1. To find the minimum cost of producing 1000 units of output, we need to find the combination of capital (K) and labor (L) that maximizes the production function Q = 100KL given the prices of capital and labor. The cost of producing Q units of output is given by: C = wL + rK where w is the price of labor and r is the price of capital. To find the minimum cost, we need to find the combination of K and L that maximizes the production function Q = 100KL. The marginal product of labor (MPL) is given by: MPL = dQ/dL = 100K The marginal product of capital (MPK) is given by: MPK = dQ/dK = 100L Setting MPL equal to MPK, we get: 100K = 100L This implies that K = L. Substituting K = L into the production function, we get: Q = 100L^2 To find the minimum cost, we need to find the value of L that maximizes Q. The cost function is given by: C = wL + rK Substituting K = L, we get: C = wL + rL To find the minimum cost, we need to find the value of L that minimizes C. The marginal cost (MC) is given by: MC = dC/dL = w + r The average total cost (ATC) is given by: ATC = C/Q = (wL + rL)/100L^2 To find the minimum cost, we need to find the value of L that minimizes MC and ATC. 2. A. The Cobb-Douglas production function exhibits increasing returns to scale. B. The short-run cost curve, C(q), is given by: C(q) = wL + rK C. A. Given the values of w, r, and α, we can derive the expressions for MC, VC, FC, ATC, AVC, and AFC as follows: MC = dC/dq = w + rαq^(α-1) VC = C - wL = rK FC = wL + rK ATC = FC/q = (wL + rK)/q AFC = FC/q = (wL + rK)/q AVC = VC/q = (C - wL)/q 3. A. To show the graphical relationship between short-run AFC, AVC, ATC, and MC, we can plot the following: AFC = FC/q = (wL + rK)/q AVC = VC/q = (C - wL)/q ATC = FC/q = (wL + rK)/q MC = dC/dq = w + rαq^(α-1) B. ATC reaches its minimum after AVC reaches its minimum because as output increases, the average variable cost (AVC) decreases. This is because the fixed cost (FC) is spread over a larger number of units of output, resulting in a lower average cost. C. The relationship between ATC and MC is that as output increases, the marginal cost (MC) decreases. This is because the average total cost (ATC) decreases, resulting in a lower marginal cost. 4. A. The expression for TFC is given by: TFC = wL + rK B. The expressions for AFC, AVC, AC, and MC are given by: AFC = TFC/q = (wL + rK)/q AVC = VC/q = (C - wL)/q AC = C/q = (wL + rK)/q MC = dC/dq = w + rαq^(α-1) C. To find the levels of output that minimize MC and AVC, we need to find the value of q that minimizes MC and AVC. To find the minimum value of MC, we need to find the value of q that minimizes MC. To find the minimum value of AVC, we need to find the value of q that minimizes AVC. By substituting the values of w, r, and α into the expressions for MC and AVC, we can find the minimum values of MC and AVC.