What Is MRTS in Cost Minimization?

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

\[ MRTS_{LK} = \frac{w}{r} \]

Why MRTS Must Equal the Input Price Ratio

1. Firms want the cheapest way to produce a given output

A firm chooses the combination of labor and capital that produces the target output at the lowest possible cost.

2. MRTS shows the technological tradeoff

MRTS tells the firm how easily it can substitute labor for capital while holding output constant.

3. The price ratio shows the market tradeoff

The ratio \( w/r \) tells the firm how expensive labor is relative to capital.

4. Tangency ensures no cheaper substitution is possible

If MRTS ≠ w/r, the firm can reduce cost by substituting toward the cheaper input.

5. Isoquant slope = isocost slope at optimum

This tangency condition ensures the firm is on the lowest possible isocost line that still touches the isoquant.

How Cost Minimization Works (Step by Step)

Step 1: Fix the output level

The firm chooses the cheapest input bundle that produces a target output \( q \).

Step 2: Draw the isoquant

The isoquant shows all combinations of labor and capital that produce output \( q \).

Step 3: Draw isocost lines

\[ C = wL + rK \]

Step 4: Find the tangency point

\[ MRTS_{LK} = \frac{w}{r} \]

Step 5: Solve the first‑order conditions

\[ \frac{MP_L}{w} = \frac{MP_K}{r} \]

Step 6: Derive conditional factor demands

Solving the system yields \( L(w,r,q) \) and \( K(w,r,q) \).

Numerical Example

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

w = 10, r = 5

Step 1: Compute MRTS

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

Step 2: Set MRTS = w/r

\[ \frac{K}{L} = 2 \]

Step 3: Plug into production function

\[ q = \sqrt{2}L \]