Extensive and Intensive Margins
Ricardo identified two channels through which rent-generating activity expands. At the extensive margin, new and less fertile land is brought into cultivation as prices rise — each extension lowers the quality of the marginal parcel and raises rents on all better land already in use. At the intensive margin, more capital and labour are applied to a fixed plot of existing land. Because land is fixed in supply, successive increments of labour face diminishing returns: each extra worker adds less to output than the previous one. Rent on a fixed plot equals the total output value minus all wage and profit payments — the surplus left after every unit of labour earns its competitive return. Both margins ultimately lead to the same measure of rent: the advantage of the best (or a given) parcel over the no-profit condition at the margin.
Extensive Margin
The extensive margin is the boundary between cultivated and uncultivated land. Cultivation expands to inferior grades when the product price $p$ is high enough to cover costs on that grade, i.e. when $pQ_m \geq wL + \pi K$ on marginal land. As the margin retreats to poorer land:
- The no-rent plot's output $Q_m$ falls.
- Rents $R_i = p(Q_i - Q_m)$ rise on all inframarginal plots.
- The corn price rises to cover higher marginal costs.
Intensive Margin
On a fixed plot of land $T$, output is a function of labour alone (holding capital constant): $Q = f(L)$, with $f'(L) > 0$ and $f''(L) < 0$ (diminishing marginal product). A profit-maximising farmer hires labour until:
$$p \cdot f'(L^*) = w \quad \Longrightarrow \quad L^* = (f')^{-1}\!\left(\frac{w}{p}\right)$$
$L^*$ = optimal labour input, $w$ = wage rate, $p$ = output price.
Once labour is paid its marginal product, the residual accrues to the landowner as rent:
$$R = p\,f(L^*) - w L^*$$
Geometrically, rent equals the area between the marginal-product curve and the wage line, integrated from $0$ to $L^*$:
$$R = p\int_0^{L^*}\!f'(L)\,dL - wL^* = p\,f(L^*) - wL^*$$
This is identical to Ricardo's differential-rent formula: the surplus of output over the wage bill.
At equilibrium, the last unit of labour applied at the intensive margin earns exactly the wage, and the last (no-rent) plot cultivated at the extensive margin earns exactly zero rent. Both conditions reflect the same no-surplus principle at the margin: every factor earns its opportunity cost, and only inframarginal units earn a surplus — which accrues to the fixed factor (land).
A plot has marginal product $MP_L = 100 - 4L$ bushels per worker. Product price $p = \$3$/bu, wage $w = \$60$/worker. Find the optimal labour input and the land rent.
- Set $p \cdot MP_L = w$:
- $3(100-4L) = 60 \Rightarrow 300-12L = 60 \Rightarrow L^* = 20$ workers.
- Total output: $Q = \int_0^{20}(100-4L)\,dL = [100L - 2L^2]_0^{20} = 2000-800 = 1200$ bu.
- Rent $= pQ - wL^* = 3(1200) - 60(20) = 3600 - 1200 = \$2{,}400$.
Using Example 1, the wage rises to $w = \$90$/worker. Find the new $L^*$ and rent.
- $3(100-4L) = 90 \Rightarrow L^* = 17.5$ workers.
- $Q = 100(17.5) - 2(17.5)^2 = 1750 - 612.5 = 1137.5$ bu.
- Rent $= 3(1137.5) - 90(17.5) = 3412.5 - 1575 = \$1{,}837.50$.
- Higher wages reduce both employment and land rent — consistent with Ricardo's wages–profits–rent inverse relationship.
Grade E land produces 50 bu per labour unit. Labour cost per unit $= \$3$/worker, profit rate requires $\$1$/worker return on capital. Should Grade E be cultivated when corn price $= \$5$/bu? What is the minimum price to justify cultivation?
- Revenue per unit $= 5(50) = \$250$. Cost (wage + profit) $= \$4$ per unit.
- Revenue exceeds cost: yes, cultivate.
- Minimum price: $p_{\min} \cdot 50 = 4 \Rightarrow p_{\min} = \$0.08$/bu. Grade E earns positive rent at $\$5$/bu.
- Wait — if Grade E is to be the no-rent margin (replacing current margin), rent must be zero: $p(50) = $ total cost ⟹ this occurs when price exactly covers costs on E, making E the marginal land.
- At $p = \$5$ Grade E already earns rent, so a still-inferior grade would become the new margin.
Practice Problems
Show Answer Key
1. $4(80-2L)=48 \Rightarrow 80-2L=12 \Rightarrow L^*=34$. $Q=80(34)-34^2=2720-1156=1564$ bu. Rent $=4(1564)-48(34)=6256-1632=\$4{,}624$.
2. $6(80-2L)=48 \Rightarrow L^*=36$. $Q=80(36)-36^2=2880-1296=1584$ bu. Rent $=6(1584)-48(36)=9504-1728=\$7{,}776$. (Higher price, more labour, much higher rent.)
3. $5(60-3L)=75 \Rightarrow L^*=15$. $Q=60(15)-1.5(15^2)=900-337.5=562.5$ bu. Rent $=5(562.5)-75(15)=2812.5-1125=\$1{,}687.50$.
4. $MP_L=100-4L$. Set $2(100-4L)=40 \Rightarrow L^*=15$. $Q=100(15)-2(225)=1500-450=1050$. Rent $=2(1050)-40(15)=2100-600=\$1{,}500$.
5. Higher wages mean the farmer must pay more per unit of labour, leaving a smaller surplus after the wage bill — rent falls.
6. At the no-rent margin: $p \cdot MP_L = w$ and rent $= pQ - wL = 0$, meaning total output value exactly equals the wage bill — no surplus for land.
7. $10(50-L)=100 \Rightarrow L^*=40$. $Q=50(40)-\frac{1}{2}(40^2)=2000-800=1200$. Rent $=10(1200)-100(40)=12000-4000=\$8{,}000$. At $w=200$: $10(50-L)=200 \Rightarrow L^*=30$. $Q=50(30)-450=1050$. Rent $=10500-6000=\$4{,}500$. Rent halved.
8. Both describe the same law of diminishing returns: whether you extend cultivation to worse land or pile more inputs on existing land, successive increments yield progressively less output. The rent on good land (or the surplus on the existing plot) is the sum of all these diminishing increments above the marginal unit.
9. $MP_L=5L^{-0.5}$. Set $9(5L^{-0.5})=15 \Rightarrow L^{-0.5}=1/3 \Rightarrow L^*=9$. $Q=10(3)=30$. Rent $=9(30)-15(9)=270-135=\$135$.
10. More labour is employed. If the $MP_L$ curve falls more slowly (later diminishing returns), the point where $p \cdot MP_L = w$ occurs at a higher $L^*$, generating larger output and higher rent.
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