Structural Change and Aggregate Productivity

Why Structural Change Matters

An economy can become more productive even when productivity within each sector is unchanged. If workers move from a sector with relatively low output per worker to one with higher output per worker, the economy’s aggregate output per worker can rise. Conversely, a shift toward lower-productivity sectors can offset productivity gains within sectors. These changes in the distribution of employment across sectors are part of structural change.

To study this effect, distinguish two questions: how productive is each sector, and how is the workforce distributed among sectors? The aggregate result depends on both. This tutorial develops the weighted-average relationship and a decomposition that separates changes within sectors from changes in employment shares.

Definition: For sector $j$, let $Y_j$ denote its output and $L_j$ its employment. Sector labor productivity is $p_j=Y_j/L_j$, output per worker. Total output is $Y=\sum_j Y_j$, and total employment is $L=\sum_j L_j$. The sector’s employment share is $s_j=L_j/L$. Aggregate labor productivity is $P=Y/L$. Thus, $P=\sum_j s_jp_j$, provided sector outputs and employment cover the same economy and period and use a comparable output measure.

The employment shares sum to one: $\sum_j s_j=1$. Aggregate productivity is therefore not generally the simple average of sector productivities. It is their employment-weighted average: a sector with a larger share of workers has more influence on the aggregate.

Worked Example: Workers Move Toward More Productive Sectors

Worked Example: Measuring a Pure Reallocation Effect

A hypothetical economy has 100 thousand workers in agriculture, manufacturing, and services. Sector productivity is measured in thousands of dollars of output per worker for the same year. In the first period, employment is 40 thousand in agriculture, 30 thousand in manufacturing, and 30 thousand in services. Productivity is respectively $20,000, $50,000, and $80,000 per worker. In the second period, employment is 30 thousand, 30 thousand, and 40 thousand, while productivity within each sector is unchanged. Calculate aggregate productivity in both periods and the change.

Method. Calculate each sector’s output, add the outputs, and divide by total employment. Since employment is recorded in thousands of workers and productivity in thousands of dollars per worker, sector output is in millions of dollars.

First period. Agriculture produces $40\times20=\$800$ million; manufacturing produces $30\times50=\$1{,}500$ million; services produces $30\times80=\$2{,}400$ million. Total output is $800+1{,}500+2{,}400=\$4{,}700$ million. Total employment is 100 thousand workers, so aggregate productivity is $4{,}700/100=\$47{,}000$ per worker.

Second period. Agriculture produces $30\times20=\$600$ million; manufacturing produces $30\times50=\$1{,}500$ million; services produces $40\times80=\$3{,}200$ million. Total output is $600+1{,}500+3{,}200=\$5{,}300$ million. With employment still at 100 thousand workers, aggregate productivity is $5{,}300/100=\$53{,}000$ per worker.

The increase is $\$53{,}000-\$47{,}000=\$6{,}000$ per worker. Sector productivities did not change; the increase came entirely from employment moving away from agriculture and toward services, which had higher output per worker. The calculation is checked by the weighted averages: $0.40(20)+0.30(50)+0.30(80)=47$, and $0.30(20)+0.30(50)+0.40(80)=53$, in thousands of dollars per worker. The common mistake to avoid is averaging sector productivities without using employment shares.

An Exact Decomposition of Change

In practice, sector employment shares and sector productivities can both change. To distinguish their contributions, compare an initial period 0 with a later period 1. Write $\Delta s_j=s_{j1}-s_{j0}$, $\Delta p_j=p_{j1}-p_{j0}$, and let a bar denote the arithmetic average across the two periods: $\bar{s}_j=(s_{j0}+s_{j1})/2$ and $\bar{p}_j=(p_{j0}+p_{j1})/2$.

Theorem: The change in aggregate labor productivity has the exact decomposition $$ \Delta P = \sum_j \bar{s}_j\Delta p_j + \sum_j \bar{p}_j\Delta s_j. $$ The first sum is the within-sector productivity component: it measures the effect of sector productivity changes, weighted by average employment shares. The second is the reallocation component: it measures the effect of employment-share changes, weighted by average sector productivity. Both components are accounting measures; by themselves, they do not establish why productivity or employment shares changed.

Using period averages makes this decomposition exact, including when productivity and employment shares change at the same time. To see why, start with the change in each sector’s contribution to aggregate productivity, $s_{j1}p_{j1}-s_{j0}p_{j0}$. For each sector, the identity

s_{j1}p_{j1}-s_{j0}p_{j0} = \frac{s_{j0}+s_{j1}}{2}(p_{j1}-p_{j0}) + \frac{p_{j0}+p_{j1}}{2}(s_{j1}-s_{j0})

holds because the right-hand side expands to $\tfrac12(s_{j0}p_{j1}-s_{j0}p_{j0}+s_{j1}p_{j1}-s_{j1}p_{j0}+p_{j0}s_{j1}-p_{j0}s_{j0}+p_{j1}s_{j1}-p_{j1}s_{j0})$. The two middle cross terms $s_{j0}p_{j1}$ and $-p_{j1}s_{j0}$ cancel, as do $-s_{j1}p_{j0}$ and $p_{j0}s_{j1}$. The remaining terms equal $s_{j1}p_{j1}-s_{j0}p_{j0}$. Summing this equality over sectors gives the decomposition because $P=\sum_j s_jp_j$.

Compute the two-period data.
For every sector, find its employment share and output per worker in each period.

Find changes and averages.
Calculate $\Delta p_j$, $\Delta s_j$, $\bar{s}_j$, and $\bar{p}_j$ sector by sector.

Calculate and check the components.
Sum $\bar{s}_j\Delta p_j$ for the within-sector component and $\bar{p}_j\Delta s_j$ for reallocation. Their sum must equal $P_1-P_0$.

Worked Example: Decomposing Simultaneous Changes

An economy has three sectors. Employment shares change from $(0.50,0.30,0.20)$ to $(0.40,0.35,0.25)$. Sector productivity, in thousands of dollars per worker, changes from $(30,60,90)$ to $(33,66,90)$. Find the exact within-sector and reallocation components.

Method. Use the exact decomposition with arithmetic-average shares and productivity levels. This avoids omitting the interaction between productivity and employment-share changes.

Within-sector component. Average shares are $(0.45,0.325,0.225)$. Productivity changes are $(3,6,0)$. The component is $0.45(3)+0.325(6)+0.225(0)=1.35+1.95+0=\$3.30$ thousand per worker.

Reallocation component. Average productivities are $(31.5,63,90)$. Share changes are $(-0.10,0.05,0.05)$. The component is $31.5(-0.10)+63(0.05)+90(0.05)=-3.15+3.15+4.50=\$4.50$ thousand per worker.

The total change is $3.30+4.50=\$7.80$ thousand per worker. Check directly: initial aggregate productivity is $0.50(30)+0.30(60)+0.20(90)=15+18+18=51$. Final aggregate productivity is $0.40(33)+0.35(66)+0.25(90)=13.2+23.1+22.5=58.8$. The difference is $58.8-51=\$7.80$ thousand per worker, as required. Here both within-sector productivity gains and the employment shift toward higher-productivity sectors raise the aggregate.

Reading the Reallocation Component

A positive reallocation component means that, on average, employment shares shifted toward sectors with higher productivity and away from sectors with lower productivity. A negative component means that the net shift went in the opposite direction. The calculation is relative: employment shares must sum to one, so an increase in one sector’s share is balanced by decreases elsewhere.

For example, if an economy has only two sectors, a worker-share shift of $0.10$ from a sector with productivity $20$ to one with productivity $60$ changes aggregate productivity by $0.10(60-20)=4$, when sector productivity is unchanged. The same-sized shift in the reverse direction changes it by $-4$. With more than two sectors, the reallocation sum combines all share changes and productivity gaps; examining individual sector terms can show which shifts matter most.

Worked Example: When Within-Sector Gains Are Not Enough

An economy has agriculture, manufacturing, and services. Initial employment shares are $(0.20,0.30,0.50)$, and productivity is $(10,30,50)$ thousand dollars per worker. Later, productivity rises by one thousand dollars per worker in every sector, but employment shares become $(0.30,0.30,0.40)$. Calculate the aggregate change and explain its sign.

Initial aggregate. $P_0=0.20(10)+0.30(30)+0.50(50)=2+9+25=36$ thousand dollars per worker.

Later aggregate. Productivity is now $(11,31,51)$, so $P_1=0.30(11)+0.30(31)+0.40(51)=3.3+9.3+20.4=33$ thousand dollars per worker. The aggregate change is $33-36=-3$ thousand dollars per worker.

Decomposition check. Average shares are $(0.25,0.30,0.45)$, so the within-sector component is $0.25(1)+0.30(1)+0.45(1)=1$. Average productivities are $(10.5,30.5,50.5)$, and share changes are $(0.10,0,-0.10)$. Reallocation is $10.5(0.10)+30.5(0)+50.5(-0.10)=1.05-5.05=-4$. The sum is $1-4=-3$, matching the direct calculation. The mistake to avoid is concluding that aggregate productivity must rise because every sector became more productive. The employment shift toward the lowest-productivity sector was large enough to outweigh those gains.

Common Mistakes and Scope

Taking an unweighted average. Sector productivity values must be weighted by employment shares. An unweighted mean treats a small sector and a large sector as equally important.
Confusing output shares with employment shares. The weighted-average formula uses $s_j=L_j/L$, not the sector’s share of total output.
Ignoring the direction of a shift. A larger employment share in a high-productivity sector tends to raise the aggregate; a larger share in a low-productivity sector tends to lower it.
Calling an accounting component a causal effect. The decomposition describes how measured sector values and employment shares combine. It does not, by itself, identify why workers moved or why sector productivity changed.
Mixing inconsistent units or coverage. Sector outputs and employment must refer to the same period and economic coverage, and output measures must be comparable across sectors. Otherwise the calculated productivity levels and their weights may not describe a meaningful aggregate.

Structural change matters because economy-wide productivity reflects both what happens within sectors and the allocation of workers among them. The decomposition helps distinguish these two sources rather than attributing every aggregate movement to technological progress or to worker reallocation alone. Its interpretation depends on the sector definitions and the productivity measure used.

Check Your Understanding

For each problem, calculate the requested aggregate or component and check that the result is consistent with the stated employment shares.

An economy has agriculture and services. Their employment shares are $0.70$ and $0.30$, and their productivities are 18 and 48 thousand dollars per worker. A shift of 0.05 of total employment moves from agriculture to services, with sector productivities unchanged. Find the change in aggregate productivity. (Answer: Increase of 1.5 thousand dollars per worker.)
Three sectors have unchanged productivities of 25, 45, and 75 thousand dollars per worker. Their shares change from $(0.40,0.40,0.20)$ to $(0.35,0.40,0.25)$. Calculate aggregate productivity in each period and the reallocation component. (Answer: 43.0 initially, 45.5 finally, and a 2.5-thousand-dollar increase from reallocation.)
For three sectors, shares change from $(0.50,0.30,0.20)$ to $(0.45,0.35,0.20)$; productivities change from $(20,40,70)$ to $(22,42,73)$, in thousands of dollars per worker. Calculate the within-sector and reallocation components, then verify their sum against the aggregate change. (Answer: Within-sector 2.20, reallocation 1.00, total increase 3.20 thousand dollars per worker.)
Two sectors have productivities of 16 and 64 thousand dollars per worker, with initial employment shares of 0.25 and 0.75. A share of 0.10 moves from the higher-productivity sector to the lower-productivity sector, and sector productivities do not change. Find initial and final aggregate productivity and the change. (Answer: 52.0 initially, 47.2 finally, a decrease of 4.8 thousand dollars per worker.)
In a two-sector economy, the lower-productivity sector’s share falls from 0.70 to 0.60 while its productivity rises from 20 to 22 thousand dollars per worker. The other sector’s share rises from 0.30 to 0.40 while its productivity rises from 60 to 63. Use the exact decomposition and verify the total change. (Answer: Within-sector component 2.35, reallocation component 4.05, total increase 6.40 thousand dollars per worker.)

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