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7 . Consider a perfectly competitive firm that produces output from capital and labor, according to the production function: q f(K,L): The production function has continuous first and second partial derivatives with the following properties: f > 0,fk > 0,fu < 0,fkk <0, fuk = fkl >0. The firm can purchase as much capital (K) and labor (L) as it wants to at their market prices: > 0 (the unit price of capital) and w > 0 (the unit price of labor), which the firm takes as given. It
can sell as much output as it likes at the market price for its product: p > 0, which the firm takes as given.
Find both first-order partial derivatives of the firm'$ profit function: T(K,L) = p f(K,L) (rK + wL): What two equations describe the first order, necessary conditions for a local max of the profit function? Find the total differentials of the two equations that describe the first order, necessary conditions for a local max of the profit function_ [dK Letting the vector of unknowns, X, be defined as x = write the system of two dL equations from part € as a matrix equation, Ax = b_ Use Cramer's rule to find the partial derivatives dK* /dr and dK* /dw of the capital demand function, K* g (r ,W,p), implicitly defined by the first order, necessary conditions for a local max of the profit function. What are the signs of dK* /dr and dK* /dw?