MRTS and Cost Minimization

Marginal Rate of Technical Substitution

Marginal Rate of Technical Substitution (MRTS) is the rate at which one input can be substituted for another while keeping output constant.

MRTS is the slope of an isoquant.

For labour and capital:

MRTS_{LK} = -\frac{\Delta K}{\Delta L}

Using marginal products:

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

Where:

MRTS shows how much capital can be reduced when labour is increased while keeping output constant.

The marginal product of labour is:

MP_L = \frac{\partial Q}{\partial L}

The marginal product of capital is:

MP_K = \frac{\partial Q}{\partial K}

Therefore:

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

For a convex isoquant, MRTS decreases as more labour is substituted for capital.

This means that as the firm uses more labour and less capital, it becomes willing to give up less capital for an additional unit of labour.

A firm wants to produce a given level of output at the lowest possible cost.

The firm chooses the combination of inputs that produces the required output at minimum cost.

At the cost-minimizing input combination:

\frac{MP_L}{MP_K} = \frac{P_L}{P_K}

Equivalently:

MRTS_{LK} = \frac{P_L}{P_K}

Where:

The firm minimizes the cost of producing Q₁ where the isoquant is tangent to the lowest attainable isocost line.

Written by

September 26, 2026 1 min read

Was this helpful?