Diminishing MRTS and Convex Isoquants

Why Isoquant Curvature Matters

An isoquant shows combinations of inputs that produce the same output. Its slope describes the production trade-off at a particular combination; its curvature describes how that trade-off changes as the firm substitutes one input for another. The key question here is whether the marginal rate of technical substitution (MRTS) diminishes as the firm uses more labor and less capital along a fixed-output isoquant.

A declining MRTS is associated with an isoquant bowed toward the origin. However, this shape does not follow just because each input has a diminishing marginal product. The relevant condition depends on how the marginal products change together, including their cross-effect.

Definition: Let output be $q=f(K,L)$, where $K$ is capital and $L$ is labor. Assume $f$ is twice continuously differentiable near the input combination of interest, and that $f_K>0$ and $f_L>0$ there. On a smooth isoquant, the marginal rate of technical substitution of labor for capital is $$ \operatorname{MRTS}_{LK}=\frac{f_L}{f_K}=-\frac{dK}{dL}. $$ It measures the amount of capital that can be reduced when labor increases by one unit, holding output fixed. MRTS is diminishing in the labor direction if this ratio falls as $L$ rises along that isoquant.

The assumptions $f_K>0$ and $f_L>0$ make the isoquant downward sloping locally and allow it to be represented as a function $K=K(L)$. The sign convention matters: $dK/dL$ is negative, while MRTS is its positive magnitude. Diminishing MRTS means the slope becomes less negative as labor rises, not more negative.

From the MRTS to Isoquant Curvature

On a fixed-output isoquant, $f(K(L),L)=q$, where $q$ is constant. Differentiating this identity gives the isoquant slope. Differentiating again gives a condition for the curve's curvature.

Differentiate output along the isoquant.
Because output is constant, $f_K K'(L)+f_L=0$. Since $f_K>0$, $K'=-f_L/f_K$.

Differentiate the slope identity.
Differentiating $f_K K'+f_L=0$ with respect to $L$ gives $f_K K''+f_{KK}(K')^2+2f_{KL}K'+f_{LL}=0$.

Interpret the signs.
Since $\operatorname{MRTS}_{LK}=-K'$, its rate of change with $L$ is $-K''$. Thus MRTS diminishes exactly when $K''>0$, under these smoothness and positive-marginal-product assumptions.

Result: Write $r=f_L/f_K$, evaluated along the isoquant. Then $$ K''(L)=-\frac{f_{LL}-2r f_{KL}+r^2 f_{KK}}{f_K}. $$ Therefore, with $f_K>0$, the isoquant is locally convex as a graph $K(L)$ when $f_{LL}-2r f_{KL}+r^2 f_{KK}\leq 0$. It has positive curvature, $K''>0$, when this expression is strictly negative. In this smooth, downward-sloping case, $K''>0$ also means MRTS is locally diminishing as labor rises.

The numerator in this condition is the second derivative of the production function in the direction tangent to the isoquant. The tangent vector in $(K,L)$ coordinates is $(K',1)=(-r,1)$. If $H$ is the Hessian matrix of second derivatives of $f$, that numerator is $(-r,1)H(-r,1)^{\mathsf T}$. This shows why the cross-partial $f_{KL}$ matters: curvature depends on how output changes when the two inputs change together, not only on each input's own second derivative.

Worked Examples

Worked Example: Measuring MRTS Along a Cobb–Douglas Isoquant

Suppose a hypothetical production process is described by $f(K,L)=\sqrt{KL}$, with positive inputs. Find the isoquant for output $q=12$, then compare the MRTS at $L=3$ and $L=6$. State whether MRTS diminishes as labor rises along this isoquant.

Method. First solve the fixed-output equation for capital as a function of labor. Then calculate the isoquant's slope and its positive magnitude.

The output condition is $\sqrt{KL}=12$. Squaring both sides gives $KL=144$, so the isoquant is $K(L)=144/L$. Differentiating, $$ K'(L)=-\frac{144}{L^2},\qquad K''(L)=\frac{288}{L^3}. $$ At $L=3$, $K=144/3=48$, and the MRTS is $-K'(3)=144/9=16$. At $L=6$, $K=144/6=24$, and the MRTS is $-K'(6)=144/36=4$. Thus labor rises from 3 to 6, capital falls from 48 to 24, and MRTS falls from 16 to 4. Also, $K''(L)>0$ for every $L>0$, confirming convexity of this isoquant.

Check. Substitution gives $\sqrt{48\cdot3}=12$ and $\sqrt{24\cdot6}=12$, so both bundles lie on the stated isoquant. The common mistake is to report the negative slope, $-16$ or $-4$, as MRTS; MRTS is the positive amount of capital given up, $-K'$.

Worked Example: Applying the Curvature Condition

Let $f(K,L)=K^{0.6}L^{0.3}$ for positive inputs. On the isoquant for $q=1$, find $K(L)$, calculate the MRTS, and determine whether MRTS diminishes as labor rises.

Method. Use the production equation to obtain the isoquant directly, and use the marginal-product ratio to check the MRTS result.

The fixed-output equation is $K^{0.6}L^{0.3}=1$, so $K^{0.6}=L^{-0.3}$. Raising both sides to the power $1/0.6$ gives $K=L^{-0.5}$. Therefore, $$ K'(L)=-0.5L^{-1.5},\qquad K''(L)=0.75L^{-2.5}. $$ Because $K''(L)>0$ for positive $L$, the isoquant is strictly convex on this range and MRTS diminishes. The marginal products are $f_K=0.6K^{-0.4}L^{0.3}$ and $f_L=0.3K^{0.6}L^{-0.7}$, so their ratio is $f_L/f_K=0.5K/L$. Substituting $K=L^{-0.5}$ gives $\operatorname{MRTS}_{LK}=0.5L^{-1.5}$, agreeing with $-K'(L)$.

At $L=1$, $K=1$ and MRTS is $0.5$. At $L=4$, $K=4^{-0.5}=0.5$, and MRTS is $0.5(4^{-1.5})=0.5/8=0.0625$. The ratio has fallen as labor increased. The common mistake is to infer curvature from the exponents alone without checking how the isoquant changes; here the explicit isoquant and MRTS confirm the conclusion.

Check. At $L=1$ and $L=4$, the bundles satisfy $K^{0.6}L^{0.3}=1$: respectively $1$ and $0.5^{0.6}4^{0.3}=1$.

Worked Example: Diminishing Marginal Products but a Nonconvex Isoquant

Consider the hypothetical production function $f(K,L)=10K+10L-K^2-L^2-3KL$. Examine the isoquant through $(K,L)=(1,1)$. Do the inputs have diminishing own marginal products there, and does MRTS diminish as labor rises?

Method. Check marginal products and own second derivatives first, then use the tangent-direction curvature condition. This separates diminishing marginal products from diminishing MRTS.

At $(1,1)$, output is $10+10-1-1-3=15$. The marginal products are $f_K=10-2K-3L$ and $f_L=10-2L-3K$, so each equals $5$ at that bundle. The own second derivatives are $f_{KK}=-2$ and $f_{LL}=-2$. Thus, locally, each input has a diminishing marginal product when the other input is held fixed.

The isoquant slope is $K'=-f_L/f_K=-1$ at $(1,1)$. The cross-partial is $f_{KL}=-3$. The numerator in the second-derivative equation is $$ f_{LL}+2f_{KL}K'+f_{KK}(K')^2=-2+2(-3)(-1)+(-2)(1)=2. $$ It follows that $K''=-2/5=-0.4$. Since $\operatorname{MRTS}_{LK}=-K'$, its local rate of change as labor rises is $-K''=0.4>0$: MRTS is increasing, not diminishing, at this point. The isoquant bends the wrong way for convexity in this neighborhood.

Check. Both marginal products are positive at the point, so the local isoquant is downward sloping and the calculation is valid. The common mistake is to see $f_{KK}<0$ and $f_{LL}<0$ and conclude the isoquant must be convex. The positive cross-partial contribution along the tangent in this example outweighs the two negative own-curvature terms.

What Guarantees Convex Isoquants?

A useful sufficient condition is that the production function be concave over the relevant region. Concavity means that its Hessian is negative semidefinite, so for every direction $z$, $zHz^{\mathsf T}\leq 0$. In particular, this holds for the tangent direction $(-r,1)$. The curvature formula then gives $K''\geq0$, provided $f_K>0$. This supports convex isoquants locally where both marginal products are positive.

Concavity of the production function is sufficient, not necessary, for a particular isoquant to be convex. The direct condition is the sign of the Hessian in the tangent direction at the points on that isoquant. Conversely, diminishing own marginal products, $f_{KK}<0$ and $f_{LL}<0$, do not determine that sign: the cross-partial can change the total.

For a twice-differentiable isoquant written as $K(L)$ on an interval, $K''>0$ throughout the interval is sufficient for strict convexity there. It is not necessary: a strictly convex function can have zero second derivative at an isolated point. For example, $g(x)=x^4$ is strictly convex even though $g''(0)=0$. A straight segment is convex but not strictly convex; strict convexity means the graph contains no line segment.

Why This Distinction Matters

Diminishing MRTS describes the changing trade-off between inputs at a fixed output level. It says that as the firm uses more labor and less capital, it gives up progressively less capital for each additional unit of labor. Convex isoquants express that same pattern geometrically under the smooth, downward-sloping conditions used here. The conclusion is about the joint behavior of inputs along an isoquant, rather than the productivity of either input considered separately.

Keep direction explicit. Diminishing MRTS here means labor rises along a fixed-output isoquant and $f_L/f_K$ falls.
Use the positive ratio. The isoquant slope is negative; MRTS is its positive magnitude.
Check the cross-partial. Own diminishing marginal products alone omit an important term in the curvature test.
Separate sufficient from necessary conditions. Concavity of production guarantees weak local convexity under the stated assumptions, but other production functions can also have convex isoquants.

Key takeaway: Along a smooth, downward-sloping isoquant, diminishing MRTS as labor rises is equivalent to positive curvature of $K(L)$. The relevant test uses the full second-derivative expression, including the cross-partial; diminishing marginal products by themselves are not enough.

Check Your Understanding

For each problem, identify the direction of movement along the isoquant and use positive MRTS values.

A production process has isoquant $K=100/L$. Find $K$ and MRTS at $L=5$ and $L=10$, and state whether MRTS diminishes as labor rises. (Answer: At $L=5$, $K=20$ and MRTS $=4$; at $L=10$, $K=10$ and MRTS $=1$; MRTS diminishes.)
On a smooth isoquant, $K'(L)=-3$ and $K''(L)=0.8$ at a bundle. What is MRTS there, and what is its local rate of change with labor? (Answer: MRTS $=3$; its local rate of change is $-0.8$, so it is diminishing.)
At a bundle, $f_K=4$, $f_L=2$, $f_{KK}=-2$, $f_{LL}=-1$, and $f_{KL}=0$. Find the local slope and curvature of the isoquant, and classify its MRTS change as labor rises. (Answer: $K'=-0.5$, $K''=0.375$, and MRTS is locally diminishing.)
Suppose $f_{KK}<0$ and $f_{LL}<0$ at a bundle. Is that enough to conclude the isoquant is locally convex? (Answer: No. The cross-partial also enters the tangent-direction curvature condition, so the own second derivatives alone are insufficient.)
Does a zero second derivative at one point rule out strict convexity of an isoquant graph? (Answer: No. Strict convexity does not require positive second derivative at every point; $g(x)=x^4$ is strictly convex despite $g''(0)=0$.)

What You'll Learn