The model utilized by Hayami and Peterson is simple but can be implemented using basic economic information and does give insights into the problem of quantifying the benefits of information. The specific equation developed by Hayami and Peterson for the social loss incurred from a forecast with a proportional error of e is:

social loss = e²(Value of Production/a)
 

where a is the price elasticity of demand. The derivation of this equation is summarized in Figure 1. The above formula should also be multiplied by a factor of (1/2) to get the annual loss.

In deriving the social loss formula, Hayami and Peterson found the annual rates of social losses for the pre-harvest and post-harvest seasons but neglected to multiply this by one half to obtain the amount each part of the season contributed to the annual loss. If there were T crops per year the formula would still only involve a factor of (1/2) because the number of crop seasons cancels out in the computation of the annual rate. Lowe, Summers and Greenblat proposed a factor of 1/(T + 1) but this appears to be incorrect for the model under analysis (2).

Since the basis for the welfare loss is the deviation of prices from their proper level all elements which enter into the determination of market price must be included. On the supply side this means that, in addition to domestic production, imports must be included and also depletion of inventories. This latter would be a major element in the short run determination of prices but would not necessarily be involved in the determination of prices in the absence of uncertainty. However government purchases for inventory or export is a major factor for some crops in the relationship between output fluctuations and price fluctuations. In effect, the computation reouires a price elasticity of total demand for a commodity to translate fluctuations in the quantity bought and sold into price fluctuations.

The analysis for the case in which the forecast under estimates the crop harvest is shown below.

The corresponding analysis for an over estimate of the crop harvest is shown below.

3. Elasticities of Demand

The Hayami-Peterson study uses an elasticity of demand for wheat of 0.02. This may be a reasonable estimate for domestic consumption but it is probably an inaccurate estimate for total demand elasticitv. This is easily seen hy comparing the variation in wheat prices to the variation in production. An elasticity of total demand of -.02 would imply that variations in production would be amplified by a fifty to one ratio in prices but the coefficient of variation in farm prices for wheat for the period 1962 to 1972 is about the same as the coefficient of variation in production. The explanation for the relatively small variation in price lies in the greater sensitivity of inventory and export demand to price. If the Federal Government is stabilizing prices government demand may be infinitely elastic at the price support level.

If the elasticities used by Hayami and Peterson are interpreted as being the domestic demand elasticities, then the corrected figure would equal to the Hayami-Peterson estimates multiplied by the factor (1/2)(a_D/a_T)(q_D/q_p), where q_D and q_p are domestic use and domestic production, respectively. This correction is in addition to the multiplication by a factor of (1/2)(a_D/a_T) mentioned previously.

The issue of which benefits should he included in assessing the value of improved information to the American economy may be examined in more detail. Clearly the benefit computed from consumer demand should be included and that computed from export demand should be excluded. Also the demand of the Federal Government in maintaininq price levels should clearly be excluded.

Intermediate or derived demand for anricultural commodities derived from consumer demand for goods is, of course, the major element of consumer demand in agriculture. It must be demonstrated that the change in consumer's surplus computed from the derived demand schedule has any relationship to the changes in consumer's surplus computed from the final products demand schedules. This issue will be treated in more detail later in the paper.

There is one situation where the domestic demand for the farm product does not represent demand derived from domestic consumption. This is when the farm product is purchased by an American firm but manufactured into a product which is exported.

Investment in inventory and seed constitutes a different type of derived demand. Increases in the price of the anricultural product should increase the cost of holding inventory and ultimately the price of consumer qoods. By this test the benefits computed from an inventory demand function should he included. If a decrease in inventory cost is not entirely reflected in retail price it positively affects the income of producers which would constitute a benefit for a part of the economy. Other components of demand must he examined individually to ascertain whether or not they are sources of benefits for the American economy of improved information systems.

4. Computation of the Quantity Traded

The quantity traded is the quantity bought and sold at market equilibrium. It could be computed by summing either the quantities demanded or supplied at the market price. For inventory change the distinction between supply and demand is blurred. The quantity sold of a commodity is the production plus imports plus the net decrease of any inventory category.

When a category of inventory increases, it is an increment to the demand side rather than a decrement to the supply side. Some categories of inventory may be increasing while others are decreasing and, therefore, in estimating the quantity bouoht and sold as much detail of inventory levels as possible should be sought. Since the transactions to and from inventories will not be revealed by the net change in inventory figures the estimates all tend to underestimate the quantity bought and sold.

When all inventory levels are decreasing the quantity bought is identical to the domestic disappearance. But when inventories are increasing the quantity bought is equal to production plus imports.

For example, consider the case in which the total level of inventories is decreasing whereas the level of one type of inventory is increasing. One category is decreasing and the amount of this decrease should be included as a part of supply. For example in 1964 total wheat inventories dropped by 84.1 million bushels whereas private inventories increased by 125.0 million bushels. This means that public inventories decreased 209.1 million bushels. This latter figure must be added to the production plus imports of 1264.5 million bushels to give the estimated quantity sold of 1493.6 million bushels.

The estimates of the quantity of the six commodities bought and sold are given in Table 1.

The price variable which should influence consumption decisions is the commodity price relative to the price of other goods. In the analysis the price variables used are the average farm values of the crops corrected for changes in the cost of living (CPI). These data are given in Table 2. Ideally the price variables should be retail prices but this would require a much more extensive analysis than is possible in this paper. This would require consideration of the value added and market structure each stage of the processing, and it would be very difficult to select out the price changes in retail prices which arise from variation in the quantity of farm products. Average farm value is a very imperfect measure of the price level of even a specific crop. It can be influenced by the geographic distribution of production and the product mix of different varieties of the crop. Statistics on prices of specific commodities at specific locations are available such as number 2 hard winter wheat at Kansas City, but the corresponding quantity information is not readily available. The correlation of specific market prices with average farm values are given in Table 3. These figures indicate the extent to which-farm value reflect the relevant price information. Because other factors beside farm price affect market price there can be negative corrections.

6. The Use of Derived Demand Versus Final Demand Functions for the Computation of Welfare Losses

The source of the social loss from inaccurate forecasts ultimately stems from the impact on households and therefore the change in consumer surplus should be evaluated from the final demand schedule. But there is a derived demand schedule for the agricultural product and it certainly would he more convenient to deal with. For exanple, it is easier to compute the change in consumer surplus for the derived demand schedule for wheat rather than have to consider the impact of wheat prices on flour, bread, cakes, sandwiches, and so forth, and compute the change in consumer surplus from each of these.

Not only would it be inconvenient but, practically speaking, it would be impossible to consider all of the direct demand functions for wheat products. On the other hand if the consumer surplus computed from the derived demand curve is unrelated to the quantity that is sought, then nothing would be accomplished by using it. Fortunately there is a relationship between the consumer surpluses computed from the derived demand and the final product demand curves.

The major complicating factor is that the prices of the products to the consumer are a function not only of the farm prices. The market structures of the intermediate sector, among other things, affect the relationship between farm price and retail prices. If a monopoly sector exists between the farmer and the consumer, then price decreases in farm products will not he fully reflected in retail prices. Oligopoly elements would have a similar effect. If there are bottlenecks in processing, an increase in final demand will not be fully reflected in the prices received by farmers.

If the market structure is perfectly competitive from the farm to the retail outlet then analysis indicates that it would not make any difference marginally whether consumer surplus is evaluated from the demand curve for the final product or the derived demand curve for the farm product.

In order to demonstrate this proposition let the following definitions hold:

• X = quantity of the farm product

• p_f = farm price of the farm product

• Q = quantity of the final or retail product

• P_r = final or retail price

• CS_f = consumer's surplus computed from the derived demand schedule.; i.e., X as a function of p_f.

• CS_r = consumer's surplus computed from the retail demand price schedule-, i.e., Q as a function of

From the definitions of consumer's surplus it holds that for infinitesimal changes (denoted by a prefix of d to a variable):

dCS_f = X dp_f
dCS_r = Qdp_r
 

Let h be the elasticity of retail price with respect to farm price and thus:

dp_r = (p_r/p_f)hdp_f.
 

A substitution of this expression into the equation for dCS_r gives

dCS_r = Q(p_r/p_f)hdp_f.
 

This will be equal to dCS_f if

Q(p_r/p_f)h = X
which will be the case if
h = (p_fX)/(p_rQ)
 

That is to say, derived consumer's surplus will equal final consumer's surplus if the elasticity of final price with respect to farm price is equal to the farm product's share of total value of the final products.

When the production function is homogeneous to the first degree the elasticity of marginal cost with respect to an input price will be equal to the share of the cost of that input to total cost providing the input combination is chosen to minimize costs. Under perfect competition the industry's production function is effectively homogeneous to the first degree and also at long run enuilibriun the final product price is equal to marginal and average cost. A sufficient condition for the equivalence of marginal changes in consumer's surplus computed from derived and final demand schedules is perfect competition in each stage of the processing of the farm products. Whether the processing or even the farm production stages are perfectly competitive is seriously in doubt but the analysis, at least, indicates what sort of assumptions are made implicitly when computing the change in consumer's surplus from the derived demand schedule.

In computing welfare changes another quantity, the area below the demand schedule for a given change in output, is often significant. The condition that dCS_r be equal to dCS_f is that the elasticity of output wtth respect to the input of the farm product be equal to the factor share of that input. This latter condition also holds for profit maximizing industries in competitive factor markets.

7. Estimates of Price Elasticities of Demand

If

• elasticity is constant over the range of variation in price
• demand is unaffected by changes in variables other than price
• supply is unaffected by price

then variations in the quantity sold and price will he directly proportional. The elasticity estimate used by Hayami and Peterson implied that proportional variation in wheat quantities would be amplified fifty times in terms of proportlional variation in wheat price. But a comparison of the variation in wheat prices and quantities reveals that there is nowheres near a fifty-tn-one ratio in the variations of quantity and price. In fact, during the period 1962 to 1972 there was only fifty percent more proportional variation in the price of wheat compared to the variation in the quantity of wheat. Table 4 gives the relative amounts of variation in quantities sold and prices for the six crops over the period 1962 to 1972.

If there are shifts in demand which are correlated positively or negatively with shifts or changes in supply, the relationship between the relative sizes of the variation in quantity and price will be more complex. When there is no positive correlation between the shifts in demand and supply, the variation in price would be even greater relative to variation in the quantity sold than the reciprocal of elasticity. Only if the correlation is substantially positive, such that years of unusually high production are accompanied by unusual increases in demand, will the variation in price be less than that indicated by the reciprocal of elasticity times the variation in production.

The estimation of parameters such as demand elasticities require a formally specified model. Even with a formal model and adequate data for statistical analysis it may not be possible to derive unique estimates of the structural parameters of the model. A crude estimate of the elasticity of demand would be the ratio of the coefficients of variatlon of quantity and prices. This gives an order of magnitude estimate that might be thought of as the zeroeth order approximation estimate of the price elasticities of demand.

These zeroeth order approximations of price elasticities are shown in Table 5 along with more proper estimates of elasticity. The estimates of elasticities are from a model in which the quantity demanded is proportional to population and a function of price and per capita real disposable income. the quantity supplied is assumed to be exogenous. These estimates, while not sophisticated econometrically, give the order of magnitude of the elasticities. the model leaves out the factors other than current price which affect demand and supply. This shows up when the implications are drawn from the elasticity estimates concerning the relative variation in prices and quantity. the quantity variation is not amplified in price variation to the extent implied by the elasticities used by Hayami and Peterson. The reason for this is the role of government price stabilization programs. When production is unusually large government purchases are increased to prevent a drastic fall in price.

Hayami and Peterson do not explicitly state the exact nature of their elasticity estimates. They searched the literature for competent estimates, but what may have been appropriate for one study may not have been appropriate for their purposes. It should be emphasized that what is being criticizes is not so much the accuracy of the elasticity estimates used by Hayami and Peterson as only that the estimates do not represent total demand elasticities.

8. Marginal Effect of Survey Results on Crop Expectations

The uncertainty in production forecast for a given year should be no greater than the historical variation in production. In lieu of any current information people could make the naive prediction that output will be equal to the historical average of past productions. the sampling error utilized by Hayami and Peterson is not explicitly defined but if it represents one standard deviation unit, then a comparison can be made with the naive predictions. Table 4 indicates that the variation in the quantitly of potatoes as measured by the coefficient of variation of quantity traded is 8.3 percent. However Hayami and Peterson indicate that the sampling error for potatoes is 18.5 percent when the typical sampling error is 3 percent. The sampling error drops to 15.5 percent when the typical sampling error is reduced to 2.5 percent. If the sampling error is so large for potatoes that it is irrelevant, then it is inappropriate to include a social benefit for reduced sampling error for potatoes. This is significant for the quantitative results of Hayami and Peterson because improved forecasts for potatoes is the second highest source of benefits, constituting slightly over 20 percent of the benefits in going from a 3 percent typical sampling error to a 2.5 per cent level.

The largest single source of benefits comes from improved forecasts for rice production, constituting 35.5 per cent of the total for the change from a 3 percent to a 2.5 per cent typical sampling errcr. The situation is similar to that of potatoes, the sampling is so large for rice as to make the forecast almost irrelevant. The coefficient of variation of the quantity traded was 13.2 percent for the period 1902 to 1972. The sampling error used by Hayami and Peterson was 15.8 percent when the typical sampling error was 3.0 and 12.6 percent when the typical error was dropped to 2.5 Percent. Therefore only the drop from 13.2 percent to 12.6 percent should be counted as a benefit of the improved accuracy of the survPy.

In actuality the sample information could probably be combined with some simple projection model to yield a forecast with a smaller margin of error than would be obtained from the sample alone. In this case there would probably be marginal improvement in the forecasts even with the higher sampling error.

The market price would be influenced by other factors besides variation in production. Product users would need not just a forecast of production but of the quantity marketed, includinn imports and inventory depletion. The variance of the quantity marketed would depend upon the variance of production and the variance of other sources of supply and their correlation. For example the coefficient of variation of production and imports might be 10% and 5%, respectively. If they are uncorrplated the coefficient of variation of their total would be 11.2 percent. If the variation in production was reduced by 40 percent to 6 percent with no change in the figure for imports, the variation in total would be 7.8 percent, a reduction of 30 percent.

If the other elements of uncertainty were included in the squared error term, then in the computation of the social loss there would be less of a reduction in the social loss when production forecast error is reduced. No attempt was made to estimate the magnitude of this correction to the Hayami-Peterson estimates. Its magnitude is likely to be significant but small compared to the size of the other corrections.

9. The Production Adjustment Model

The corrections which must be made in the lHayami-Peterson production adjustment model are much more complex because there are two sets of supply and demand curves involved.

The effect of the adjustment on the relative magnitude of the production and inventory components of social benefit cannot be determined without estimates of demand and surpply elasticities. These parameters were not estimated but the magnitudes seem more reasonable approximations of total demand elasticities than is the case for the inventory model.

The Hayami-Peterson production adjustment model may involve a conceptual shortcoming. In their model supply is responsive to price and, therefore it would be inappropriatp to give a forecast without an indication of the price at which the forecast is applicable. Since the forecast itself may affect the market price the public agency should not only forecast production but also the short run equilibrium price. Of course, if, as shown in Figure 3, the forecast is in error, then a welfare loss will arise, which is equal to

(1/2)(�_T/a_T)²)(e²pq/(1+(�_T/a_T)))
 

The formula derived assumes that the forecasters know precisely the short run responsiveness of supply to expected price. Figure 3 applies to an overestimate of supply. The analysis for an underestimate of supply is given in Figure 4.

The Hayami -Peterson estimates can be put into the form

(1/2)(�/a)²(1 + �/a)(e²pQ)
 

The similarity is striking and suggests that there is some relationship involved between the two estimates.

When the above formula is applied to the data used by Hayami and Peterson in their production adjustment model the marginal social benefits turn out to be approximately one-half those computed by Hayami and Peterson.

10. Estimates of Social Benefits

The adjustment of the Hayami and Peterson estimates of the benefits of improved forecast accuracy is made in three stages. First the neglected factors of one half and the ratio of the elasticities of domestic and total demand are introduced. Second, the relative elasticity ratio is again applied and the ratio of domestic use to domestic production are applied to the estimates. This limits benefits to American users. In Table 6 the total benefits from reducing the typical error of crop forecasts from 3 percent to 2.5 percent are, for the six commodities analyzed, lowered from the $270.8 million found by Hayami and Peterson to $31.2 million by the first adjustment and to $11.5 million by the second adjustment. The third stage of the adjustment is to eliminate the benefits for those crops for which thr sampling error is larger than the coefficient of variation of quantity. These crops are rice and potatoes. This adjustment drops the total benefit figure to $2.1 million. If this were the only benefit then the benefit/cost ratio would be about 5.8. The other components of benefits Hayami and Peterson estimated to be $18.0 million for other crop inventories and $8.0 million for the production adjustment. The first and second types of adjustment would apply for the other inventory adjustments and if the same proportional relationship held as for the commodities wheat, corn, oats and barley, then the $18.0 million would fall to about $0.3 million. The total benefits would be $2.4 million and the ratio of inventory benefit-cost would be about 6.7. The production adjustnent benefits appear to be much less affected by the corrections introduced and much of the $8.0 million benefit for a reduction in the typical sampling error from 3.0 percent to 2.5 percent would carry over to the corrected estimates. While in the Hayami-Peterson estimates the production adjustment benefits are insignificant compared to the inventory adjustment benefits the the two components are much closer in the corrected estimates. The total benefits are $5.9 million and hence the lenefit-cost ratio is 16. Much smaller than the 824 ratio found by Hayami and Peterson, but still a fantastic investment comparvd to most public investment projects.

The estimates for the other improvements in the typical sampling error are given in Table 6. The benefit-cost ratio is higher for the reduction from 2.5 percent to 2.0 percent than for the previous reduction largely because the sampling error for rice is reduced significantly below the historical coefficient of variation for rice. The same thing occurs for potatoes in the reduction of the typical sampling error from 1.5 percent to 1.0 percent. The benefit-cost ratio is above one for the reduction of the typical sampling error down to 1.0 percent.

11. Conclusions

This paper considered the correction of the Hayami-Feterson estimates to allow for the following factors:

• 1. the computation of the benefits at an annual rate

• 2. the use of a total elasticity of demand for translation of quantity variation into price variation

• 3. the limitation of benefits to private domestic users

• 4. the exclusion of benefits from forecasts with a margin of error creater than the historical coefficient of variation of the crop.

The first factor involves the introduction of the factor of one half into the Hayami-Peterson estimates because they, in effect, computed social loss for a two year period. This, of course, reduced the magnitude of their estimates by 50 percent. The second correction when applied to the estimates for wheat, corn, oats and barley, reduced their value by approximately an additional 80 percent. The third factor brings about an additional reduction of 80 percent. The fourth factor wipes out most of the benefits for potatoes, which alone had accounted for about one-fifth of the social benefit of improved forecasts. The net result of the corrections was to scale down the benefit-cost ratios computed by Hayami and Peterson by as much as a factor of 50.

It was determined that there is justification for using the derived demand curve of agricultural commodities rather than the final demand curve for consumer products manufactured from agricultural products. the assumptions involved in this procedure are perfect competition in the industries processing the agricultural commodities and fixed technical coefficients for the consumer goods utilizing agricultural products.

It appears that the policy implications of Hayami and Peterson original study still hold; i.e., that social investment in improved crop forecasts is extremely beneficial relative to cost.

References

• Y. Hayami and W. Peterson, "Social returns to Public Information Services: Statistical Reporting of U.S. Farm Commodities," American Economic Review, Vol. LXII, No. 1, March, 1972, pp.119-130.
• D.S. Lowe, R. P. Summers and E. J. Greenblat; An Econowic Evaluation of the Utility of ERTS Data for Developing Ccuntries, August, 1974.