Show me the steps to solve Under constant returns to scale, average cost as the quantity produced increases. Over this range of output, the marginal cost curve is the average cost curve.
State the constant returns to scale (CRS) property: for any scalar α > 0, the production function satisfies f(αK, αL) = α f(K, L).
Write the cost-minimization problem for output q:
C(q) = min_{K,L} [rK + wL] subject to f(K,L) = q,
where r and w are input prices.
Use scaling: if (K*, L*) is cost-minimizing for output 1, then (qK*, qL*) produces q. By CRS, costs scale linearly, so C(q) = q · C(1).
Compute average cost:
AC(q) = C(q) / q = [q · C(1)] / q = C(1),
so average cost is constant as quantity increases.
Compute marginal cost:
MC(q) = dC/dq = d[q · C(1)]/dq = C(1),
hence MC(q) = AC(q) for all q in this range.
Conclusion: Under CRS, average cost remains constant as output increases, and the marginal cost curve coincides with the average cost curve over this range.
Leave a Reply