Alternative Expectations Models

Supply Response: Expectations Formation and Partial Adjustment

4.6. Alternative Expectations Models

Given the generally unsatisfactory performance of Nerlovian models, other specifications of the formation of product price expectations have been sought. Alternative approaches basically fall into three categories: those that rely on econometric techniques to identify a lag structure from past prices; those that make use of additional information available to producers about future prices; and the rational expectations approach that relies on economic theory to specify the mechanism of future price formation.

Econometric techniques specify a forecasting equation for as an autoregressive moving-average (ARMA) process of order (p, q) in past prices as follows: pt

e

(5) pt ,

e _i

i1



p ^pti _j

j1



q ^tj

where  is white noise. Substituting in (1), the parameters of the ARMA model (5) are estimated jointly with the other structural parameters using time series estimation techniques (see Judge et al., 1988; for an application, see Antonovitz and Green, 1990).

Additional information on prices typically combines futures prices (p^f), government price support programs (p^s), and cash prices (p) in a weighted average. In (1), expected price becomes:

pt e ₁pt

f ₂pt

s₃pt1 with



_i= 1.

Estimation of these weights allows to determine the relative importance of futures, support, and cash prices in the formation of farmers’ expectations. Chavas, Pope, and Kao (1983) find that price support programs explain much of the supply response for corn in the United States, while Gardner (1976) shows that futures prices are a good substitute for cash price lagged one year. However, futures prices do not fully capture government decisions, indicating that support price needs to be added separately.

Anderson, Dillon, and Hardaker (1977) suggest using a conditional price expectations model where the expected price is derived from a joint distribution of market and support prices. The mean support price, E(p^s), the standard deviations of market and support prices,

_p and _ps, and the correlation between market and support price, corr(p, p^s), are calculated each year on the basis of observed data for the previous five years. If p and p^s are jointly normally distributed, the mean of the conditional distribution of market prices for a given announced support price pt

s is:

pt

e E p p



^sp^s



^{E(p)corr(p,ps)}_^p

p^s

pt

sE(p^s)

 

^.

The expected market price E(p) is approximated by either the lagged cash price or the

futures price pt1

pt

f (see Shideed and White, 1989).

Empirical tests of the relative predictive power of these alternative specifications of price expectation show that not one model dominates the rest. To the contrary, since different market participants seem to use different ways of forming expectations, expectations are heterogenous and the best approach consists of combining these different approaches (Shonkwiler and Hinckley, 1985; Antonovitz and Green, 1990).

The rational expectations formulation, originally developed by Muth (1961), is the third option.

4.7. The Rational Expectations Approach*

4.7.1. General model

As seen above, the Nerlovian specification of adaptive expectations is based on the history of past prices with weights declining geometrically over time. The expectation formation model is thus:

pt

e f(past prices)

i1



 ^{(1)i1p}ti.

This approach has been criticized on the following grounds:

a. Price weights are ad hoc as opposed to being the explicit outcome of an optimization process.

b. Price predictions underuse the information available to the decision maker: (1) on the structural process of price formation, for which one should use knowledge of both supply and demand or whatever more complete structural model is the best available predictor; (2) on available forecasts about the exogenous variables that affect this process; and (3) on anticipated policy changes that affect price formation, a process that corresponds to the

“Lucas critique” (1976).

Rational expectations, by contrast, use the model’s prediction of the endogenous variables, including price, to form expectations. Instead of being based on past prices, forecasts are thus based on knowledge of a structural model of price determination, exogenous forecasts of the independent variables in this model, and expectations about the policy instruments in the model (Fisher, 1982; Eckstein, 1984). The expectations formation model is thus:

(model predictionsexogenous variable forecasts and expected policy changes).

pt e f

The general model we want to estimate is:

(6) Byt Ayt

e ₁x_1t ₂x_2t ut,

where y_t is a vector of observable endogenous variables, is a vector of unobservable

expected variables, yt

e

x1t is a vector of uncertain exogenous variables (including policy variables), and x2t is a vector of certain exogenous variables. In this model, structural parameters will be identified if the number of exogenous variables is greater than the number of expected variables (Wallis, 1980). The rational expectations hypothesis consists in postulating that expectations are given by the model predictions at the time when the predictions are formed, given information yt

e yt

e

x1t on the exogenous and policy variablesx1tat that time: yt

e E(y_t|information on x_1t^ and x2t).

The model can be rewritten as:

Byt Ayt

e ₁x_1t ₂x_2t ut,

or, taking expected values at the time when the prediction is made, E(BytAyt

e)(BA)yt

e ₁x_1t^  ₂x_2t. Solving for yt gives:

e

yt

e (B A)^1(₁x_1t^  ₂x_2t).

Substituting in (6),

Byt  A(BA)^1₁x_1t^  ₁x_1t A(BA)^1₂x_2t  ₂x_2tut,

where all variables are either directly observed (x_1t, x_2t) or predicted (x_1t^) by, for example, an autoregressive moving average of the type x1t x_1,t1 _t, E(_t) 0, where  is known.

4.7.2. Estimating Supply Response

Like in the cobweb model, the supply and demand equations for an agricultural commodity are (omitting the subscript t, which is common to all variables):

Supply : q₁₃₁₂p^e₁₁xu₁, Demand : p₂₃₂₁q₂₂yu₂,

where x includes either input prices (with a negative sign) or policy variables such as a fertilizer quota (with a positive sign), and y is income.

Under the rational expectations hypothesis, the expected price is the model equilibrium price at the time for which the prediction is made:

q^e ₁₃₁₂p^e ₁₁x^,

₂₁q^e ₂₃ p^e ₂₂y^,

where x^ and y^ are predicted exogenous variables as seen at the time for which the price expectation is made.

We solve this system by multiplying the supply function by – ₂₁ and adding the two equations:

23 21 13 21 12 21 11 22 ,

0 (    ) (1   )p^e  x^ y^ or p^e 1

1₂₁₁₂



(₂₃₂₁₁₃)₂₁₁₁x^₂₂y^



^.

Thus, if the input price x^ is expected to fall (or the fertilizer quota to rise), the expected commodity price p^e falls as supply is expected to rise. Replacing in the supply function, we obtain:

q ₁₃₁₂(₂₃₂₁₁₃) 1₂₁₁₂



 

 ₁₂₂₁₁₁

1₂₁₁₂ x^  ₁₂₂₂

1₂₁₁₂y^ ₁₁xu₁, or q₀ ₁x^₂y^₁₁xu₁,

where all variables are either observed or predicted exogenously.

The policy variable x thus has both a direct effect ₁₁ and an indirect effect,

 (₁₂₂₁₁₁) /(1₂₁₁₂)  ₁₁, which is of opposite sign but smaller than the direct effect.

The exogenous variables are predicted as:

x^₁x_t1₂x_t2, y^₃y_t1,

where the ’s have been estimated separately. The system of equations to be estimated is:

Supply : q₀ ₁₁x_t1₁₂x_t2₂₃y_t1₁₁xu₁, Demand: p₂₃₂₁q₂₂yu₂.

The supply equation contains lagged values of the exogenous variables in the demand equation. Note, however, that, in contrast to the Nerlovian adaptive expectations model, it does not contain lagged values of the endogenous variable. Nonlinear cross-equation restrictions on the supply equation parameters should be imposed in order to allow identification of the structural parameters of the supply function, particularly identification of the price response coefficient ₁₂.

4.7.3. Criticisms of the Rational Expectations Approach

While rational expectations offers a more logical approach to the formation of expectations than adaptive expectations, it suffers from both conceptual and empirical drawbacks. Theoretically, the approach tends to exaggerate the rationality of the decision- making process through which expectations are formed. Specifically:

a. Agents may not use all the information that could be available to them because acquiring it is costly. They also may appear not to use all the information available to them because change is costly and may exceed the resulting gains.

b. Agents may not use this information as “intelligently” as the model; that is, they do not know the model, or they have an incomplete understanding of the mechanisms of price determination. They may, however, be buying these predictions from economists like us or from specialized forecasting services that presumably have complete information and are using the model for predictive purposes.

c. Agents may not know how to forecast the exogenous variables and policy changes.

Empirically, rational expectations has, to this date, not proved its superiority to more ad hoc specifications of expectations formation such as the Nerlovian adaptive adaptations (Lovell, 1986). However, the approach suggests a rich research agenda to transcend the informality of adaptive expectations. To make progress in specifying the mechanism of expectations formation, what is needed is a more accurate understanding of how agents actually form their opinions about expected prices based on who they are, how they think about the future, their cost of accessing information, the quality of that information, and their expected benefits from using it.

Exercise 4

Supply Response for Groundnuts in Sub-Saharan Africa

In this exercise (file 4SUPPLY) you will estimate groundnut acreage responses to groundnut and millet producer prices using a data set based on the agricultural situation in Senegal from 1960 to 1988. Groundnuts are extremely important to the Senegalese economy, representing the major source of cash income for farmers and the principal source of export earnings for the country as a whole. A government-controlled marketing board determines

the producer price for the crop and extracts the difference between the world price and the producer price as government revenue. Traditionally, because world prices have been higher than producer prices, the export surplus has been a major source of government revenue.

However, the dramatic decline in world groundnut prices in the late 1980s forced the government to become a net subsidizer of the sector. The impact on farmers from the removal of this subsidy became a central issue in the policy debate over whether to continue government control of the sector or to let the private market determine producer prices.

Braverman and Hammer, in a study examined later, construct a multimarket model which predicts the effects of groundnut pricing policies on groundnut production and other crop prices and quantities, as well as on consumer and producer welfare. The price elasticities estimated in this exercise form the basis for such a model. The exercise will estimate long- and short-run elasticities.

Model specifications are based on the price determination mechanisms observed in Senegal. Groundnut prices are set by the marketing board and known before production.

Cereal prices de facto fluctuate with supply and demand and are not known to the farmer at the time of planting. Therefore, the effect of cereal prices on groundnut acreage is modeled as a lagged expectation of previous years’ prices. Rainfall is also not known before planting and is modeled as an expectation based on rainfall in the previous years. Two alternative lagged-expectation functions are compared for this variable in the exercise. We will also look at the effect on production of the structural adjustment policies implemented since 1979.

To improve the significance of the results, some marginal modifications have been made on the original data.