Problems 1-3 concern cotton, which can be produced all over the world using the same basic long-run production function: where q is output, L is labor, and K is capital. For example, labor is relatively inexpensive in India, and cotton is produced with this combination of inputs: _ = 36 and K = 9. In the United States, labor is relatively expensive and it produces cotton with this combination of inputs: _ = 4 and K = 81. 1. (5 points) If we were to map this production function using isoquants, is it possible for India and the United States to be on the same isoquant? 2. (4 points) Based on how much Z and X they actually use, do India and the United States have the same marginal rate of technical substitution? Calculate them based on the formula that we went through in class. 3. (4 points) If the price of capital (X) in each country is 50, what does this imply that the cost of labor is in each country if cotton producers are minimizing costs in each country? Solve for wage (w) in each country, the units of which are in dollars per week. Problems 4-6 concern a dairy operation that is trying to determine the optimal number of workers per day (I). It wants to hire workers up to the point where the extra output from hiring workers achieves its maximum. If q is the amount of milk produced per day and K is the number of cows, the amount of milk produced is: q = 600K'L' - K'D4. (4 points) In the short run, the number of cows (X) is fixed at 10, but the operation can vary (1) how many hours its employees work. Write the formula for the short-run production function by replacing K with its fixed short run value (10). 5. (4 points) A graph of this short run production function will show it to be shaped like an upside down U. We want to find the number of workers (1) that will maximize output. Before this maximum, each additional worker will provide more output than the previous worker did. But after this point, each additional worker makes output q decline. Start by finding the slope of the production function, which is MP, or marginal product of labor. 6. (4 points) Output is maximized at the very top of the inverted U shaped production function. This is where the slope of the production function is zero, that is, where MP, = 0. With this information, find the number of workers (Z) that gives the maximum amount of output from the last worker