Nerlovian Models of Supply Response

Supply Response: Expectations Formation and Partial Adjustment

4.2. Nerlovian Models of Supply Response

A central problem in the estimation of the supply response equation derived in Chapter 3, q = q(p, z),

is that producers respond to expected as opposed to actual prices. Usually, the observed prices are market or effective farm-gate prices after production has occurred, while production (planting) decisions have to be based on the prices farmers expect to prevail several months later at harvest time. Because of the time lag involved in agricultural production, modeling the formation of expectations is thus an important issue in the analysis of agricultural supply response.

Another problem is that the observed quantities may differ from the desired ones because of adjustment lags in the reallocation of variable factors. When the price of a crop changes, it may take several years before farmers can reach their desired production patterns at the new price. Therefore, before applying the model to the actual data, one has to specify these adjustment lags explicitly. In the following sections, we discuss Nerlovian models which are built to handle these two dynamic processes.

4.2.1. The General Nerlovian Supply Response Model

Models of supply response can be formulated in terms of yield, area, or output response of individual crops, for instance, the desired area to be allocated to a crop in period t is a function of expected relative prices and a number of shifters:

(1) q_t^d ₁₂p_t^e₃z_tu_t.

In this equation, q_t is the desired cultivated area in period t; is the expected price (or, more generally, a vector of relative prices including the price of the crop itself, prices of competing crops, and factor prices, with one of these prices chosen as numéraire); z_t is a set of other exogenous shifters, principally private and public fixed factors and truly exogenous variables such as weather; u_t accounts for unobserved random factors affecting the area under cultivation and has an expected value of zero; and the _i’s are parameters (or elasticities if the variables are expressed logarithmically) with ₂ the long-run coefficient (elasticity) of supply response.

d p_t^e

Because full adjustment to the desired allocation of land may not be possible in the short run, the actual adjustment in area will be only a fraction  of the desired adjustment:

(2) q_t q_t1(q_t^dq_t1)v_t, 01,

where q_t is the actual area planted of the crop,  the partial-adjustment coefficient, and v_t a random term with zero expected value.

The price the decision maker expects to prevail at harvest time cannot be observed.

Therefore, one has to specify a model that explains how the agent forms expectations based on actual and past prices and other observable variables. For example, in a formulation that represents a learning process, farmers adjust their expectations as a fraction  of the magnitude of the mistake they made in the previous period, that is, of the difference between the actual price and expected price in t – 1:

p_t^e p_t1^e (p_t1p_t^e_1)w_t, 0 1, or (3) p_t^e p_t1(1)p_t1^e w_t,

where is the price that prevails when decision-making for production in period t occurs,

 is the adaptive-expectations coefficient, and w_t is a random term with zero expected value.

An alternative interpretation of this learning process is that the expected price is a weighted sum of all past prices with a geometrically declining set of weights:

p_t1

p_t^e



1



^i1

i1



 ^pti,

where the right-hand side geometric series is the solution to equation (3), which gives the certainty equivalent to p_t^e.

Since p_t and q_t are not observable, we eliminate them from equations (1), (2), and (3).

Substitution from equations (1) and (3) into equation (2) and rearrangement give the reduced form:

e d

(4) q_t ₁₂p_t1₃q_t1₄q_t2 ₅z_t ₆z_t1e_t,

where:

₁₁,

₂ ₂ , the short-run coefficient (elasticity) of supply response,

₃(1)(1),

₄  (1)(1),

₅₃,

₆  ₃(1),

e_t v_t(1)vt1u_t(1)ut1₂w_t.

Equation (4) is the estimable form of the supply response model defined by equations (1), (2), and (3). This reduced form is overidentified, since there are six reduced- form coefficients  but only five structural parameters (₁, ₂, ₃, , and ). To obtain a unique solution for the latter, a nonlinear constraint must be imposed on the parameters of the reduced form:

₆² ₄₅² ₃₅₆ 0.

The model should be estimated using nonlinear, maximum-likelihood techniques, and correction needs to be made for serial correlation in the error terms (see Table 4.1 for the structure of the residuals displaying serial correlation). The structural coefficients can be solved for with the following equations: .

²(₃2) 1₃₄ 0,

 1₄/(1),

₁₁/,

₂ ₂ /, the long - run coefficient (elasticity) of supply response,

₅ ₅/.

Table 4.1 approximately here

The short-run price response is estimated by ₂, and the long-run price response is calculated as ₂, where ₂ = ₂/≥ ₂ since both and  ≤ 1. As expected, the long-run supply response exceeds the short-run supply response.

4.2.2. Restricted Nerlovian Supply Response model

If exogenous shifters (z) are not included in the model, then ₃ = 0 in the structural form and ₅ = ₆ = 0 in the reduced form. The reduced-form equation then becomes:

q_t ₁₂p_t1₃q_t1₄q_t2 e_t.

Because  and  enter the remaining reduced-form coefficients symmetrically, the model is underidentified, and no solution can be found for  and . However, the short-run and long- run price responses can be calculated as ₂ and ₂ = – ₂/(₃ + ₄ – 1), respectively. The

error term is the same as with the general model and indicates the presence of serial correlation (see Table 4.1). This model can be estimated with ordinary least squares (OLS) or generalized least squares to correct for serial correlation.

4.2.3. Simplified Models with either No Partial Adjustment or No Expectations Formation We give in Table 4.1 a number of additional estimable versions of the Nerlovian model where there is either no partial adjustment ( = 1) or no expectations formation (  The former would apply to crops where there are no specialized fixed factors of any significance and adaptation can be complete in one period, implying q_t The latter applies to situations where administered prices are announced at planting time, such as in the case of Egypt, which we analyze later in this chapter. In this case, This latter model is exactly identified and has frequently been used when forward guaranteed prices are announced. Note that in all models with either  = 1 or   , the long-run elasticity of supply response is

q_t^d. p_t^e p_t1.

₂ ₂/(1₃). All these models have and q_t1 as exogenous variables and must consequently be carefully distinguished on the basis of the z variables included.

Note that, when no z variables are present, the restricted models with either  = 1 or

  cannot be distinguished from one another at the level of the reduced form.

p_t1