ECON6023 · International Trade
The Dornbusch-Fischer-Samuelson Continuum Model
Week 3 extends Ricardian logic to a continuum of goods z ∈ [0,1], replacing the price band with two curves: the downward-sloping relative-efficiency curve A(z) = a*(z)/a(z) and the upward-sloping trade-balance curve B(z'), whose intersection gives the marginal good z' and the relative wage ω = w/w*. The DFS derivation and its comparative statics (a productivity shock, a change in country size) are a staple exam theory question — in one recent year it was the 25-mark Q1.
What this chapter covers
- 01Continuum of goods z ∈ [0,1], one factor (labour), endowments L and L*, unit labour requirements a(z), a*(z)
- 02Goods reordered so Home's comparative advantage is decreasing in z (Home makes low-z goods)
- 03Relative-efficiency curve A(z) = a*(z)/a(z), downward sloping
- 04Marginal good z': w·a(z') = w*·a*(z') ⟺ ω = w/w* = A(z'); Home makes [0, z'], Foreign makes [z', 1]
- 05Cobb-Douglas continuum preferences U = ∫₀¹ ln c(z) dz: uniform expenditure share across goods
- 06Trade-balance curve B(z'): wL = z'(wL + w*L*) ⟹ ω = [z'/(1−z')]·(L*/L), upward sloping
- 07Equilibrium = intersection of A(z') and B(z'); Home exports [0, z'], imports [z', 1]
- 08Comparative statics: larger Home L shifts B, lowering ω and raising z'; a uniform productivity fall shifts A down
Solving the DFS equilibrium: marginal good and relative wage
- +1A-curve (specialisation). The marginal good z' is where costs are equal, w·a(z') = w*·a*(z'), i.e. ω = w/w* = a*(z')/a(z') = A(z') = 2(1 − z'). Home makes the goods [0, z'] where it is cheaper; Foreign makes [z', 1].
- +1B-curve (trade balance). Home income equals world spending on Home goods, wL = z'(wL + w*L*), which rearranges to ω = w/w* = [z'/(1 − z')]·(L*/L). With L*/L = 1, ω = z'/(1 − z').
- +1Equilibrium: set A = B. 2(1 − z') = z'/(1 − z') ⟹ 2(1 − z')² = z' ⟹ 2z'² − 5z' + 2 = 0. The roots are z' = (5 ± 3)/4 = 2 or 1/2; reject z' = 2 (outside [0,1]), so z' = 1/2.
- +1Relative wage. ω = z'/(1 − z') = (1/2)/(1/2) = 1 (and A(1/2) = 2·1/2 = 1 confirms it). Home produces the lower half of goods, z ∈ [0, 1/2], Foreign the upper half; wages are equal.
Key terms
- Continuum of goods
- An index z ∈ [0,1] labelling a smooth range of goods, reordered so Home's comparative advantage falls as z rises. Replaces the two-good Ricardian model's discrete goods with a marginal-good cutoff.
- Relative-efficiency curve A(z)
- A(z) = a*(z)/a(z), Home's productivity in good z relative to Foreign's. Decreasing in z by the chosen ordering; at the margin it equals the relative wage, ω = A(z').
- Marginal good z'
- The threshold good where the two countries' costs are equal, w·a(z') = w*·a*(z'). Home produces everything to the left (cheaper at Home), Foreign everything to the right.
- Trade-balance curve B(z')
- B(z') = [z'/(1−z')]·(L*/L), the relative wage consistent with balanced trade when Home captures a share z' of world spending. Upward sloping in z'.
- Relative wage (ω)
- ω = w/w*, Home's wage relative to Foreign's, determined where A(z') = B(z'). A larger country tends to have a lower relative wage but produces a wider range of goods.
- Cobb-Douglas continuum preferences
- U = ∫₀¹ ln c(z) dz, which spreads expenditure uniformly over goods so that a fraction z' of world spending falls on Home's range — the demand assumption behind the B-curve.
The Dornbusch-Fischer-Samuelson Continuum Model FAQ
What is the difference between the A(z) curve and the B(z') curve?
A(z) is the supply/cost side: it says which country can make good z more cheaply and, at the marginal good, ties the relative wage to relative efficiency, ω = A(z'). It slopes down. B(z') is the demand/trade-balance side: it says what relative wage keeps trade balanced when Home makes a share z' of goods, ω = [z'/(1−z')]·(L*/L). It slopes up. Their intersection sets both z' and ω.
What happens to the relative wage if Home gets bigger?
A rise in Home's labour force L lowers L*/L, shifting the B-curve down (a lower ω at each z'). Since A slopes down and B slopes up, the new intersection is at a higher z' and a lower ω: the larger country produces a wider range of goods but has a lower relative wage — a signature DFS comparative static.
Why does a uniform doubling of Home's a(z) lower its relative wage?
If every a(z) doubles, Home becomes uniformly less productive, so A(z) = a*(z)/a(z) halves at every z and the A-curve shifts down. Against an unchanged B-curve the intersection moves to a lower ω — a fall in own productivity lowers own relative wage, the expected result the exam asks you to explain on the diagram.
Can AI help me with the DFS model?
Yes. Sia can derive the A(z) and B(z') curves with you, solve the intersection, and run the standard comparative statics (country size, productivity shocks, taste shifters) one step at a time, checking your algebra and your case-elimination. It explains the method and never does graded assessment for you; confirm exam details on Canvas.
Exam move
Practise deriving A(z) and B(z') from scratch every time, then intersecting them, because the exam rewards the full derivation, not just the picture. Memorise the two shapes (A down, B up) and the size result (bigger country → lower ω, wider range). Drill the three comparative statics the unit emphasises: a rise in L (B shifts right), a uniform productivity/cost shock (A shifts down), and the taste-shock variant where a(z) = a*(z) makes A flat at ω = 1. Always reject roots outside [0,1] explicitly — that case-elimination earns marks. Confirm the exam timing on Canvas.
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