A firm is a business that takes inputs (like labour, capital, and raw materials) and turns them into outputs (goods or services) to sell in the market. Think of a firm as a kitchen that mixes ingredients (inputs) to bake a cake (output). 🍰
The production function shows the relationship between the quantity of inputs used and the quantity of output produced. It is written as:
\$Q = f(L, K, R)\$
Productivity measures how efficiently inputs are turned into outputs. Two common types are:
In a Cobb‑Douglas production function, TFP is represented by \$A\$:
\$Q = A K^\alpha L^\beta\$
Imagine a bakery that uses 2 bakers (\$L=2\$) and 1 oven (\$K=1\$). If the bakery’s technology level \$A\$ is 1.5 and the exponents are \$\alpha=0.4\$ and \$\beta=0.6\$, the production function becomes:
\$Q = 1.5 \times 1^{0.4} \times 2^{0.6} \approx 1.5 \times 1 \times 1.52 \approx 2.28 \text{ cakes/day}\$
If the bakery invests in a new mixer (increasing \$K\$ to 2), the output rises to:
\$Q = 1.5 \times 2^{0.4} \times 2^{0.6} \approx 1.5 \times 1.32 \times 1.52 \approx 3.01 \text{ cakes/day}\$
| Production Function Type | Key Features |
|---|---|
| Linear | Constant returns to scale; output increases proportionally with inputs. |
| Cobb‑Douglas | \$Q = A K^\alpha L^\beta\$; flexible returns to scale; widely used in economics. |
| Leontief (Fixed‑Proportions) | Inputs must be used in fixed ratios; output limited by the scarcest input. |
| Metric | Formula | Example (Bakery) |
|---|---|---|
| Labour Productivity | \$Q / L\$ | \$2.28 / 2 = 1.14\$ cakes per baker per day |
| TFP (Total Factor Productivity) | \$Q / (K^\alpha L^\beta)\$ | \$2.28 / (1^{0.4} \times 2^{0.6}) = 1.5\$ (matches \$A\$) |