## _wp1164

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---

### Supply: Production and Storage — Key mechanics and observations
- Total supply ("amount on hand") in any period = new harvest + storage from previous period.
- New harvest depends on area planted and current yield; yield reflects technological advances and idiosyncratic factors (e.g., weather, fertilizer cost).
- Storage carried forward reflects an optimal decision from the previous period; storage volatility reflects speculative behavior.
- Long-run and trend observations:
  - Annual global wheat harvest has consistently trended upward since the 1960s.
  - Area planted has been fluctuating around 220 million hectares, implying rising yield per hectare drives output growth.
  - Trend growth in yield has slowed since the mid-1990s.
  - Large yield shocks in the 1970s; moderate shocks in the 1980s; volatility rose again after the mid-1990s.
- Modeling assumptions for yield and expectations:
  - (1) Market participants form expectations via a backwards ten-year moving-window detrending regression of log yields.
  - (2) Suppliers decide area planted at beginning of each harvest year; area decision is known to every market participant and no area surprises occur during the harvest year.
  - Consequences under these simplifications:
    - Next-year yield follows a log-normal distribution with mean and variance from recursive regressions.
    - The only uncertainty in the model is on future yield.
    - Realized supply shocks = area planted × (realized yield − expected yield).
- Storage motives:
  - Precautionary storage: public grain stocks, relatively stable.
  - Speculative storage: endogenous, volatile, based on expected price changes.
  - Empirical/simulation evidence: without storage food prices would become more volatile in both the original Deaton and Laroque (1992) model and the augmented model.

### Prototype competitive storage model — framework and implications
- Aggregate structure:
  - Total demand = current consumption demand (depends only on current price) + speculative demand (function of current and expected next-period price).
  - Total supply = new crops + carryover storage.
- Accounting and nonarbitrage condition:
  - Notation: p_t (actual price), z_t (new harvest), I_t (inventory), x_t (supply available this trading season).
  - Storage with constant linear depreciation rate δ: x_t = z_t + (1 − δ) I_t.
  - Discount factor β = 1/(1 + r).
  - Nonarbitrage equilibrium price: p_t = Max [ β(1 − δ) E_t[p_{t+1}], P(x_t) ].
  - Inventory rule: I_t = Max[ x_t − D(p_t), 0 ].
- Key model insights:
  - Price is a function of availability; expected future prices are functions of future availability.
  - If storage is costly (β(1 − δ) < 1) and new harvest z_t is i.i.d., market price functions converge to a stationary rational expectations equilibrium (SREE).
  - Nonnegativity constraint on storage generates asymmetry: positive shocks absorbed by speculators; severe negative shocks can exhaust storage and trigger rapid price spikes.
- Example calibration (baseline parameters based on wheat price sample 1969–2008):
  - Real Interest Rate r: 1.1%
  - Discounting Rate β = 1/(1 + r): 0.989
  - Std Dev of Shocks to Log Output σ: 0.037
  - Mean of Shocks to Log Output μ: 0
  - Depreciation Rate δ: 0.05
  - Consumption Price Elasticity ρ: 0.5

### Augmented model — extensions and interpretation
- Augmentations:
  - Output process decomposed: harvests = area planted A_t × trending yield Y_t × random shock z_t; shock series estimated from a 10-year recursive detrending model.
  - Harvests follow time-varying log-normal distributions with a deterministic, constantly updating trend.
  - Consumer demand includes trend growth approximated by population growth λ^D_t; augmented demand: λ^D_t D(p_t); inverse demand: P(x_t / λ^D_t).
  - Monetary policy shock: replace constant real interest rate with actual U.S. interest rate series.
- Augmented nonarbitrage condition and accounting:
  - p_t = Max [ β(1 − δ) E_t[p_{t+1}], P(x_t / λ^D_t) ]
  - p_t = f_t(x_t / λ^D_t); p_{t+1} = f_{t+1}(x_{t+1} / λ^D_{t+1})
  - x_{t+1} = A_{t+1} Y_{t+1} z_{t+1} + (1 − δ) I_t
  - I_t = Max[ x_t − λ^D_t D(p_t), 0 ]
- Interpretation:
  - Each period has its own converged SREE price function because adaptive expectations and time-varying parameters make beliefs period-specific.
  - Computational burden: need many price functions (in the sample, 40 price functions vs a single function in Deaton and Laroque (1992)).

### Estimation approach (Method of Simulated Moments) and results
- Free parameters estimated: consumption price elasticity ρ and storage depreciation rate δ.
- Estimation approach:
  - For each candidate (ρ, δ), simulate calibrated market price functions and generate artificial prices for 1969 to 2008 by inputting actual shocks.
  - Objective function matches first three autocorrelations:
    - min Gap(ρ, δ) = ∑_{i=1}^{3} ( AC(i) − ÂC(i) )^2
    - AC(i): i-th order autocorrelation of actual prices
    - ÂC(i): i-th order autocorrelation of simulated prices
- Estimation results:
  - Estimated constant decay rate: 2.3 percent
  - Estimated price elasticity of consumption: about 0.19
  - Minimized value of objective function: about 0.008 (implies maximal deviation of any of first three autocorrelations about 0.09)
  - If restricting objective function ≤ 0.01 (max deviation 0.1 in autocorrelation), parameter ranges:
    - decay rate between 1.8 and 2.8 percent
    - elasticity between 0.12 and 0.22
- Rationale for not using GMM:
  - Forecasting errors may not be mean-zero under adaptive expectations, invalidating efficient market moment conditions.
  - Conversion problem between real price (US$ per metric ton) and simulated index series; no clear preferable conversion.
  - GMM can yield unstable or implausible estimates (e.g., negative decay rates) and is sensitive to grid resolution.
- Numerical notes:
  - With the endogenous gridpoints algorithm and hardware described, a trial for a particular (δ,ρ) pair takes 3–5 minutes; overall MSM search process required about eight hours.

### Model performance versus data (1969–2008) — successes and limitations
- Matched patterns with MSM-estimated parameters:
  1) Declining long-run trend of real prices with slight reversal after mid-1990s.
  2) Large variations with general alignment in ups-and-downs though magnitudes vary.
  3) High autocorrelations (absent in Deaton and Laroque (1992, 1996)), replicated here mainly due to trending output and demand.
  4) Asymmetric price movements with close to 1 skewness and positive excess kurtosis.
- Limitations:
  - Model does not generate extreme price hikes of magnitude similar to 1973–74.
  - Simulated excess kurtosis: 0.7; actual series excess kurtosis: 2.6.

### Comparative statics — drivers of price dynamics
- Effect of output trend:
  - Upward per-capita output trend implies downward pressure on real prices.
  - As output grows, p* falls, and investors delay stock-taking until higher availability (x* increases); p* decreases.
- Effect of demand trend:
  - Growing consumer demand pushes market price functions upward; p* increases for given supply.
  - Quantitative examples:
    - Demand trend alone would cause p* to rise from 0.8 in 1970 to 24 in 2008.
    - Supply trend alone would have resulted in p* of 0.017 in 2008 and around 0.70 in 1970.
  - Conclusion: output and demand trend effects dwarf all other factors combined; demand trend accelerating while supply trend decelerating increases price impacts since mid-1990s.
- Effect of yield shocks:
  - Yield shocks are log-normally distributed from 10-year recursive regressions; smaller relative to long-run trend.
  - Higher yield risk raises the market price function due to convexity; speculative demand begins at a higher current price level.
- Effect of interest rates:
  - U.S. real interest rate varied between -3.84 percent (1974) and 5.19 percent (1983) over 1969–2008.
  - Pure interest rate movements generally have small price impacts relative to demand and output trends.
  - Nonlinear threshold behavior:
    - Estimated threshold lies at -2.3 percent.
    - If real rate falls below -2.3 percent, stocking can become a one-way bet and prices can runaway.
  - Historical illustration:
    - 1973–74: real interest rate -3.84 percent, price hike of 75 percent (largest in sample) with one of the largest negative yield shocks.
    - Simulation: two-thirds of the 1973–74 price hike may be explained by the real rate passing the threshold; moving from -1.35 percent (1973) to -3.84 percent (1974) raises price by around 50 percent.
  - Sensitivity within normal ranges is modest: e.g., 1983 (5.19 percent) vs 2008 (-0.026 percent) more than 500 basis points reduction yields around 5 percent rise in stock-taking point price.
- Effect of depreciation rates (storage cost):
  - Varying storage cost between 0 and 15 percent shows reduced storage cost induces earlier stock-taking (x* smaller) and smoothes prices.
  - Overall effect smaller than demand/output trends unless dramatic industrial innovation occurs.
- Effect of consumption elasticity ρ:
  - More inelastic demand (lower ρ) produces more convex consumer demand and market price, more frequent violent price moves, and lower x*.
  - Example: ρ = 0.50 implies less violent price movement relative to more inelastic cases.

### Overall effects and narrative on long-run vs short-run
- Combined simulation using estimated (δ,ρ) and actual time-varying parameters for 1970, 1980, 1990, 2000, 2008 yields:
  1) 1970 → 1980 → 1990: market conditions improved (per capita output grew and price dropped).
  2) mid-1990s → 2008: market conditions deteriorated; 2008 marginally more relaxed than 2000.
- Drivers of tightening since mid-1990s:
  - Accelerating consumer demand, relative output slowdown, negative interest rate shock, and inelastic demand — potentially contributing to the 2007–08 food crisis.
- Short-run vs long-run:
  - Risk of medium- to long-term rising prices because the system remains in a low output/demand ratio regime observed since mid-1990s.
  - Trend output/demand ratio largely determines long-run price movements; short-run yield shocks and other factors cause small divergences.
  - Conceptual link: resembles permanent income hypothesis; nonarbitrage condition akin to Euler equation in precautionary savings.

### Role of storage and model variants
- Storage relevance when per-capita supply has a trend:
  - Yes — year-to-year trends move little; yield shocks and interest rate volatility dominate short-term decisions; speculators with one-season horizons still find storage relevant.
- Comparative assessment of five model settings (I–V) using same estimated output shock:
  - (I) Barebone (no storage, no trends, i.i.d. output, constant interest rate): very small serial correlations, smallest skewness and kurtosis.
  - (II) Barebone plus trend: significantly increases autocorrelations but still below actual data.
  - (III) Deaton and Laroque (1992): adds storage to barebone — better first-order autocorrelation and skewness; higher-order autocorrelations remain small.
  - (IV) Augmented model: integrates storage, trends, time-varying interest rates — best overall fit to actual data. Trend and storage together needed; nonnegative storage constraint is key for asymmetric price movements.
  - (V) Augmented plus oil (ad hoc oil price adjustment assuming complete pass-through): improves fit for excess kurtosis and skewness, at cost of slightly reduced autocorrelations; helps capture extreme price episodes like 1973–74 and 2007–08.
- Limitation: augmented model cannot fully reproduce extreme peakedness seen in crisis periods; oil price shocks are a plausible additional factor.

### Other food commodities — similarities, calibration, and caveats
- Maize, rice, and soybeans display supply and demand characteristics similar to wheat.
  - Rice: output/demand ratio increasing while real price falls.
  - Maize: price behavior aligns with expectations except for bio-fuel demand after 2000s.
  - Soybeans: output growth outpaces GDP and population growth; output accelerated after mid-1990s.
- Key findings:
  - Output/demand ratio combined with intertemporal storage explains long-run food commodity price movements and high autocorrelations.
  - Short-run abrupt fluctuations caused by small-probability severe events (e.g., drought or oil price shocks), not by large-variance shocks.
  - Yield shock distributions calibrated from data are quite stable with small variances.
  - Monetary policy has limited role in normal times but can have nonlinear significant impact when the real rate becomes deeply negative.
- Model calibration applied to other commodities:
  - Uses estimated depreciation rate 2.3% and price elasticity 0.19; MSM grid-search gap = 0.008.
- Limitations and omitted factors:
  - Model omits oil/energy price effects, trade policy responses (export bans), substitution among food commodities.
  - Application to energy prices is not straightforward due to different elasticity and supplier-instrument identification issues.
- Historical/data conventions:
  - Annual price convention: average monthly price over the 12 months in an international trade year; for wheat the trade year is July 1–June 30.
  - The “2008 food crisis” refers to price hikes following low harvests in 2006 and 2007; consecutive bad harvests left very low storage before 2008 new crops.
  - Since July 2008, a positive 2008 harvest shock led world staple prices to decline from their late-2006/early-2008 highs.

*Source: IMF working paper — section 2.1 "Supply: Production and Storage" (content excerpt).*

### 2.1    Supply:  Production and Storage .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .7

### 2.1    Supply:  Production and Storage

### Overview
- Total supply ("amount on hand") in any period = new harvest + storage from previous period.
- New harvest depends on area planted and current yield; yield reflects technological advances and idiosyncratic factors (e.g., weather, fertilizer cost).
- Storage carried forward reflects an optimal decision from the previous period; storage volatility reflects speculative behavior.

### Long-run and trend observations
- The annual global wheat harvest has consistently trended upward since the 1960s.
- Area planted has been fluctuating around 220 million hectares, which implies the rising yield is driven mainly by yield per hectare.
- Although the trend in yield is clearly upward, trend growth has slowed since the mid-1990s.
- Large shocks to yields were observed during the 1970s; shocks were moderate in the 1980s but again became volatile after the mid-1990s.

### Modeling assumptions for yield and expectations
- To capture trend and shock components of actual yield the following simplifying assumptions are made:
  - (1) market participants form their expectations on the distribution of next year’s yield through a backwards ten-year moving-window detrending regression of log yields.
  - (2) suppliers at the beginning of each harvest year make optimal decisions about areas to be planted, taking into account expected profits; such area decision is known to every market participant and no area surprises to the market throughout the harvest year.
- Under these two simplifications:
  - the yield next year will follow a log-normal distribution, with mean and variance defined by the recursive regressions at each point of time;
  - the only uncertainty in the model is on future yield;
  - realized supply shocks are defined as areas planted times deviations of realized from expected yields.
- Appendix A contains the sources for data on wheat output, area, yield, stock, and etc.

### Storage: precautionary vs. speculative
- Storage comprises two parts: precautionary and speculative.
  - Public grain stocks are usually taken for precautionary purpose; they are relatively stable, barring revisions in regulations.
  - Speculative storage is an endogenous decision incorporating expected price changes over time and is much more volatile.
- Empirical evidence (Figure 1(e) as cited) suggests a fair amount of stock volatility when both motives are included.
- A common perception is that speculation drives up price and increases volatility; however, the simulation results reported indicate that without storage food prices would become more volatile in both the original Deaton and Laroque (1992) model and the augmented model.

### Implications for identifying supply shocks
- Supply shocks are interpreted as deviations of realized yields from expectations (e.g., due to inclement weather or fertilizer cost).
- Because area planted decisions are treated as known at the beginning of the harvest year, shocks originate primarily from yield deviations rather than area surprises.
- The time-varying nature of the yield distribution (via the adaptive learning/detrending rule) differs from the i.i.d. harvest processes assumed in earlier work.

*Source: IMF working paper — section 2.1 "Supply: Production and Storage" (content excerpt).*

### 3.1    The Prototype Competitive Storage Model

### 3.1    The Prototype Competitive Storage Model

### Model framework
- Total demand = current consumption demand (depends only on current price) + speculative demand (function of current and expected next period price).
- Total supply = new crops + storage from previous season.
- Consumer shadow price pc_t equates total supply with demand for current consumption.
- Speculator shadow price function ps_t renders speculators indifferent about taking stocks; ps_t is a decreasing function of the total amount stored and has a maximal value p*_t corresponding to zero stock-taking.
- Two equilibrium regimes:
  - If pc_t ≥ p*_t: no new storage; market price determined solely by consumer demand.
  - If pc_t ≤ p*_t: speculators take new stock until speculative demand drives price up to break-even.

### Accounting identities and price as nonarbitrage condition
- Notation:
  - p_t: actual price
  - z_t: new harvest
  - I_t: inventory
  - x_t: supply available this trading season
- Storage with constant linear depreciation rate δ:
  - x_t = z_t + (1 − δ) I_t
- Consumer demand D(p_t), inverse demand P(x_t), market price function f(x_t).
- Speculators are risk-neutral; discount rate β = 1/(1 + r).
- Nonarbitrage equilibrium price:
  - p_t = Max [ β(1 − δ) E_t[p_{t+1}], P(x_t) ]
- Market price functions:
  - p_t = f_t(x_t)
  - p_{t+1} = f_{t+1}(x_{t+1})
  - x_{t+1} = z_{t+1} + (1 − δ) I_t
  - I_t = Max[x_t − D(p_t), 0]

### Key model insights
- Price is treated as a function of availability (a function), not a single number; expected future prices are functions of future availability, which is uncertain.
- Under conditions (storage costly: β(1 − δ) < 1; new harvest z_t i.i.d.), the sequence of market price functions converges to a stationary rational expectations equilibrium (SREE) — a time-invariant market price function (Deaton and Laroque (1992)).
- With i.i.d. output, the only linkage between consecutive years is previous storage; generated price series are stationary allowing computation of mean, variance, skewness, kurtosis as for a cross-sectional sample.
- Numerical methods (e.g., endogenous grid points algorithm) are used to solve model complications from output uncertainty and nonnegative storage constraint.

### Price dynamics implications
- A kink point (x*, p*) separates consumer-determined prices (when pc_t > p*) from prices influenced by speculative storage.
- Two notable features due to speculative storage:
  - Asymmetry and peakedness of price movements arise from nonnegativity constraint: positive output shocks absorbed by speculators; severe negative shocks may exhaust storage leading to rapid and large price increases.
  - Price series may lack trend and exhibit low autocorrelation if harvest shocks are stationary (i.i.d.), resulting in stationary price movements.

### Example calibration (baseline parameters based on wheat price sample 1969-2008)
- Real Interest Rate r: 1.1%
- Discounting Rate β = 1/(1 + r): 0.989
- Std Dev of Shocks to Log Output σ: 0.037
- Mean of Shocks to Log Output μ: 0
- Depreciation Rate δ: 0.05
- Consumption Price Elasticity ρ: 0.5

---

### 3.2    The Augmented Model

### Augmentations introduced
- Output process:
  - Harvests z_t in prototype assumed i.i.d.; augmented model decomposes harvests into area planted A_t, trending yield Y_t, and random shock z_t.
  - Shock series estimated from a 10-year recursive detrending model.
  - Annual harvest A_t Y_t z_t follows time-varying log-normal distributions and a deterministic, constantly updating trend.
- Consumer demand:
  - Introduce trend growth approximated by population growth λ^D_t.
  - Augmented consumer demand: λ^D_t D(p_t); inverse demand: P(x_t / λ^D_t).
  - Time-varying trend assumed not to affect price elasticity.
- Monetary policy shock:
  - Replace constant real interest rate with actual U.S. interest rate series.
  - Lower real interest rates reduce opportunity cost of holding stocks, encouraging speculative storage and potentially sustaining or raising prices.

### Augmented equilibrium conditions
- Augmented nonarbitrage condition:
  - p_t = Max [ β(1 − δ) E_t[p_{t+1}], P(x_t / λ^D_t) ]
- Market price functions and accounting:
  - p_t = f_t(x_t / λ^D_t)
  - p_{t+1} = f_{t+1}(x_{t+1} / λ^D_{t+1})
  - x_{t+1} = A_{t+1} Y_{t+1} z_{t+1} + (1 − δ) I_t
  - I_t = Max[ x_t − λ^D_t D(p_t), 0 ]

### Interpretation of augmented model
- Each period has its own converged SREE price function due to adaptive expectations and time-varying parameters.
- Market participants update beliefs period-by-period; at any point they believe expected future price functions equal today’s, yielding a SREE for that period.
- Computational burden increases: in the sample, compute 40 price functions (one per year) versus a single price function in Deaton and Laroque (1992). Memory and computation challenges arise during estimation.

---

### 4    Matching Theory And Data — Estimation (Method of Simulated Moments)
- Free parameters estimated: price elasticity of consumption ρ and storage depreciation rate δ.
- Estimation approach:
  - For each candidate (ρ, δ), simulate calibrated market price functions and generate artificial prices for 1969 to 2008 by inputting actual shocks.
  - Compare simulated and actual prices using method of simulated moments (MSM).
- Objective function to match first three autocorrelations:
  - min Gap(ρ, δ) = ∑_{i=1}^{3} ( AC(i) − ÂC(i) )^2
    - AC(i): i-th order autocorrelation of actual prices
    - ÂC(i): i-th order autocorrelation of simulated prices
- MSM advantages:
  - Convenient when sample moments are complex non-linear functions of parameters.
  - Objective function intuitively assesses matching of first three autocorrelations in order and magnitude.

### Estimation results and parameter ranges
- Estimated constant decay rate: 2.3 percent
- Estimated price elasticity of consumption: about 0.19
- Minimized value of objective function: about 0.008 (implies maximal deviation of any of first three autocorrelations about 0.09)
- If restricting objective function ≤ 0.01 (max deviation 0.1 in autocorrelation), parameter ranges:
  - decay rate between 1.8 and 2.8 percent
  - elasticity between 0.12 and 0.22
- Notes on comparability:
  - Average price elasticity for breads and cereals = 0.30 (ERS, USDA); typical ranges for advanced/emerging markets: 0.10 to 0.30.
  - Annual storage cost anecdotal evidence: around 2-5 percent depending on wheat prices; survey by Kenkel (2008) implies annual depreciation rates of 3.5 and 2.4 percent in 2005 and 2008 respectively given listed bushel prices and variable costs.

### Rationale for not using GMM
- Two obstacles to GMM:
  1. Forecasting errors μ_{t+1} = p_{t+1} − p^{Pred}_{t+1} may not be mean-zero because adaptive expectations make p^{Pred}_{t+1} resemble simulated series p^{Sim}_t, and p^{Sim}_t close to observed p_t, while observed series display asymmetric non-mean-zero movements — voiding efficient market moment conditions.
  2. Conversion problem between real price (US$ per metric ton) and simulated index series; no conversion choice is clearly preferable.
- GMM implementation can lead to unstable or implausible estimates (e.g., negative decay rates) and sensitivity to grid resolution in numerical solutions.

---

### 4.2    Model performance in matching data
- With MSM-estimated parameters, simulated series compared to actual 1969–2008 series generate several matched patterns:
  - 1) Declining long-run trend of real prices with slight reversal after mid-1990s.
  - 2) Large variations with general alignment in ups-and-downs though magnitudes vary.
  - 3) High autocorrelations (feature missing in Deaton and Laroque (1992, 1996)); replicated here mainly due to trending output and demand.
  - 4) Asymmetric price movements with close to 1 skewness and positive excess kurtosis.
- Limitations:
  - Model does not generate extreme price hikes of magnitude similar to 1973-74.
  - Simulated excess kurtosis: 0.7; actual series excess kurtosis: 2.6.

*Source: _wp1164 - 3.1    The Prototype Competitive Storage Model*

### 4.2    Comparative Statics

### 4.2    Comparative Statics

### The Effect of Output Trend
- Upward per-capita output trend implies downward pressure on real prices.
- In simulations (Figures 5(a)–5(c)):
  - p* falls as output grows over the past four decades.
  - Magnitude of price changes is proportional to output growth: large reductions in real prices when output grew strongly between 1970 and 1990; marginal reductions when output stagnated after the mid-1990s.
  - With growth of per capita output:
    - market price function becomes increasingly relaxed;
    - investors delay stock-taking until total supply reaches higher levels (x* increases);
    - investors wait until current prices approach bottom (p* decreases).

### The Effect of Demand Trend
- Growing consumer demand continuously pushes market price functions upward (Figure 6(a)).
- For any given supply, rising price functions imply increasing prices and more stringent market conditions: p* keeps increasing and market price rises for a given market supply.
- Quantitative dominance of trends:
  - Demand trend alone would cause p* to rise from 0.8 in 1970 to 24 in 2008 (a thirty fold price run-up in less than four decades).
  - Supply trend alone, if left alone, would have resulted in p* of 0.017 in 2008 and around 0.70 in 1970 (a paltry one fortieth of the 1970 price).
- Combined assessment:
  - Output and demand trend effects dwarf all other factors combined.
  - Demand trend is accelerating while supply trend is decelerating, increasing the price impact of consumer demand and decreasing the impact of supply trend over the past four decades.

### The Effect of Yield Shocks
- Yield is subject to unexpected shocks, assumed log-normally distributed and derived from 10-year recursive regressions.
- Compared to long-run trend, yield shocks are smaller in magnitude and thus have relatively less impact.
- Figure 6(b) comparison of extremes:
  - Maximal yield shock forecasted in 1978 for 1979 versus minimal shock forecasted in 2007 for 2008.
  - Higher yield risk induces a higher market price function due to convexity: expected next-period price rises and speculative demand begins at a higher current price level, indicating tighter current market conditions.

### The Effect of Interest Rates
- U.S. real interest rate varied between -3.84 percent (1974) and 5.19 percent (1983) over 1969–2008.
- Pure interest rate movements generally have small price impacts relative to demand and output trends and are smaller than yield shocks for reasonable cross-year variation.
- Nonlinear threshold behavior:
  - Estimated threshold lies at -2.3 percent.
  - Under estimated cost of taking storage, condition β(1−δ)<1 holds; if real rate goes further negative than -2.3 percent, stocking becomes a one-way bet and prices can runaway.
- Historical instance:
  - 1973-74: real interest rate -3.84 percent, price hike of 75 percent (largest in sample) and one of the largest negative yield shocks.
  - Simulation: two-thirds of the 1973-74 price hike may be explained by the real rate passing the threshold. When real rate moves from -1.35 percent (1973) to -3.84 percent (1974), price rises by around 50 percent (Figure 6(c)).
- Sensitivity within normal ranges:
  - If real interest rate reduced from around zero (2008) to around the threshold (1979), the price at the corresponding stock-taking point changes by less than 10 percent.
  - Example: 1983 (5.19 percent) vs 2008 (-0.026 percent): more than 500 basis points reduction yields around 5 percent rise in stock-taking point price.
- Asymmetry:
  - When real rate plunges beyond the threshold, marginal impact on prices is increasingly large; when real rate remains positive or close to zero, impact is almost negligible.

### The Effect of Depreciation Rates (Storage Cost)
- Assumed constant decay rate in literature; possibility of declining storage cost due to upgrade in storage/distribution and bulk purchases.
- Simulation with storage cost varying between 0 and 15 percent shows:
  - Reduced storage cost induces earlier stock-taking (intervention point x* becomes smaller).
  - Reduced storage cost smoothes prices: market price function curve becomes smoother with lower storage decay.
  - Example: Magenta line (0 storage cost) vs Purple line (15 percent storage cost) in Figure 7(a): overall price process under Magenta is much smoother, indicating less dramatic price movements when per capita availability varies.
- Overall effect of changing storage cost is smaller than that of other factors unless dramatic industrial innovation occurs.

### The Effect of Consumption Elasticity
- Price elasticity of food demand (ρ) is a key determinant of computed market price.
- For more inelastic demand (lower ρ):
  - consumer demand and market price are more convex;
  - price hikes and plunges are more frequent;
  - threshold of stock-taking x* is lower.
- Example contrast: ρ = 0.50 implies a less violent price movement relative to more inelastic cases (Figure 7(b)).

### 4.3    Overall Effects
- Combined simulation using newly estimated (δ,ρ) and actual time-varying parameters (Figure 8(a)) for years 1970, 1980, 1990, 2000, and 2008 yields:
  1) 1970 → 1980 → 1990: market conditions improved (per capita output grew and price dropped).
  2) mid-1990s → 2008: market conditions deteriorated; 2008 market condition only marginally more relaxed than 2000.
- Drivers of tightening since mid-1990s:
  - constant and perhaps accelerating growth in consumer demand;
  - intrepid growth in output (shift in balance between demand and supply);
  - negative interest rate shock and inelastic demand — these may have produced the 2007-08 food crisis.
- 2008–2009: market became somewhat less stringent due to a positive yield shock in 2008 following negative ones in 2006 and 2007.
- Risk: medium- to long-term rising prices because still in middle of low output/demand ratio regime observed since mid-1990s.
- Dominant role of supply-demand imbalance:
  - trend output/demand ratio largely determines long-run price movements.
  - Figure 8(b): x* series closely tracks per capita output/demand ratio; higher x* with lower p* implies more relaxed market conditions.
  - Over long run, price trend is dominated by per capita output (output/demand ratio); in short run, yield shocks and other factors create small divergences.

- Conceptual link:
  - Story resembles permanent income hypothesis (Friedman (1957)): permanent trends dominate transitory shocks.
  - Nonarbitrage condition akin to Euler equation in precautionary savings studies; output/demand ratio predominates price movements.
  - Augmented model addresses autocorrelations that Deaton and Laroque (1992) could not generate.

### 4.4    The Role of Storage
- Question: Is competitive storage relevant when per capita supply contains a trend?
  - Answer: Yes. Year-to-year trends move little; yield shocks and interest rate volatility dominate short-term decisions. Speculators with one-season planning horizons still find storage relevant when next-period expected price can be higher.
- Storage in the model:
  - Determined endogenously by risk-neutral speculators.
  - Acts as demand this period and supply next period.
  - Competitive storage can smooth prices by buying cheap and selling dear and can increase autocorrelation, but Deaton and Laroque (1996) found it insufficient to reach observed high autocorrelations.
- Comparative analysis of five model settings (I–V), feeding same estimated output shock and comparing simulated vs actual price summary statistics (Table 2):
  - (I) Barebone: no storage, no trends, i.i.d. output disturbance, constant interest rate → very small serial correlations, smallest skewness and kurtosis.
  - (II) Barebone plus trend: adds trends → significantly increases autocorrelations but still below actual data.
  - (III) Deaton and Laroque (1992) version: adds storage to barebone → better match for first-order autocorrelation and skewness; higher-order autocorrelations remain small because shocks have no persistence.
  - (IV) Augmented model: integrates storage, trends, and time-varying interest rates → best overall fit to actual data. Trend and storage together needed; natural nonnegative constraint on storage is key for asymmetric price movements. Without trend and time-varying interest rate, volatility (coefficient of variation) is underestimated.
  - (V) Augmented plus oil: ad hoc oil price adjustment (complete pass-through assumption) improves fit for excess kurtosis and skewness at cost of slightly reduced autocorrelations; helps capture egregious price movements like 1973-74 and 2007-08.
- Limitations:
  - Augmented model cannot reproduce peakedness (huge crisis-period price spikes) fully; oil price shocks are a plausible additional factor.
- Visualization:
  - Figure 4 displays simulated prices for model II, IV, and V for vivid comparison.

*Source: _wp1164 - 4.2    Comparative Statics*

### 4.5    Other Food Commodities

### 4.5    Other Food Commodities

### Similarities with Wheat and General Patterns
- Maize, rice, and soybeans display supply and demand characteristics similar to wheat (Figure 10).
- Rice: output/demand ratio is increasing while the real price is falling.
- Maize: price behavior aligns with expectations, except for an additional demand effect from bio-fuel after the 2000s.
- Soybeans: output growth clearly outpaces that of GDP and population; soybean output accelerated significantly after the mid-1990s.
- Anecdotal evidence links soybean demand growth to emerging market consumers upgrading from staple foods into meats, especially pork; soybeans are the primary feed for pork and other livestock.

### Key Findings on Price Dynamics and Storage
- The predominant role of the output/demand ratio, combined with intertemporal storage, explains long-run food commodity price movements and the high autocorrelations observed in actual prices.
- Short-run price movements are driven mostly by the realization of small-probability events (e.g., a yield shock larger than two standard deviations), not by shocks with a large variance.
- Yield shock distributions calibrated from the data are quite stable and have small variances.
- Abrupt short-run fluctuations are caused by small-probability events such as severe drought or oil price shocks.
- Monetary policy plays a limited role in normal times but could have nonlinear and significant impact when the real rate becomes deep negative.

### Model Calibration and Estimation Results
- The simulation utilizes the estimated depreciation rate 2.3% and price elasticity 0.19.
- The Method of Simulated Moments grid-search arrived at estimates: depreciation 2.3% and elasticity 0.19, at which point the gap is 0.008.
- Numerical solution notes: with endogenous gridpoints algorithm and hardware described, a trial for a particular (δ,ρ) pair takes 3-5 minutes; overall the MSM search process required about eight hours.

### Limitations, Omitted Factors, and Caveats
- The model omits several factors that may help explain occasional large price hikes:
  - Oil/energy prices affecting fertilizers, pesticides, and transportation, and unexpected inflation.
  - Deterioration of world trade during food crises; the model assumes all global output is available for global consumption and does not internalize export bans or raised regulatory restrictions that countries may impose to stabilize domestic supply.
  - Substitution among different food commodities is not accounted for.
- Application to energy prices is not straightforward because:
  - Food staples have a lower and fairly stable price elasticity of consumer demand.
  - It is relatively easy to identify an instrument for suppliers’ response to food prices in agriculture (area planted); a similar concise instrument cannot be found for oil and industrial raw materials.
  - The model approximates suppliers’ optimal decision by observable area planted; area decision for t+1 is known by the market at t and affects current price indirectly through total supply at t+1 and hence speculative demand at time t.

### Historical and Data Conventions Relevant to Interpretation
- Annual price calculation convention: annual price is the average monthly price over the 12 months in an international trade year; for wheat the international trade year is July 1-June 30.
- The “2008 food crisis” refers to price hikes in response to low harvests in 2006 and 2007 harvest years; the negative supply shocks in those years were not particularly large individually, but consecutive bad harvests imply very low storage before the 2008 new crops.
- Since July 2008, when the good harvest due to a positive supply shock gradually reached the market, world staple prices declined from their highs between late 2006 and early 2008.

*Source: 4.5 Other Food Commodities — _wp1164 - 4.5    Other Food Commodities*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp1164.pdf_