GIt = DPt + LDPt + CCPt
Where DPt = 0.85PD × 0.9Et-1(Yt),
LDPt = Et-1(Yt) max(0, LR - Pt),
CCPt = 0.85 × 0.935 Et-1(Yt) max[0, PT - PD - max(Pt ,LR)]
where NCt is the net income from producing and selling the crops in the cash market; CIt is the
net income from purchasing yield-based Multiple Peril Crop Insurance (MPCI); FIt is the net
income from hedging in the futures market; and GIt is the net income from government programs.
Pt and Yt represent cash prices1 and yields for winter wheat at harvest time respectively,
with PCt as the production cost. Ft is the futures price at time t and the futures market is treated
as unbiased. xt-1 is the hedging amount determined at a previous time period which is positive
for a long position and negative for a short position. xt-1 is in bold face to indicate its status as a
choice variable. TCt is the transaction cost of trading futures. Pb is the base price used to
calculate the indemnity from crop insurance with Pret as the premium2. zt-1 is the coverage
selection of the insurance and is also in bold face to indicate a choice variable. DP is the direct
payment program which gives a constant payment to farmers, LDP is the loan deficiency
payment, and CCP is the counter cyclical payment. PD is the direct payment rate, LR is the loan
rate, and PT is the target price. The formulation of DP, LDP, and CCP is specified according to
the 2002 Farm Bill and calibrated to PNW wheat growers, the chosen area for the empirical
analysis.
Due to the nonlinearity in the objective function and the random interrelationships
among variables, closed-form optimal solutions are unavailable in the dynamic optimization.
Therefore empirical solutions are obtained by numerical methods. For the dynamic optimization,
1 Cash price is a farm gate price after transportation cost is deducted from the spot market cash price.
2 The premium of the current year’s crop insurance is paid at harvest time.