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Consider a firm with the following Cobb-Douglas production function: Y = KLSH, where K is capital (number of machines), L is the number of unskilled workers, and H the number of skilled workers

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Consider a firm with the following Cobb-Douglas production function: Y = KLSH, where K is capital (number of machines), L is the number of unskilled workers, and H the number of skilled workers. (a) (1 point) Suppose that 52,000 units of capital are in place and 1 unit of skilled labor is already hired. Derive an expression for the marginal product of unskilled labor. (b) (2 points) Suppose that the wage rate for unskilled labor is 10. What is the amount of unskilled labor this firm should hire? (c) (1 point) At this input combination, what is the marginal product of skilled labor? We know that at a competitive equilibrium, wages are equal to marginal products. The difference between marginal products of different kinds of labor determines their income difference. (d) (2 points) Suppose that a technology progress changes the production process; the new production function is given by the following: Y =K3LH Suppose that the wage rate for unskilled labor stays at 10. What is the optimal amount of unskilled labor this time? (e) (1 point) At the new input combination what is the marginal product of skilled labor? What happens to the income difference between skilled and unskilled workers?