Question 1 In class we have discussed the different properties of production functions and especially the Cobb Douglas production function Here let s use that for a real life example Specifically assume you are running a business where you use capital and labor You know that increasing each of these factors of production increases your output However you don t know how much increasing one factor of production affects the productivity of the other one So you asked your CFO to check that for you She reports back that We increased our capital stock by one unit and we measured two things i the productivity of labor did not change following this addition and 2 the productivity of capital fell as we added this unit 1 Write down a production function that is consistent with these two observations Answer We know that capital does not affect the productivity of labor So unlike the examples in class we need to come up with an example where labor is independent of capital We also know that as we add capital the MPK of capital fell So here is an example that satisfies these two conditions Y K N where 0 1 You then ask your second CFO yes your business is doing so well that you have two CFOS to redo the analysis She reports that We increased our labor by one worker and we measured two things i the productivity of capital did not change following this addition and 2 the productivity of labor fell as we added this unit 2 Assuming that both of your CFOS are competent so their answers are consistent with each other how would you if it all modify the production function you wrote in 1 Answer Now we see that indeed capital and labor are separate from each other But what we also learned is that labor also exhibits decreasing MP So overall we need to modify the production function to Y K N where 0 1 0 1 Now assume that a year passes and you ask your first CFO to run the experiment again This time she reports that We increased our capital stock by one unit and we measured two things i the productivity of labor did not change following this addition and 2 the productivity of capital did not change as well after we added this unit 3 Write down a production function that is consistent with these two observations Answer Now we see that capital and labor still don t interact but that capital does not exhibit a decreasing MP any longer 1 So we need to modify the production function to Y K N where 0 1 4 Do you need to ask your second CFO to run again her experiment where she adds another unit of labor given the new information you got Answer Yes we want to know if labor exhibits also decreasing MP I e we want to know if 1 or 1 Question 2 In class we discussed the Cobb Douglas production function Y AK N 1 We argued in class that is the elasticity of output with respect to capital and that 1 is the elasticity of output with respect to labor You are going to prove these statements Step 1 Remind yourself from your micro class what an elasticity is Answer Recall that an elasticity measures the percentage change in as you change X by 1 percent So letting denoting an elasticity and by the change in a variable we have that the elasticity is Y X Y Y X X Y X X Y Step 2 Convince yourself that the definition you got in Step 1 can be used to express the elasticity as a statement about the derivatives of the natural log i e LN of one variable with respect to the natural log of another variable ln Y Y X Answer Recall that ln Y Y1 then we get ln X ln Y Y Y X ln X 1 Y 1 Y X Y X ln X X Y and note that the right hand side is exactly the X definition of an elasticity Step 3 Armed with your results from Step 2 take a good hard look at the production function above take the natural log of both sides and use the property of the natural log of a power function Answer Taking the LN of both sides we get that ln Y ln K 1 ln N Step 4 Now go back to Step 2 and explain what indeed is the elasticity of output with respect to capital and that 1 is the elasticity of output with respect to labor Answer Taking the derivative of ln Y with respect to ln K we see that this equals The same for labor where the derivative equals 1 2 Question 3 Argue whether the following statements are True False or Uncertain and explain your answers 1 Nominal GDP must always grow faster then real GDP Answer False If prices fall i e there is deflation then nominal GDP will grow slower than real GDP 2 In a production of the type Y AK N a doubling of labor and capital will always double output Answer False It depends on the sum of If this sum is less than 1 than doubling of the inputs would less than double output If it is exactly 1 then doubling of the inputs will double output And finally if the sum is more than 1 than doubling of the inputs will more than double output Question 4 Assume that there are two countries that have 1 The same population size 2 The same capital stock 3 Both produce with the same type of production function i e Y AK N 1 and you know they have the same Given this information 1 If you know that country 1 produces twice as much as country 2 then what do you know about the ratios of the TFP values in both countries 1 A1 K1 N1 Answer The ratio of output is given by YY21 A K2 N2 2 Given the information is has to be that the ratio of the TFP is also 2 2 What can you say about the ratio of the marginal productivities of labor in both countries 1 A1 P L1 K1 N1 Answer The ratio of the MPL is given by M M P L2 1 A2 K2 N2 Given the information is has to be that the ratio of the MPL is also 2 3 Now you learn that country two just grew its population size by a factor of 2 but that capital has not changed There is no change in economy 1 Then given this information 3 If you know that country 1 produces twice as much as country 2 then what do you know about the ratios of the TFP values in both countries Answer With this we have the new information 1 that 1 ratio of Y1 A1 K1 N1 A1 K1 N1 21 A2 K2 So we output is Y2 A2 K2 2N2 N2 see that that ratio of 2 in output can come …