1. The Farm-Retail Price Spread in a Competitive Food Industry
Bruce L. Gardner (1975)
Journal of Agricultural Economics, 57: 399–409
Summarized by: Pham Thi Phuong Loan

2. SUMMARY Basic framework for analyzing farm-retail price spread:
Sources of variation in the retail-farm price spread: shifts in the retail food demand, in the farm product supply, in the supply of marketing services.
Consequences of retail price ceilings and farm price floor, the elasticity of price transmission
Determinants of change in the farmer’s share of the food dollar

3. THE MODEL
Consider a competitive food marketing industry using two factors of production: purchased agricultural commodities (a) and other marketing inputs (b) to produce food sold at retail (x).
For example: a (wheat), b (labor, transportation, packaging, etc.), x (bread)
The marketing industry’s production function
x = f(a,b)
The retail food demand function
x = D(Px,N)
with Px is the retail price, N is demand shifter, for example population

4. THE MODEL
Firms are assumed to buy the profit-maximizing quantities of b and a (implies that the value of marginal product equals price). Demand functions of b and a:
(3) Pb = Px . fb
(4) Pa = Px . fa
Where fb , fa are the partial derivatives of x with respect to b and a.

5. THE MODEL The supply function of b to the food marketing industry
Pb = g(b, T)
T: the shifter of marketing input (for example tax)
The supply function of agricultural output a
(6) Pa = h(a, W)
W: the shifter of farm product supply (for example a weather variable for which higher value increases Pa (e.g. index of drought). That means higher value of W will make supply shift to the left)

6. THE MODEL
There are six endogenous variables (a, b, a, Px, Pb, Pa) in six equations. Under normal condition, there is a unique equilibrium. Six variables and the farm-price spread are determined.
Measure the price spread:
Px - Pa (difference)
Px /Pa (ratio)
aPa / xPx (the farmer’s share of the food dollar)
(Px - Pa)/Pa or Px/Pa - 1 (percentage margin)

7. EFFECT OF A FOOD DEMAND SHIFT ON THE RETAIL-FARM PRICE RATIO
The question: how Px/Pa change when demand for food shifts
The answer: elasticity of Px/Pa with respect to N.
Where
ηN : elasticity of demand for x with respect to N
Sa and Sb : relative shares of a, b, e.g. Sa = aPa / xPx
ea and eb : own price elasticities of supply of a and b
D is a function of σ, η, ea , eb , Sa and D is positive in normal cases (ea and eb >0)
σ: elasticity of substitution between a and b
η: price elasticity of demand for x

8. EFFECT OF A FOOD DEMAND SHIFT ON THE RETAIL-FARM PRICE RATIO
ea < eb so EPx/Pa, N <0: When the demand for food shifts to the right, Px / Pa falls (eg. the retail-farm price ratio is expected to decline when population (or other food demand shifters) increases.
ea = eb so EPx/Pa, N = 0, Px / Pa is unchanged.
ea > eb so EPx/Pa, N >0: When the demand for food shifts to the right, Px / Pa increases (eg the retail-farm price ratio is expected to increase when population (or other food demand shifters) increases.

9. EFFECT OF A FARM PRODUCT SUPPLY SHIFT ON THE RETAIL-FARM PRICE RATIO
The answer: elasticity of Px/Pa with respect to W.
Where
eW : elasticity of Pa with respect to W
Sa and Sb : relative shares of a, b, e.g Sa = aPa / xPx
ea and eb : own price elasticities of supply of a and b
D is a function of σ, η, ea , eb , Sa and D is positive in normal cases (ea and eb >0)
σ: elasticity of substitution between a and b
η: price elasticity of demand for x

10. EFFECT OF A FARM PRODUCT SUPPLY SHIFT ON THE RETAIL-FARM PRICE RATIO
In normal case η<0, eb >0, that means EPx/Pa, W <0
When Pa rises as a result of a decline in the supply of agricultural output, retail-farm price ratio falls.
Conversely, an event such as a technical improvement in crop production that increases a and reduces Pa will widen retail-farm price ratio.

11. EFFECT OF A MARKETING INPUT SUPPLY SHIFT ON THE RETAIL-FARM PRICE RATIO
The answer: elasticity of Px/Pa with respect to T.
Where
eT : elasticity of Pb with respect to T
Sa and Sb : relative shares of a, b, e.g Sa = aPa / xPx
ea and eb : own price elasticities of supply of a and b
D is a function of σ, η, ea , eb , Sa and D is positive in normal cases (ea and eb >0)
σ: elasticity of substitution between a and b
η: price elasticity of demand for x
In normal case EPx/Pa, T >0
When Pb rises as a result of a specific tax on marketing input, retail-farm price ratio increases.

12. PRICE SUPPORTS AND PRICE CEILINGS
Price control on x
Question: If a price ceiling lower than the market-clearing price is impose on a food product at the retail level (not at the farm level), what effect will this have on the retail-farm price ratio?
Where
Sa and Sb : relative shares of a, b, e.g Sa = aPa / xPx
ea and eb : own price elasticities of supply of a and b
σ: elasticity of substitution between a and b
ceiling price

13. PRICE SUPPORTS AND PRICE CEILINGS
Price control on x
: price ceiling on retail food will always reduces farm level prices. The reason is that price ceiling on x always reduces derived demand for a.
If ea = eb then , farm-retail price ratio is unchanged, eg. a reduction in Px will reduce Pa by the same percentage.
If ea < eb then , farm-retail price ratio widens, eg. Pa falls by a greater percentage than Px

14. PRICE SUPPORTS AND PRICE CEILINGS
Price control on a
Question: If the price of a is kept at the support level, what effect will this have on the retail-farm price ratio?
Where
Sa and Sb : relative shares of a, b, e.g Sa = aPa / xPx
ea and eb : own price elasticities of supply of a and b
σ: elasticity of substitution between a and b
η: price elasticity of demand for x
price support level

15. PRICE SUPPORTS AND PRICE CEILINGS
Price control on a
In normal case: eb > η then
A production control program that raises Pa will raises Px by a smaller percentage.

16. ELASTICITY OF PRICE TRANSMISSION AND THE ELASTICITY OF DERIVED DEMAND
1. When price changes are caused by a shift in product demand
2. When price changes are caused by a shift in the supply curve of a

17. ELASTICITY OF PRICE TRANSMISSION AND THE ELASTICITY OF DERIVED DEMAND
Elasticity of derived demand for a
σ: elasticity of substitution between a and b
η: price elasticity of demand for x
If σ < |η| : derived demand function for a will be less elastic than the retail demand function for x.
If σ = |η| : the retail and farm level elasticities are equal.
If σ > |η| : derived demand function is more elastic than the retail level demand.

18. THE FAMRER’S SHARE OF THE FOOD DOLLAR
Effects of demand shifts
ηN : elasticity of demand for x with respect to N
D >0
If either ea = eb or σ =1 then ESaN = 0 (the Sa constant): A shift in demand for food at the retail level will have no effect on farmer’s share.
If the numerator >0 then ESaN > 0: An increase in demand for food will increase farmer’s share.
If the numerator < 0 then ESaN < 0: An increase in demand for food will decrease farmer’s share.

19. THE FAMRER’S SHARE OF THE FOOD DOLLAR
Effects of supply shifts
σ: elasticity of substitution between a and b
eW : elasticity of Pa with respect to W
η: price elasticity of demand for x
In normal case: D >0, η < eb
If σ < 1 then ESaW >0 (the Sa constant): A shift in supply function of a increases Pa and then will increase the farmer’s share (for example, W is a drought)
If σ = 1 then ESaW = 0: A shift in supply function will have no effect on the farmer’s share.
If σ > 1 then ESaW < 0: A shift in supply function of a will decrease farmer’s share.

20. SUMMARY AND CONCLUSION
No simple markup pricing rule can accurately depict the relationship between the farm and the retail price.
Sources of variation in the retail-farm price spread: shifts in the retail food demand, in the farm product supply, in the supply of marketing services.
Consequences of retail price ceilings and farm price floor, the elasticity of price transmission.
Determinants of change in the farmer’s share of the food dollar.