What Does It Mean That Isoquants Are Convex?

An isoquant shows all combinations of labor and capital that produce the same output. Convexity means the curve bends inward toward the origin.

Formally:

\[ MRTS_{LK} = -\frac{MP_L}{MP_K} \]

Diminishing MRTS means:

\[ \frac{MP_L}{MP_K} \text{ decreases as } L \text{ increases and } K \text{ decreases} \]

Why Isoquants Are Convex

1. Diminishing marginal product of each input

As a firm uses more labor while holding capital fixed, the marginal product of labor falls. The same is true for capital.

2. Substitution becomes harder as one input becomes scarce

If the firm has very little capital, replacing it with labor becomes increasingly difficult. This makes MRTS fall.

3. MRTS diminishes along the isoquant

Convexity is the geometric expression of diminishing MRTS.

4. Firms prefer balanced input bundles

Convexity reflects the idea that extreme input ratios are inefficient.

5. Convexity ensures well‑behaved optimization

Convex isoquants guarantee unique cost‑minimizing input bundles.

How Diminishing MRTS Works (Step by Step)

Step 1: Start with a balanced bundle

Suppose the firm uses moderate amounts of labor and capital. Both inputs have relatively high marginal products.

Step 2: Increase labor and reduce capital

As labor increases and capital decreases, the marginal product of labor falls, while the marginal product of capital rises.

Step 3: Compute MRTS

\[ MRTS_{LK} = \frac{MP_L}{MP_K} \] As MP_L falls and MP_K rises, MRTS decreases.

Step 4: Substitution becomes more difficult

Each additional unit of labor replaces less and less capital while keeping output constant.

Step 5: The isoquant bends inward

This diminishing MRTS creates the convex shape of the isoquant.

Step 6: Connect to cost minimization

Convexity ensures a unique tangency point between the isoquant and the isocost line.

Numerical Example

Consider a Cobb–Douglas production function:

\[ q = L^{0.5}K^{0.5} \]

Marginal products:

\[ MP_L = 0.5L^{-0.5}K^{0.5}, \quad MP_K = 0.5L^{0.5}K^{-0.5} \]

MRTS:

\[ MRTS_{LK} = \frac{MP_L}{MP_K} = \frac{K}{L} \]

Interpretation

• If L increases and K decreases → K/L falls → MRTS falls.