Suppose that a firm is facing perfectly competitive markets for its product and factor inputs. This means that the firm

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Suppose that a firm is facing perfectly competitive markets for
its product and factor inputs. This means that the firm take the
price per unit of the product (p), the unit cost of capital input
(r) and the unit coast of labor (w) as given. Suppose that the
production function of the firm is given by Y = F(K,L), where Y is
output, K is capital, and L is labor. Further, FK > 0 (Marginal
product of capital is positive), FL > 0 (Marginal product of
labor is positive, and there are diminishing returns in both factor
inputs. Finally, the firm also receives wage subsidy from the
government of s per unit of labor.
(a) Write the profit function of the firm.
(b) Based on the profit function, what is the impact of increasing
wage subsidy on capital and labor, i.e. 𝜕𝐾𝜕𝑠 and 𝜕𝐿𝜕𝑠?