Final answer:
The student's question involves cost minimization and the derivation of input demand functions, the expansion path, and the total cost function for a company producing computers with given labor and capital costs. The input demand functions and total cost function are derived from the first-order conditions of a given production function, applying economic principles of cost minimization.
Step-by-step explanation:
The student question involves the derivation of input demand functions and the total cost function in the context of a company, Hewlett Packard (HP), using labor and capital as inputs for producing computers. To address part (a), we need to set up the cost minimization problem where HP seeks to minimize total costs of labor and capital while meeting the demand for Q units of computers. The cost function can be written as C = wL + rK, where w and r represent the cost of labor and capital, respectively, and L and K are the quantities of labor and capital. The production function is given as F(K, L) = 3LK. To find the input demand functions, L(w, r, Q) and K(w, r, Q), we would derive the first-order conditions for minimizing C subject to the constraint of producing Q units.
For part (b), the equation of the expansion path, K = K(L), indicates the optimal combination of capital and labor as output expands. This can be found by applying the optimal ratio of capital to labor derived from the first-order conditions of the cost minimization problem.
Lastly, for part (c), the total cost function C(w, r, Q) would be derived using the input demand functions and substituting back into the cost function, reflecting the minimum cost to produce Q units given the wage rate w and rental rate r of capital.