Mr Keynes and the Classics: A Suggested Interpretation," Econometrica, vol. 5, no. 2, 147-159.
Hotelling, Harold (1925). "A General Mathematical Theory of Depreciation," Journal of the American Statistical Association, September.
Mosser, P. (1992). "Changes in Monetary Policy Effectiveness: Evidence from Large Macroeconometric Models," Federal Reserve bank of New York Quarterly Review, Spring, p. 42.
Renshaw, Edward (1976). Capital Budgeting and Economic Theory(Morristown, N.J.: General Learning Press).
Smith, Vernon (1961). Investment and Production(Cambridge, MA: Harvard University Press).
Terborgh, George (1949). Dynamic Equipment Policy(New York: McGraw-Hill).
Date of Duration of Business Expansion Actual
--------------------- Lag in ------------------------------ Minus
NBER Prime Rate Months for Predicted Actual Predicted
Trough Trough Prime Rate --------in Months------ Duration
(1) (2) (3) (4)n (5) (6)n
Oct. 1945 Nov. 1947 25 54 37 -17
Oct. 1949 Aug. 1950 10 31 45 14*
May 1954 July 1955 14 37 39 2
Apr. 1958 Aug. 1958 4 21 24 3
Feb. 1961 Nov. 1965 57 104 106 2*
Nov. 1970 Mar. 1972 16 40 36 - 4
Mar. 1975 Apr. 1977 25 54 58 4
July 1980 Aug 1980 1 17 12 - 5
Nov. 1982 Mar. 1987 52 97 92 - 5
Mar. 1991 Feb. 1994 35 70 ?
(4)n. The predicted duration of the business expansion is equal to 15 months plus 1.57 times the monthly lag for the prime rate in column (1). This formula was developed at the Columbia University's Center for International Business Cycle Research under the direction of its Director, Geoffrey Moore, and widely publicized by Lindley Clark in "A Slump Predictor Clinton Should Love," The Wall Street Journal, December 28, 1993, p. A10.
(6)n. Column (5) minus column (4).
Source of basic data: The Federal Reserve Bulletin.
*Business expansions which may have been prolonged by wars.
N 3% 5% 7% 9% 11% 13% 15% 1 0.9709 0.9524 0.9346 0.9174 0.9009 0.8850 0.8696 2 2.8843 2.8118 2.7426 2.6765 2.6134 2.5531 2.4953 3 5.7130 5.5350 5.3669 5.2078 5.0571 4.9142 4.7785 4 9.4301 9.0810 8.7541 8.4476 8.1596 7.8887 7.6335 5 14.0098 13.4105 12.8543 12.3372 11.8555 11.4059 10.9856 6 19.4270 18.4862 17.6209 16.8231 16.0860 15.4035 14.7701 7 25.6572 24.2725 23.0102 21.8561 20.7982 19.8261 18.9305 8 32.6769 30.7357 28.9815 27.3909 25.9443 24.6248 23.4179 9 40.4630 37.8436 35.4967 33.3861 31.4814 29.7565 28.1894 10 48.9932 45.5653 42.5203 39.8038 37.3706 35.1827 33.2082 11 58.2459 53.8717 50.0189 46.6090 43.5771 40.8697 38.4419 12 68.1999 62.7350 57.9616 53.7697 50.0695 46.7873 43.8625 13 78.8348 72.1285 66.3193 61.2566 56.8194 52.9091 49.4457 14 90.1309 82.0272 75.0647 69.0428 63.8012 59.2116 55.1702 15 102.0688 92.4068 84.1727 77.1035 70.9921 65.6740 61.0175 16 114.6299 103.2446 93.6193 85.4160 78.3713 72.2779 66.9718 17 127.7961 114.5187 103.3825 93.9597 85.9201 79.0070 73.0189 18 141.5496 126.2083 113.4416 102.7153 93.6217 85.8469 79.1469 19 155.8734 138.2936 123.7772 111.6654 101.4610 92.7849 85.3451 20 170.7508 150.7558 134.3712 120.7939 109.4243 99.8096 91.6045
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