Mini-Project #2 - Manufacturing Math 241, Winter '16 The purpose of this project is to expose you to determining how changes in the distribution of worker labor and capital aect the creation of a product. The Cobb-Douglas production function for manufacturing total annual output, Y , units of a product given L units of labor and K units of capital is Y = AL K 1 , where A is a constant real number. There are numerous reasons to believe this model is insu cient for sophisticated economic analysis, but it remains popular due to its historical signicance and ease of use. For the purpose of this project, the units on Y will be trillions of dollars, L million workers (full-time equivalent), and K trillion dollars of capital (the value of durable goods in manufacturing). Charles Cobb and Paul Douglas originally used manufacturing data from around the turn of the 20th century to determine that 0.75. Let's assume that value holds. In 20141 , the US had 7.742 million manufacturing workers, 0.953 trillion dollars in capital and produced an annual manufacturing output of 6.178 trillion dollars. 1. Use the 2014 data to nd the value of the constant A (with three decimal places of accuracy), then write the US manufacturing production equation only in terms of variables Y , L and K. /4 1 American Fact Finder and the Bureau of Economic Analysis 2 Mini-Project #2 - Manufacturing Math 241, Winter '16 2. Assume that the total output is xed (constant) at the value given by the 2014 data, and rewrite the Cobb-Douglas equation as K in terms of L. Let's call this function K = M (L). /6 3. Referring to the function M from the previous exercise, nd a formula for M 0 (L). In economics, this function is referred to as the marginal rate of technical substitution. /3 4. What does the fact that M 0 (L) is always negative imply about labor and capital? (Remember that we are assuming total output Y is constant) /2 3 Mini-Project #2 - Manufacturing Math 241, Winter '16 5. Compute and write a sentence interpreting the value of the function M 0 (7.742), including units. /4 6. In the past decade, manufacturing labor decreased at a rate of around 0.20 million workers per year. Let t be time in years after 2014 and let's treat L as a function of time, L = f (t). Use the 2014 data from the rst page, along with the assumption that the rate of change is as stated in the previous sentence, to write L as a linear function of t. 2 /3 7. Compute the derivative of M (f (t)) with respect to time t. (Hint: if you didn't earlier, you may want to rst simplify the expression for M (L) before nding the derivative) /6 2 For values of t close to 0. 4 Mini-Project #2 - Manufacturing Math 241, Winter '16 8. At what rate is capital predicted to be changing with respect to time in 2018? Include units. /3 Presentation: /5 Total: / 36 5 \f\f\f