Summary of Firm Production: Isoquants and Isocost Lines
Firm Production: Isoquants & Isocost Lines Explained
Introduction
Costs of production determine how firms choose inputs and organize production. This study guide explains short-run and long-run cost differences, production isoquants, isocost lines, and how firms pick the least-cost combination of inputs. Examples and formulas are included to help you apply the concepts.
Short run vs Long run costs
- The short run is a time horizon in which at least one input is fixed (often capital). Some costs are fixed and cannot be changed immediately.
- The long run is a time horizon in which all inputs are variable and the firm can change its scale of production.
Note: Fixed costs are costs that do not vary with output in the short run; variable costs change with output.
Key consequence:
- When the firm changes output, average total cost may rise more in the short run than in the long run because fixed inputs cannot be adjusted quickly.
Production isoquants
Definition
A production isoquant is the set of all combinations of inputs that yield the same level of output.
- Isoquants look like contour lines for production; each curve corresponds to a particular output level $Q$.
- Points along an isoquant are technically equivalent in the sense they produce the same $Q$.
Example (pizza factory)
- Suppose isoquant for $Q=600$ includes combinations: 5 hours labour and 1 hour machine; 2 hours labour and 2 hours machine; 1 hour labour and 4 hours machine.
Marginal Rate of Technical Substitution (MRTS)
- The slope of an isoquant is the MRTS, the rate at which labour can be substituted for capital while holding output constant.
Definition: $\text{MRTS} = \dfrac{MP_L}{MP_K}$ where $MP_L$ is the marginal product of labour and $MP_K$ is the marginal product of capital.
Example: On an isoquant $Q=1050$, moving from $(L,K)=(5,6)$ to $(3,9)$ changes labour by $\Delta L=-2$ and capital by $\Delta K=+3$. The MRTS (absolute slope) is $\dfrac{3}{2}=1.5$, meaning capital must increase by 1.5 units per 1 unit of labour reduced to keep $Q$ constant.
Isocost lines
Definition
An isocost line shows all combinations of inputs that cost the same given input prices and a budget (total cost) constraint.
If the price of capital is $P_K$ and the price of labour is $P_L$, and total cost is $TC$, then the isocost equation is:
$$TC = P_K K + P_L L$$
Rearranged for $K$:
$$K = \frac{TC}{P_K} - \frac{P_L}{P_K} L$$
- The vertical intercept is $\dfrac{TC}{P_K}$, and the slope is $-\dfrac{P_L}{P_K}$.
Example (pizza factory with prices)
- Price of machine hour $P_K = 100$, price of labour hour $P_L = 60$, and consider $TC=840$.
- Equation: $840 = 100K + 60L$ which rearranges to
$$K = 8.4 - 0.6 L$$
- Using $L=6$ gives $K=8.4-0.6\cdot6=4.8$.
Table of combinations that satisfy $K = 8.4 - 0.6L$:
| K | L |
|---|---|
| 8.4 | 0 |
| 7.8 | 1 |
| 7.2 | 2 |
| 6.6 | 3 |
| 6.0 | 4 |
| 5.4 | 5 |
| 4.8 | 6 |
| 4.2 | 7 |
| 3.6 | 8 |
| 3.0 | 9 |
| 2.4 | 10 |
| 1.8 | 11 |
| 1.2 | 12 |
| 0.6 | 13 |
| 0.0 | 14 |
Note: The slope of the isocost line equals $-\dfrac{P_L}{P_K}$, the ratio of input prices.
Least-cost input combination
- Firms choose the input combination where an isoquant is tangent to an isocost line. At that tangency, the firm produces a given output at minimum cost.
Tangency condition written with marginal products and prices:
$$\frac{MP_L}{MP_K} = \frac{P_L}{P_K}$$
This condition can also be written as:
$$\frac{MP_L}{P_L} = \frac{MP_K}{P_K}$$
- The left equation states the MRTS equals the ratio of input prices.
- The right equation says the marginal product per dollar spent is equalized across inputs; no reallocation of spending can raise output for the same cost.
Graphical intuition
- Any point along an isocost is affordable. Among those points, the one on the highest possible isoquant (highest output) that
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Isoquants and Isocosts
Klíčová slova: Copyright, Costs of Production
Klíčové pojmy: Short run has fixed inputs; long run all inputs variable, Average total cost can rise more in short run than long run, Isoquant: set of input combinations producing same output, MRTS = $MP_L / MP_K$, slope of isoquant, Isocost equation: $TC = P_K K + P_L L$, Isocost slope = $-P_L / P_K$, Least-cost tangency: $MP_L/MP_K = P_L/P_K$, Equivalently $MP_L/P_L = MP_K/P_K$, Change in input prices rotates isocost lines, Practical effects: outsourcing and automation