Isoquants

This scene draws the isoquant map of a production function.

An isoquant joins input mixes that make the same output. Move the reference input point and the tangent slope changes. That slope is the marginal rate of technical substitution below. It says how much of one input replaces a unit of the other.

For example, one fewer 🧑 worker plus one more ⚙️ machine still bakes 100 🍞 bread rolls. The scarcer workers get, the more each worker is worth.

q=z1αz2βq = z_1^{\alpha} z_2^{\beta}
MRTS=∂f/∂z1∂f/∂z2=αβz2z1MRTS = \frac{\partial f/\partial z_1}{\partial f/\partial z_2} = \frac{\alpha}{\beta}\frac{z_2}{z_1}

Part of these learning paths

Examples & Exercises

Let q=z11/2z21/2q = z_1^{1/2} z_2^{1/2}.

Compute the MRTS at the reference input (z1,z2)=(4,9)(z_1, z_2) = (4, 9) by direct differentiation.

MRTS=∂q/∂z1∂q/∂z2=(1/2)z1−1/2z21/2(1/2)z11/2z2−1/2=z2z1MRTS = \frac{\partial q / \partial z_1}{\partial q / \partial z_2} = \frac{(1/2) z_1^{-1/2} z_2^{1/2}}{(1/2) z_1^{1/2} z_2^{-1/2}} = \frac{z_2}{z_1}
MRTS(4,9)=94=2.25MRTS(4, 9) = \frac{9}{4} = 2.25
α=β=1/2\alpha = \beta = 1/2, reference (4,9)(4, 9) — the MRTS tangent.
Show proof

  1. Take each marginal product: ∂q/∂z1=12z1−1/2z21/2\partial q/\partial z_1 = \tfrac{1}{2} z_1^{-1/2} z_2^{1/2}, ∂q/∂z2=12z11/2z2−1/2\partial q/\partial z_2 = \tfrac{1}{2} z_1^{1/2} z_2^{-1/2}.

  2. The ratio kills the constants, giving MRTS=z2/z1MRTS = z_2/z_1, which is 9/4=2.259/4 = 2.25 at the reference point. That is why the dashed tangent in the figure has slope −2.25-2.25.

  3. Interpretation: giving up one unit of z1z_1 requires (to first order) 2.252.25 more units of z2z_2 to hold output constant. The point is relatively abundant in z2z_2, so z1z_1 is valued highly there.

Determine the returns to scale of the Cobb-Douglas production function with α=0.6\alpha = 0.6, β=0.6\beta = 0.6, and compute by what factor output grows when inputs double.

f(tz1,tz2)=(tz1)0.6(tz2)0.6=t1.2f(z1,z2)f(tz_1, tz_2) = (tz_1)^{0.6}(tz_2)^{0.6} = t^{1.2} f(z_1, z_2)
f(2z1,2z2)=21.2f(z1,z2)≈2.297 f(z1,z2)f(2z_1, 2z_2) = 2^{1.2} f(z_1, z_2) \approx 2.297\, f(z_1, z_2)
α=β=0.6\alpha = \beta = 0.6 (increasing), reference (5,5)(5, 5).
Show proof

  1. Since α+β=1.2>1\alpha + \beta = 1.2 > 1, returns to scale are increasing — scaling inputs by tt scales output by t1.2t^{1.2}.

  2. 21.2=e1.2ln⁡2≈2.2972^{1.2} = e^{1.2 \ln 2} \approx 2.297: doubling inputs multiplies output by about 2.302.30 — more than double, as increasing returns requires.

6. Production