The level curves of a production function play the role that indifference curves played for the consumer. Building on Production Functions and Marginal Products and the logic of The Marginal Rate of Substitution, the slope of a level curve says how one input can replace another.
Key idea
An isoquant is every combination of inputs that makes the same output, $f(K, L) = q$. With labor across and capital up, the size of its slope is the marginal rate of technical substitution $$MRTS = \frac{MP_L}{MP_K} = -\frac{dK}{dL}\Big|_{Q \text{ fixed}},$$ the capital that one more worker can replace with output unchanged. For $Q = AK^\alpha L^\beta$, $MRTS = \dfrac{\beta}{\alpha}\cdot\dfrac{K}{L}$.
Why it works
Along $f(K, L) = q$, differentiate with respect to $L$: $MP_K\dfrac{dK}{dL} + MP_L = 0$. So $\dfrac{dK}{dL} = -\dfrac{MP_L}{MP_K}$. In words: removing $\Delta K$ machines loses about $MP_K\,\Delta K$ units; adding $\Delta L$ workers gains about $MP_L\,\Delta L$; output stays put when they match. With diminishing marginal products, each extra worker replaces less capital, so isoquants bow toward the origin.
Worked example: MRTS of a Cobb-Douglas technology
$Q = K^{1/2}L^{1/2}$. Find the MRTS at $K = 16$, $L = 4$.
- $MP_L = \tfrac{1}{2}K^{1/2}L^{-1/2}$ and $MP_K = \tfrac{1}{2}K^{-1/2}L^{1/2}$.
- $MRTS = \dfrac{MP_L}{MP_K} = \dfrac{K}{L} = \dfrac{16}{4} = 4$.
Answer: 4.
Worked example: Staying on an isoquant
$Q = \sqrt{KL}$. The mix $K = 12$, $L = 3$ makes 6 units. With $L = 9$, how much capital keeps output at 6?
- On the isoquant, $KL = 36$.
- $K = 36/9 = 4$.
- Six more workers replaced 8 machines: an average rate of $8/6 \approx 1.33$, between the MRTS at the two ends (4 and $4/9$).
Answer: $K = 4$.
Worked example: Two special shapes
What is the MRTS for $Q = 2L + K$, and what does an isoquant of $Q = \min\{K, L/2\}$ look like?
- $Q = 2L + K$: $MP_L = 2$, $MP_K = 1$, so $MRTS = 2$ everywhere. Straight isoquants: a worker always replaces two machines.
- $Q = \min\{K, L/2\}$: each unit needs one machine and two workers. Isoquants are L-shaped, with corners where $L = 2K$; extra of one input alone adds nothing.
Answer: $MRTS = 2$; L-shaped isoquants with corners on $L = 2K$.
Common mistake
Flipping the ratio to $MP_K/MP_L$, or reading an isoquant as a set of input mixes with equal cost. With labor on the horizontal axis, labor's marginal product goes on top. And an isoquant holds output fixed; the lines of equal cost are isocost lines, which depend on input prices and come in Cost Minimization. Check: at a bundle with lots of capital and few workers, a worker should be able to replace a lot of capital, so the MRTS should be large.