ECON6023 · International Trade
The Specific-Factors Model (Short Run)
Week 4 develops the short-run model where labour is mobile between two sectors but each sector has its own immobile specific factor (capital, land), so a change in relative prices redistributes income. The wage-rental (labour-allocation) diagram and the magnification chain that signs who gains and loses are the Q2 setup in the mid-semester test and a favourite exam political-economy question.
What this chapter covers
- 01Setup: labour mobile between two sectors; each sector has its own specific factor (capital, land); short run
- 02Labour-market equilibrium W = P_M·MPL_M = P_A·MPL_A; the back-to-back p·MPL diagram
- 03Diminishing MPL and the bowed-out (concave) PPF with slope −P_M/P_A
- 04A rise in P_M: labour moves into M, wage rises by less than the price (ΔW/W < ΔP_M/P_M)
- 05Real wages: W/P_M falls, W/P_A rises → the mobile factor's welfare is ambiguous
- 06Magnification chain ΔR_T/R_T < 0 < ΔW/W < ΔP_M/P_M < ΔR_K/R_K (for a rise in P_M)
- 07Rental-change formulas from revenue shares: ΔR_K/R_K = [ΔP_M/P_M − θ_LM·ΔW/W]/θ_KM
- 08Political economy: the specific factor in the expanding sector gains, the other loses; workers ambiguous
Signing rentals with the magnification formula
- +1Rental on capital (specific to M). Zero-profit accounting gives ΔR_K/R_K = [ΔP_M/P_M − θ_LM·ΔW/W]/θ_KM = [8% − 0.5·3%]/0.5 = (8 − 1.5)/0.5 = 13%. Capital in the rising-price sector gains more than the price rose.
- +1Rental on land (specific to A). With ΔP_A/P_A = 0, ΔR_T/R_T = [ΔP_A/P_A − θ_LA·ΔW/W]/θ_TA = [0 − 0.6·3%]/0.4 = (−1.8)/0.4 = −4.5%. Land in the sector whose price did not rise loses, because labour is bid away and the wage bill it must cover rose.
- +1Real wage of the mobile factor is ambiguous. In terms of manufactures W/P_M changes by 3% − 8% = −5% (falls); in terms of agriculture W/P_A changes by 3% − 0% = +3% (rises). Workers' welfare depends on their consumption basket.
- +1Magnification chain. Ordering the results: ΔR_T/R_T (−4.5%) < 0 < ΔW/W (3%) < ΔP_M/P_M (8%) < ΔR_K/R_K (13%). The specific factor in the expanding sector gains most, the specific factor in the other sector loses, and the mobile factor sits in between.
Key terms
- Specific factor
- A factor tied to one sector and immobile in the short run (capital in manufacturing, land in agriculture). Its real return moves sharply with that sector's price, unlike the mobile factor.
- Mobile factor
- Labour, which moves freely between sectors so a single wage clears the labour market: W = P_M·MPL_M = P_A·MPL_A. Its real wage change is ambiguous when a relative price shifts.
- Wage-rental (labour-allocation) diagram
- Back-to-back plot with each sector's p·MPL curve declining toward its own origin; their intersection sets the common wage W and the split of labour L_M | L_A.
- Magnification effect
- For a rise in P_M, ΔR_T/R_T < 0 < ΔW/W < ΔP_M/P_M < ΔR_K/R_K: factor prices move by more (in percentage terms, at the ends) than the goods price, so distribution effects exceed price effects.
- Rental-change formula
- ΔR_K/R_K = [ΔP_M/P_M − θ_LM·ΔW/W]/θ_KM, where θ_LM and θ_KM are labour's and capital's revenue shares in the sector. The same form (with land's share) gives ΔR_T/R_T.
- Short-run Stolper-Samuelson
- The specific-factors result that a rise in a good's relative price makes the factor specific to that sector gain and the factor specific to the other sector lose in real terms, with the mobile factor ambiguous.
The Specific-Factors Model (Short Run) FAQ
Why is the effect on workers ambiguous but the effect on the specific factors clear?
Labour is mobile, so its wage is anchored by both sectors' value of marginal product; a relative-price change moves the wage up in one good and down in the other, and the net effect depends on what workers consume. The specific factors are stuck: the one in the sector whose price rose enjoys more labour working with it (higher return), while the one in the other sector loses labour (lower return) — no ambiguity.
How is the specific-factors model different from Heckscher-Ohlin?
Specific factors is short run: some factors cannot move, so income distribution turns on which sector you are tied to. Heckscher-Ohlin is long run: all factors are mobile, so distribution turns on which factor you own (Stolper-Samuelson), regardless of sector. Mixing up the two — for instance applying Stolper-Samuelson's clean factor result in a short-run question — is a classic exam trap.
Can AI help me with a specific-factors political-economy question?
Yes. Sia can set up the wage-rental diagram, shift the right curve for a price or migration shock, and sign every entry (employment, output, wage, rentals) with W = p·MPL and R = p·MPK, then help you write the 'who lobbies for/against' paragraph. It teaches the method and checks your reasoning; it does not do graded work, and USyd academic-integrity rules apply.
Do I need the revenue shares to sign the rentals, or just the direction?
For the direction (who gains, who loses) the magnification chain is enough. For a numerical answer you need the sector revenue shares, because ΔR/R = [ΔP/P − θ_L·ΔW/W]/θ_specific. If the shares are not given, state the signs via the chain and note the magnitude depends on the shares — do not invent them.
Exam move
Draw the back-to-back labour-allocation diagram from memory and practise shifting one p·MPL curve up or down for a price shock, a technology shock, or a migration shock — this single diagram powers most of the second third of the unit. Memorise the magnification chain for a rise in P_M and be able to state it instantly, then attach the rental-change formula when revenue shares are given. Always separate the clear result (specific factors) from the ambiguous one (labour's real wage in each good). Keep specific factors (short run) firmly distinct from Heckscher-Ohlin (long run). Confirm the mid-semester test timing on Canvas.
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