Multi-Sector Models and Marketing Margins

Multi-Sector Models

The basic supply & demand model focuses on one sector of the marketing channel:

  • Could be the farm level (price & quantity of raw inputs like barley).

  • Could be the retail level (price & quantity of final goods like beer).

But food systems have multiple linked sectors.

  • What happens at the farm affects processors and consumers.

  • What happens at retail feeds back to processors and farmers.

To capture these linkages, we need a multi-sector model.

Example: Beer Industry

  • Inputs (Farm Level): barley, hops, yeast, water.
  • Processor (Brewers): brewers transform inputs into beer.
  • Output (Retail): beer sold to consumers (ignoring wholesalers/retailers).

There is both:
- A supply & demand for beer (retail).
- A supply & demand for barley (farm).

But how do we connect them?

Linking Farm and Retail Markets

We need two simplifying assumptions to tie the beer and barley markets together.

  • These assumptions let us draw one diagram showing both markets.

  • Without them, the analysis gets messy (technology choice, variable costs, substitution across inputs, etc.).

Assumption 1: Fixed Proportions Technology

  • Each gallon of beer requires a fixed amount of barley.

  • Brewers can’t substitute away from barley if its price rises.

  • This lets us say:

\[ Q^F = \alpha Q^R \] Where: - \(Q^F\): farm output (bushels of barley)
- \(Q^R\): retail output (gallons of beer)
- \(alpha\): conversion factor (e.g., 1.5 lbs barley per gallon beer)

If we know retail output, we know farm output exactly.

Assumption 2: Constant Marketing Bill

  • The cost of processing barley into beers is constant (fixed) per unit.

  • Denote this as \(MB\) (per-unit marketing bill).

  • Retail price is then just:

\[ P^{Beer} = P^{Barley} + MB \]

This assumption creates a fixed wedge between the farm price and the retail price.

Derived Demand

Brewers only demand barley because they can sell beer.

Thus the derived demand for barley is just the consumer demand for beer, shifted down by \(MB\).

Numerical Example:

  • Consumers will pay $1.00 for a gallon of beer.
  • Marketing bill = $0.30.
  • Max brewers will pay for barley = $1.00 − $0.30 = $0.70.

Market Equilibrium

  • Equilibrium in the barley market occurs where supply of barley meets derived demand for barley.
  • At this quantity, the consumer price of beer equals the farm price of barley plus the marketing bill.

Shocks

1. Higher Marketing Bill (e.g., new tax or regulation)

  • Derived demand shifts down.
  • Barley price falls.
  • Beer price rises.
  • Quantities fall.

2. Higher Consumer Demand for Beer

  • Both beer demand and derived demand for barley shift out.
  • Prices and quantities rise in both markets.

Cost Transmission

New regulations raise \(MB\). Who bears the cost?

  • If demand more elastic than supply → farm price falls more.
  • If supply more elastic than demand → retail price rises more.
  • General rule: the side with less elasticity bears more of the burden.