Question 1In class we have discussed the different properties of production functions, andespecially the Cobb-Douglas production function. Here let’s use that for a “reallife” example.Specifically, assume you are running a business where you use capital andlabor. You know that increasing each of these factors of production increasesyour output. However you don’t know how much increasing one factor of pro-duction affects the productivity of the other one. So you asked your CFO tocheck that for you.She reports back that “We increased our capital stock by one unit and wemeasured two things: (i) the productivity of labor did not change following thisaddition, and (2) the productivity of capital fell as we added this unit”.1. Write down a production function that is consistent with these two obser-vations.Answer: We know that capital does not affect the productivity oflabor. So unlike the examples in class we need to come up with anexample where labor is independent of capital. We also know that aswe add capital the MPK of capital fell. So here is an example thatsatisfies these two conditions: Y = Kα+ N, where 0 < α < 1.You then ask your second CFO (yes, your business is doing so well that youhave two CFOS!) to redo the analysis. She reports that “We increased our laborby one worker and we measured two things: (i) the productivity of capital didnot change following this addition, and (2) the productivity of labor fell as weadded this unit”.2. Assuming that both of your CFOS are competent so their answers areconsistent with each other how would you (if it all) modify the productionfunction 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 exhibitsdecreasing MP. So overall we need to modify the production functionto Y = Kα+ Nβ, where 0 < α < 1,0 < β < 1.Now assume that a year passes and you ask your first CFO to run theexperiment again. This time she reports that “We increased our capital stockby one unit and we measured two things: (i) the productivity of labor did notchange following this addition, and (2) the productivity of capital did not changeas well after we added this unit”.3. Write down a production function that is consistent with these two obser-vations.Answer: Now we see that capital and labor still don’t interactbut that capital does not exhibit a decreasing MP any longer.1So, we need to modify the production function to Y = K + Nβ,where 0 < β < 1.4. Do you need to ask your second CFO to run again her experiment whereshe adds another unit of labor given the new information you got?Answer: Yes, we want to know if labor exhibits also decreasingMP. I.e. we want to know if β < 1 or β = 1Question 2In class we discussed the Cobb-Douglas production functionY = AKαN1−αWe argued in class that α is the elasticity of output with respect to capital andthat 1 − α is the elasticity of output with respect to labor. You are going toprove these statements.Step 1: Remind yourself from your micro class what an elasticity is....Answer: Recall that an elasticity measures the percentage changein as you change X by 1 percent. So letting η denoting an elasticityand by ∆ the change in a variable we have that the elasticity is ηY,X=∆YY∆XX=∆Y∆XXYStep 2: Convince yourself that the definition you got in Step 1 can be usedto 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.Answer: Recall that∆ln(Y )∆Y=1Ythen we get:∆ln(Y )∆ln(X)=∆ln(Y )∆Y∆Y∆X∆X∆ln(X)=1Y∆Y∆X1∆ln(X)X=∆Y∆XXYand note that the right hand side is exactly thedefinition of an elasticity.Step 3: Armed with your results from Step 2 take a good hard look atthe production function above....take the natural log of both sides and use theproperty of the natural log of a power function.Answer: Taking the "LN" of both sides we get that: ln(Y ) =αln(K) + (1 − α)ln(N )Step 4: Now go back to Step 2 and explain what indeed α is the elasticityof output with respect to capital and that 1 − α is the elasticity of output withrespect to labor.Answer: Taking the derivative of ln(Y ) with respect to ln(K) wesee that this equals α. The same for labor where the derivative equals1 − α.2Question 3Argue whether the following statements are “True, False, or Uncertain.” andexplain your answers.1. Nominal GDP must always grow faster then real GDP.Answer: False. If prices fall (i.e. there is deflation) then nominalGDP will grow slower than real GDP.2. In a production of the type Y = AKαNβa doubling of labor and capitalwill always double output.Answer: False. It depends on the sum of α + β. If this sum isless than 1 than doubling of the inputs would less than doubleoutput. If it is exactly 1 then doubling of the inputs will doubleoutput. And finally, if the sum is more than 1 than doubling ofthe inputs will more than double output.Question 4Assume that there are two countries that have:1. The same population size2. The same capital stock3. Both produce with the same type of production function, i.e. Y = AKαN1−αand you know they have the same α.Given this information:1. If you know that country 1 produces twice as much as country 2 then whatdo you know about the ratios of the TFP values in both countries?Answer: The ratio of output is given byY1Y2=A1A2K1K2αN1N21−α.Given the information is has to be that the ratio of the TFP isalso 2.2. What can you say about the ratio of the marginal productivities of laborin both countries?Answer: The ratio of the MPL is given byMP L1MP L2=(1−α)A1(1−α)A2K1K2αN1N2−α.Given the information is has to be that the ratio of the MPL isalso 2.3Now you learn that country two just grew its population size by a factor of2 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 whatdo you know about the ratios of the TFP values in both countries?Answer: With this new information we have that the ratio ofoutput isY1Y2=A1A2K1K2αN12N21−α=A121−αA2K1K2αN1N21−α. So wesee that that ratio of 2 in output can come only from the ratio 2 =A121−αA2implying that 22−α=A1A2.4. What can you say about the ratio of the marginal productivities of laborin both countries?Answer:The ratio