a: The short-run cost function is C short (q) = wL + min{rK, (q/L)} The short-run cost function, Cshort(q), is derived by holding the capital stock constant and minimizing the cost of producing a given output level, q, by selecting the optimal labor usage rate, L. As K is fixed at K¯ = 1, the production function is q = L ∗ K¯ = L. As a result, the short-run cost function can be expressed as: Cshort(q) = wL + min{rK¯, (q/L)} = wL + min{r, (q/L)}. Part b: The long-run cost function is C long (q) = w(q/L)^1/2ExplanationIn the long run, both labor and capital are variable. As a result, we will begin by solving the cost minimization problem by selecting the optimal input combination for producing a given output level, q.
The optimal combination is determined by equating the marginal productivities of labor and capital to their respective input prices: MPL/ w = MPK/r => L = K = (q/L)^1/2 => L = K = q/ L ^1/2.Substituting L and K into the production function, we obtain the optimal quantity of inputs for a given output level: q = L * K => q = (q/L)1/2 * (q/L)1/2 => q/L = K/L = (q/L)^1/2 => L= (q/L)1/2 => K/L = (q/L)1/2 => K = (q/L)^1/2 * L= q/L^1/2 * L = q^(1/2) * L^(1/2) => L = (q/K)^2. Now we can substitute this value of L in the production function q = LK to get q = (q/K)^2 * K or K = q/K => K = q^(1/2) => L = q/K => L = q^(1/2)/q^(1/2) = 1. Finally, we can substitute L and K in the cost function C = wL + rK to get the long-run cost function: Clong(q) = w(q/L)1/2 + r(q/K) = w(q/ q^(1/2))1/2 + r(q/ q^(1/2))1/2 = w(q)1/2 + rq^(1/2).Part c: We can find the level of q where the short-run cost and the long-run cost coincide by equating the two functions: Cshort(q) = C long (q) => wL + min{r, (q/L)} = w(q/L)1/2 + rq^(1/2) => L + min{rL/q, q/L} = w(q/L)1/2/L + rq^(1/2)/L => L + min{rL/q, q/L} = w(q/L)−1/2 + rq^(1/2)/q^(1/2). As L = 1 (fixed), the equality is achieved at q = r*w^2. This is the level of q where the short-run cost and long-run cost functions are equal. Part d: It is true that for levels of q not equal to the answer in Part c, the short-run cost is strictly higher than the long-run cost.