Agricultural prices

Mathematical formulation of price-making factors is necessary in order to know when extraordinary or strategic considerations are influencing the market. The mathematical methods are highly technical, and in order to explain most clearly we shall follow a specific problem thru from beginning to end.

The problem is to determine the price of hogs from hog receipts (supply) and from business conditions (demand). To represent business conditions, we are using bank clearings outside of New York City. The actual figures for heavy hog prices at Chicago are given in the Appendix. Hog receipts at Chicago and bank clearings outside of New York City are given on pages 81 and 82. The problem is to evolve from these figures the law of hog prices.

The first step is to determine the secular or long-time trend of these figures. Find, for example, the secular trend of such a series as:

From looking at these figures, we know that the secular trend slopes upward, starting with about 2 in 1901, reaching 3 or 4 by 1905, and 5 or 6 by 1909. To express the matter with mathematical accuracy, the method as applied to this series is as follows: First add all the figures together. Answer in this case, 36. Then divide by the number of figures—in this case 9. Thirty-six divided by 9 gives 4, which is the value of the secular trend for 1905, which is the central year.

The year 1904 is the −1 year, 1903 the −2 year, 1902 the −3 year, 1901 the −4 year, and in like manner 1906 is the +1 year, 1907 the +2 year, 1908 the +3 year and 1909 the +4 year. Multiply the minus years by their respective values: −1 by 5, −2 by 2, −3 by 3 and −4 by 2, and also the plus years, +1 by 6, +2 by 4, +3 by 6 and +4 by 6. The totals are −26 and +56, or a net of +30. Now the sum of the squares of −1, −2, −3, −4, +1, +2, +3 and +4 is 60. Sixty divided into 30 gives .5, which is the rate of movement of the secular trend each year, or if, as we found, 4 is the secular trend value for 1905, then 3.5 is the value for 1904, 3.0 for 1903, 2.5 for 1902, and 2.0 for 1901, and in like manner 4.5 for 1906, 5.0 for 1907, 5.5 for 1908 and 6 for 1909. The secular trend is a straight line, and the actual goes above and below the secular trend in more or less wave-like fashion. In Chart I, the straight line is the secular trend of heavy hog prices at Chicago for 1903–1916, and the irregular line fluctuating above and below is the actual price of heavy hogs.

The next problem is to eliminate the normal seasonal variation. For example, hog prices have a normal tendency to go down in the fall of the year, whereas bank clearings have an equally normal tendency to go up. Obviously, seasonal trends must be eliminated if such series as hog prices and bank clearings are to be compared.

As an average of the fourteen years from 1903 to 1916, inclusive, heavy hog prices at Chicago averaged in January, $6.54; February, $6.83; March, $7.22; April, $7.30; May, $7.10; June, $7.10; July, $7.18; August, $7.14; September, $7.29; October, $7.08; November, $6.65; December, $6.55; average for the entire year, $7. On this basis, January is 93 per cent of the yearly average; February, 98 per cent; March, 103 per cent; April, 104 per cent; May, 101 per cent; June, 101 per cent; July, 103 per cent; August, 102 per cent; September, 104 per cent; October, 101 per cent; November, 95 per cent, and December, 94 per cent. The December average for 1902–1915 is $6.30, or 90 per cent. Obviously, the seasonal variation as just stated in percentages is affected to some extent by the secular trend, for the Decembers of 1902–1915 average 90 per cent, and those of 1903–1916 average 94 per cent. Taking the secular trend out of our seasonal, or adding 2 points to the early months of the year and subtracting 2 points from the last months of the year, we get approximately: January, 95; February, 99; March, 104; April, 105; May, 102; June, 101; July, 103; August, 102; September, 103; October, 100; November, 94, and December, 92.^[6]

Hog receipts at Chicago, in the same manner, have a modified seasonal factor of January, 132 per cent; February, 117 per cent; March, 102 per cent; April, 85 per cent; May, 99 per cent; June, 99 per cent; July, 87 per cent; August, 87 per cent; September, 74 per cent; October, 86 per cent; November, 103 per cent; December, 129 per cent.

For bank clearings outside of New York City, the modified seasonal factors are: January, 109; February, 93; March, 104; April, 100; May, 99; June, 97; July, 98; August, 90; September, 93; October, 108; November, 103, and December, 106.

After securing normal seasonal variation, the next step is to modify secular trend for seasonal variation. Secular trend of hog prices, as modified seasonally, is portrayed in Chart II. The secular trend price of hogs in January, 1903, is $5.19, which sum, multiplied by the seasonal factor 96, gives $4.98 as the secular price of hogs modified seasonally for January, 1903. The actual price was $6.60, or $1.62 above the secular modified seasonally, or 31 per cent greater than the secular price of $5.19. In this way the percentage of departure for each month from 1903 thru 1916 may be figured. This has been done for hog prices, hog receipts and bank clearings outside of New York City.^[7]

Now, as it happens, hog receipts are a much more violently fluctuating series than bank clearings outside of New York City. To put the series on an even footing, resort is made to what is known as the standard deviation. To secure the standard deviation of hog price percentage departures, add up the squares of these departures. The total for the 168 months from 1903 thru 1916 is 31,894, or, dividing by 168, we get 190. The square root of 190 is 13.8, which is the standard deviation of hog prices. Standard deviation means that the probabilities are that on the average not more than one out of three of the series of figures under consideration will exceed the standard deviation. Standard deviation for hog receipts is 15, and for bank clearings 8.7. This indicates that hog receipts depart from the secular trend as modified seasonally with nearly twice as great violence as do bank clearings.

To put all three series on the same footing, we divide the percentage departures by the standard deviation, 13.8 in the case of hog prices, 15 in the case of hog receipts, and 8.7 in the case of bank clearings. In January of 1903, for example, hog prices were greater than the secular modified seasonally by 2.3 times the standard deviation; hog receipts were less by .3 of the standard deviation, and bank clearings were over by .9 of the standard deviation. The cycles of the hog prices, hog receipts and bank clearings, as secured in this way by reducing for standard deviation, are comparable. The results are charted in Charts III, IV, V.

It may be seen from examining these charts that hog prices seem to be related directly to bank clearings and inversely to hog receipts. The problem is: Blend hog receipts and bank clearings together in such a way as to secure hog prices. The mathematical method of approach is by correlation coefficients and lines of regression.

First, a simple illustration of the method of securing correlation coefficients:

Take the two series, A and B, which deviate from their respective means by the amounts stated in Columns 2 and 3. In Column 1 is the year, which has nothing to do with the mathematics of the case. Column 4 is A squared, Column 5 is B squared, and Column 6 is A multiplied by B.

The standard deviation of A is the square root of the sum of the A squares, or 18, divided by 4. The square root of 18 divided by 4 is 2.1. Standard deviation of B, in like manner, is 3. The sum of AB divided by 4, or +22 divided by 4, equals +5.5. The correlation coefficient is +5.5 divided by the standard deviation of A multiplied by the standard deviation of B, or 5.5 divided by 6.3, which gives +.87. A correlation coefficient of .87 is very high, perfect correlation being 1. Correlation over .5 is considered fairly good, especially if there is a long list (fifty or more) of figures in each series.

The formula for determining A in terms of B is:

A equals r(σ_a
σ_b)B

In this formula, r is the correlation coefficient and σa is the standard deviation of A, and σb is the standard deviation of B. Substituting for the specific problem, we get:

A equals .87(2.1
3.0)B or

A equals .609 B

When B is −5 we would expect A to be 3.05; when B is +1 we would expect A to be +.609; when B is +3, we would expect A to be 1.827.

Suppose now, in addition, that there are three series: A, B and C, and that the object is to determine A in terms of B and C. The three series stand:

We already know that the standard deviation of A is 2.1, and of B is 3.0, and that the correlation coefficient between A and B is +.87. Using the customary method, we find that the standard deviation of C is 2.55 and that the correlation coefficient of A and C is −.89, and of B and C −.59. To find A in terms of B and C, we use the following formula:

A equals (r_ab − r_acr_bc
1 − r²_bc)(σ_a
σ_b)B + (r_ac − r_abr_bc
1 − r²_bc)(σ_a
σ_c)C

In this formula r_ab means correlation coefficient between A and B, etc.; σ_a means standard deviation of A.

Substituting, we get:

A equals (+.87 − .53
.65)(2.1
3.0)B − (−.89 + .51
.65)(2.1
2.55)C or, A equals .37B − .49C

Applying this formula, we find that when C is +2 and B is −5, as in the year 1901, we would expect A to be −2.83, and when C is +3 and B is +1, as in 1902, we would expect A to be −1.1. In like manner, in 1903, we would expect A to be +2.60 and in 1904 +1.35.

The results expressed in a table are:

The practical problem is to express hog prices in terms of hog receipts and bank clearings. Practically the same method is used with the 168 months from 1903 thru 1916, as with the four years which have just been used for illustration.

The standard deviations are 10.1 for hog receipts, 10.5 for hog prices and 9.8 for bank clearings. The correlation coefficients are +.39 between hog prices and bank clearings, +.26 between hog receipts and bank clearings, and −.4 between hog receipts and hog prices.

Using the formula:

A equals r(σ_a
σ_b)B

and allowing A to represent hog prices and B to represent bank clearings, we get:

Hog prices equal .39(10.5
0.8) bank clearings, or

Hog prices equal .417 bank clearings

This formula is converted back into percentage departures from secular trend modified seasonally, and finally into hog prices as affected by bank clearings. The demand, or bank clearing, price, of hogs as compared with the actual is shown in Chart VI.

In like manner we get:

Hog prices equal −.4(10.5
10.1) hog receipts, or

Hog prices equal −.426 hog receipts

This formula is converted back into percentage departures from the secular trend modified seasonally, and finally into hog prices as affected by hog receipts. The supply price of hogs as compared with the actual is shown in Chart VII.

Using the longer formula on page 89, we get: Hog prices equal .56 bank clearings minus .56 hog receipts. Or converted into percentage departures from the secular trend corrected seasonally: .90 of bank clearings in percentage departures minus .51 of hog receipts in percentage departures equals the percentage which hog prices depart from their secular corrected seasonally. For instance, in January, 1903, bank clearings were 8 per cent above the secular corrected seasonally, and hog receipts were 5 per cent below. Eight times .90 plus 5 times .51 gives 9.7 as the percentage which we would expect hog prices to be over their secular corrected seasonally. The secular for January, 1903, was $5.19; 9.7 per cent of $5.19 gives 50 cents. The secular corrected seasonally for January, 1903, is $4.98. Add 50 cents to $4.98 and we get $5.48 as the price which we would have expected heavy hogs to sell at Chicago in January, 1903, on the basis of good business and small hog receipts. Actually, hogs sold for $6.60, or $1.12 over the price predicted by formula.

This is done for all the months from 1903 to 1916, and the supply-and-demand price of hogs, as derived from hog receipts at Chicago and bank clearings outside of New York is charted in Chart VIII, in comparison with the actual prices.

We assume that at the present time, and probably for some time to come, we are on a basis of 90 per cent above 1913 for hog prices, and 100 per cent over 1913 in bank clearings. This conclusion is based to some extent on the reasoning presented in the June monthly supplement of the Harvard Review of Economic Statistics for the year 1919.

On this basis, the secular trend of heavy hog prices at Chicago, modified seasonally, should be roughly as follows for the several years beginning with 1919: January, $14.35; February, $15.07; March, $15.82; April, $15.67; May, $15.22; June, $15.22; July, $15.52; August, $15.22; September, $15.52; October, $15.07; November, $14.16, and December, $13.86.^[8] This is on the assumption that hog prices and prices generally will have for their normal mean a level 90 per cent above the 1913 level. It is expected that in a rough way hog prices will depart from this level according to the size of hog receipts and the condition of general business as expressed by bank clearings. (During 1920, and possibly 1921, heavy exports will doubtless have influence.)

The secular trend of bank clearings outside New York, modified seasonally, for the year beginning with 1919, is taken as: January, $13,952,000,000; February, $11,648,000,000; March, $13,056,000,000; April, $12,800,000,000; May, $12,416,000,000; June, $12,416,000,000; July, $12,544,000,000; August, $11,648,000,000; September, $12,032,000; October, $13,824,000,000; November, $13,440,000,000, and December, $13,824,000,000.

The secular trend of hog receipts at Chicago in millions of pounds, modified seasonally, for the period beginning with 1919, is taken as: January, 184; February, 163; March, 143; April, 118; May, 139; June, 139; July, 121; August, 121; September, 103; October, 120; November, 144, and December, 180.

Based on the formula as secured in the preceding chapter (hog price equals .56 bank clearings minus .56 hog receipts), we would expect the following scale of hog prices in January, when receipts follow the secular trend (184,000,000 pounds at Chicago), but bank clearings are variable:

In like manner, tables may be made up for each month of the year, the idea being that for each $500,000,000 the bank clearings outside of New York are above or below the secular trend seasonally modified, fifty cents is added to or subtracted from the secular trend hog price seasonally modified. Thus for April the tables would be:

Taking the tables as worked out for bank clearings and hog prices, we next modify for hog receipts. An excess of 33,000,000 pounds of hog receipts at Chicago in a month means on the average $1.80 lower prices, and vice versa. Thus, in January, with bank clearings at $13,952,000,000, we would expect the following prices with various sizes of hog receipts:

The tables herewith give this problem worked out in detail for the various months. It is realized that at this writing, in early 1920, financial matters are still so deranged by the great war that our secular trend for bank clearings may be wide of the mark. This is the best prediction we can offer at this writing, and we are offering it fully aware of its weakness, but fully believing that predictions of this sort will stimulate more thoro research. It is believed that better measures of demand may eventually be found than bank clearings outside of New York City, and that better measures of supply may be found than receipts at Chicago. Also there is a possibility that the varying size of exports of hog products should be taken into account.