The hat-algebra technique in the Stolper-Samuelson Theorem

1. The Starting Point (Levels) 

In a competitive economy, the price of a good ($P$) equals the cost of the labor ($L$) and capital ($K$) used to make it. For two goods (e.g., Textiles and Machines), we have: 

$P_{T}=wL_{T}+ rK_{T}$ $𝑃_{M}=𝑤𝐿_{M} +𝑟𝐾_{M}$ where, $w$ is wage(return to labor), $r$ is rent(return to capital)

2. Applying Hat Algebra (The Change) 

If we want to see how a change in prices ($\hat{P}$) affects wages ($\hat{w}$), we take the total derivative of the equations and divide by the initial values. This transforms the “Level” equation into a “Hat” equation: $\hat{P}_{T} = \theta_{LT}\hat{w} + \theta_{KT}\hat{r}$Where: 

• $\hat{P}_{T}$ is the percentage change in the price of Textiles.
• $\theta_{LT}$ is the income share of labor in the textile industry (e.g., if 60% of textile revenue goes to workers, $\theta_{LT} = 0.6$).

3. The “Magnification Effect” 

This is where the technique reveals something powerful. If Textiles is a labor-intensive industry ($\theta_{LT} > \theta_{LM}$), and the price of Textiles rises by 10%, hat algebra allows us to solve for the changes in wages and rent. 

The resulting inequality usually looks like this:
$\hat{w} > \hat{P}_{T} > \hat{P}_{M} > \hat{r}.$

The Insight: 

• The “Magnification”: If the price of the labor-intensive good ($P_{T}$) rises, the wage ($\hat{w}$) rises more than proportionally.
• Real Income: Because wages rose faster than prices ($\hat{w} > \hat{P}_{T}$), workers are now strictly better off—they have more “real” purchasing power.
• The Losers: Conversely, capital owners see their returns ($\hat{r}$) grow slower than prices (or even fall), making them worse off. 

Summary of the Transformation Steps 

1. Write the Equilibrium: Start with the standard equation (e.g., Price=Cost).
2. Differentiate: Take the derivative to find how small changes affect the whole system.
3. Divide by Levels: This turns the changes into ratios (Hats).
4. Substitute Shares: Replace complex units with $\theta$(income shares) or $\lambda$ (employment shares)**. 

Why this matters today 

In modern “New Quantitative Trade” (like the ACR model), economists use this to calculate the “Gains from Trade”. Instead of measuring every factory’s productivity, they use: $\hat{W} = \hat{\lambda}^{1/e}_{ii}$ This tells them that a country’s welfare change ($\hat{W}$) is tied directly to the change in its “home-trade share” ($\hat{\lambda}_{ii}$). If you stop buying from yourself and start buying from abroad, your welfare is increasing.