1. Julia operates a biotechnology firm that manufactures COVID-19 vaccines using labor (L) and capital (I\\) as inputs. Julia's production function for vaccines is given by f(L, K) = 10(vV/L + 2V/K). The wage rate is w and the per-unit price of capital is r. (40 points) (a) Compute the marginal product of labor and the marginal product of capital. Explain, in words, what it means to have diminishing marginal products. Are the marginal products diminishing for Julia's firm? (6 points) (b) Compute the marginal rate of technical substitution (M RT'S) at the input bundle where L =9 and K = 36. In words, provide an economic interpretation of your answer. (6 points) () Explain, in words, what it means for a production function to have increasing, decreas- ing, or constant returns to scale. What are the returns-to-scale of Julia's production function? Justify your answer mathematically. (6 points) (d) Give the equation for the 40 unit of output isoquant and graph it (place K on the vertical axis and L on the horizontal axis). Make sure your isoquant has the correct shape, and label the intercepts. (6 points). For the remainder of the question assume w = 4 and r = 2. (e) What is the minimum cost of producing y > 0 units of output? What is the marginal cost curve? (8 points) (f) The NHS in Lancaster offers to pay Julia's firm a fixed price, p, per dose of the vaccine. Assume Julia's firm acts as a profit-maximizer. If the NHS wants 1000 doses of the vaccine from Julia's firm, what price must they offer for each dose? Explain, in words, how you arrived at your answer. (8 points)