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A production function exhibits constant returns to scale if f(tK, tL) = t·f(K, L) for any t > 0 (doubling inputs doubles output), increasing returns if the multiplier exceeds t, and decreasing returns if less than t. Returns to scale affect long-run costs: constant returns yield flat long-run average cost, increasing returns yield declining, and decreasing returns yield rising long-run average cost.
Isoquants, which you've already studied, show the input combinations that produce a given output level. Returns to scale asks a more global question: what happens to output when you scale all inputs up proportionally? If a factory uses 100 workers and 50 machines to produce 1,000 units, and you double everything to 200 workers and 100 machines, do you get exactly 2,000 units, more than 2,000, or less? The answer depends on the technology embedded in the production function.
Constant returns to scale (CRS) means proportional input scaling produces proportional output scaling: f(tK, tL) = t·f(K, L). This is intuitive under the "replication argument" — if you can perfectly duplicate the factory with the same inputs and management practices, you should get exactly double the output. Increasing returns to scale (IRS) means doubling inputs more than doubles output. IRS arises from specialization (larger scale enables finer division of labor), indivisibilities (a specialized machine can't operate at half capacity), and geometric relationships (doubling a pipeline's diameter more than doubles flow capacity since volume scales with radius squared). Decreasing returns to scale (DRS) means doubling inputs less than doubles output — typically from coordination costs, managerial complexity, or fixed resources like land that can't be proportionally increased.
The link to long-run costs is direct. If technology exhibits CRS, long-run average cost is flat: producing 2,000 units costs exactly twice as much as 1,000. IRS implies declining long-run average cost — each additional unit is cheaper to produce as scale expands. This is the source of economies of scale and explains why IRS industries tend toward concentration (large firms have lower costs, undercutting smaller rivals). DRS implies rising long-run average cost, supporting a competitive industry structure with many firms of moderate size.
A formal shortcut: a homogeneous production function of degree k satisfies f(tK, tL) = tᵏ · f(K, L). The degree k directly encodes returns to scale — k = 1 is CRS, k > 1 is IRS, k < 1 is DRS. For the Cobb-Douglas function f(K, L) = KᵅLᵝ, the degree of homogeneity is α + β. You can check returns to scale by simply summing the exponents: if α + β = 1 you have CRS, greater than 1 gives IRS, less than 1 gives DRS. This is one reason Cobb-Douglas is so widely used — it lets you parameterize returns to scale with a single number and gives closed-form expressions for cost curves.