A Simple Farm Model for Saskatchewan

Motivation: A Model is Essential for Policy Analysis

Impact of policies

Farm's

Prodcution

Crop

Crop

1. Cost is fixed

   • Cost is based on the farmers' operation and the situation of farm or soil. For cerntain area and time period, we assume it fixed.

2. Yield is normally distributed

   • Yiled depends on something like weather or other random factors.

3. Price following a stochastic process.

   • Price changes frequently, even in short term, like other commodity price.

• Five major crops in SK: Wheat, Oats, Barley, Flax, Canola

• The monthly price data is from Statistics Canada "Estimated areas, yield, production, average farm price and total farm value of principal field crops, in imperial units, annually 001-0017".

  • The data covers 1985-12-01 to 2014-03-01

• The annual yield data is from Statistics Canada "Farm product prices, crops and livestock, monthly Description 002-0043"

  • The data covers 1991 to 2013

• The cost per acre data is from Government of Saskatchewan "Crop Planning Guide"

  • The data covers different soil zones, rotations, estimated yield, variable cost, fix cost.
    

• The farm data from Statistics Canada The 2011 Census of Agriculture

  • The data covers Farm numbers, type, size, receipts, area.

• A farm in black soil zone in SK.
  • Farm size: 1668 acre ( The 2011 Census of Agriculture);
    
  • Average price over three years per bussels ( Statistics Canada );
    
  • Estimated yield ("Crop Planning Guide");
  • Variable cost ("Crop Planning Guide").
  • Land use (Statistics Canada)

• Calibration method: Postive Mathematical Programming

• Advantage:

  • Minimal data requirement
  • Calibrate MP models exactly to observed behaviour
  • Optimum: combination of binding constraints and first-order conditions
  • Policy analysis: prediction of consequences and sensitivity analysis

• Three stages ( formalized by Howitt (1995a) )

  • Estimate output marginal cost
  • Estimate cost function
  • Policy analysis

• Maximize farmer's profit including a set of calibration constraints.

\[Max: R = \sum_{k=1}^n ( p_k x_k y_k - c_k x_k)\] \[Subjet \; to: \sum_{k=1}^n x_k \le 1668 \;\; (1)\] \[x_k \ge 0\] \[x_k \le x_k^{obs} + 0.01,\; \forall k; \; \; [\lambda_k] \;\; (2)\]

• \(p_k\) is price, \(x_k\) is land use, \(y_k\) is yield, \(c_k\) is cost.
  
• Where \((1)\) is nature resource constraint (land 1668),
• \((2)\) is the calibration constraint. Solve the problem in GAMS \(\to\) the associated shadow price \(\lambda_k\) for each crop.
  

• Assumption: a quadratic cost function: \(c_k = a x_k + b x_k^2\) for SK.
  \[\hat b_k = 2 \times \frac{\lambda_k}{x_k^{obs}} \; \text{and} \; \hat a_k=c_k^{obs} - 0.5 \times \hat b_k \times x_k^{obs} \]

• Replace the \(c_k x_k\) in the objective functioin \(\to\) solve the revised problem in GAMS \(\to\) duplicate the observed results \(\to\) cost functions for SK are calibrated .

• Based on these cost functions \(a_k x_k - b_k x_k^2\), if we want to evaluate the impact of policy,need to look at \(p_k x_k y_k\)

  • Historial information
  • Monte Carlo simulation for the price \(p_k\) and yield \(y_k\). (Turvey, 2012). \[R_{ij} = P_{ij} Y_{ij} - C_i\] \[Y_{ij} \sim N(E[Y_i], \sigma(Y_i))\] \[P_{ij} = P_{i0} e^{((\mu - \frac{1}{2}) \frac{7}{12} + \sigma N(0,1) \sqrt{\frac{7}{12}})}\]

• where \(C_i\) is the variable cost associated with each crop

• \(Y_{ij}\) is crop yield generated from a normal distribution

• Since yield data is a time series, in order to find the \(\sigma\) of yield, we need to detrend the yield data (Coble, 2013).
  • The easist way to detrend: \[Y_i = \beta_0 + \beta_1 t_i + \beta_2 t_i^2+ \; ... \; +\epsilon_i\]
  • Run a regression of yield on time using a polynomial form
  • Then the predict yield \(\hat Y_i\) is time trend
  • The standard deviation \(\sigma\) of residual \(\epsilon\) is what we want
  • For example, on next page the left graph is time trend for Wheat yield; the right is the residual.

• Recall \[P_{ij} = P_{i0} e^{((\mu - \frac{1}{2}) \frac{7}{12} + \sigma N(0,1) \sqrt{\frac{7}{12}})}\]

  • \(P_{i0}\) is the initial spring price as of March 2014;
  • Price generation is based on a 7-month growing season;
  • \(P_{ij}\) is the random commodity harvest price generated by a log normal (Brownian) process with:
    • drift \(\mu\),
    • volatility \(\sigma\),
    • random deviate drawn from a normal distribution with zero mean and variance of 1.0;

• Geometric Brownian Model follow this equation: \[dS_t = \mu S_t dt + \mu S_t dW_t\]

  • \(\mu S_t dt\) is deterministic part; \(\mu S_t dW_t\) is stochastic part.
  • \(dW_t\) is the Brownian motion, which follows random normal distributioin \(N(0,t)\).
  • \(\mu\) is drift; \(\sigma\) is diffusion. \(\sigma\) increases the amount of randomness entering the system.

• With the cost, yield, and price data, the SK farm model can be structured as follows:

\[Min: \sigma_p^2 = \frac{1}{m} \sum_{j=1}^m ( \pi_j - E[\pi] )^2\] \[Subjet \; to: \sum_{i=1}^n x_k = 1668 \; \] \[\sum_{i=1}^n E[R_i] x_i = K \; (2)\]

• \(\sigma_p^2\) measures the unstability of farmers' income ; \(\pi\) is the profit for one state
• (2) \(K\) is the target profit
• \(E[R]\) is expect the average profit per acre
• \(m\) is 1000 simulation state

• The impact of policy can be added to model. For example,whole farm insurance:

\[\sum_{i=1}^n R_{1,i} x_i + Max[Z - \sum_{i=1}^n R_{1,i} x_i, \; 0]- \frac{\delta}{m} \sum_{j=1}^m Max[Z - \sum_{i=1}^n R_{j,i} x_i, \; 0] - \pi_1 = 0 \] \[\sum_{i=1}^n R_{m,i} x_i + Max[Z - \sum_{i=1}^n R_{m,i} x_i, \; 0]- \frac{\delta}{m} \sum_{j=1}^m Max[Z - \sum_{i=1}^n R_{j,i} x_i, \; 0] - \pi_m = 0 \]

• \(Z\) is the income coverage level to be protected by insurance such as 70% farmers history income
• \(Max[Z - \sum_{i=1}^n R_{1,i} x_i, \; 0]\) is indemnity payout that farmer can get from insurance
• \(\frac{\sigma}{m} \sum_{j=1}^m Max[Z - \sum_{i=1}^n R_{j,i} x_i, \; 0]\) is permium that farmers need to pay.
• \(\frac{\sigma}{m}\) is subsidy rate. If \(\delta\) = 0.50, the premium is subsidized by 50 percent