Homework 2 - Neoclassical Growth Theory1. Suppose that the function Y = F (K, L)=K.5L.5describes the pro-duction technology for an economy with fixed population. Can this economyobtain an arbitrarily high income level over time simply by building up thecapital stock, given that capital depreciates? Explain.2. Suppose that there is an economy that follows the logic of the Solowgrowth model. This economy is initally in steady state with an unchangingpopulation and an unchanging technology. If this economy experiences a one-time increase in population, then what happens over time to (a) total outputand (b) output per capita?3. Suppose that three countries have the same technology and follow thelogic of the Solow growth model. Country 1 always saves 10 percent of output(i.e s = .1), country 2 always saves 20 percent of output (i.e s = .2) and country3 always saves 30 percent of output. What is GDP per unit of labor input ineach country in steady state? What is the marginal product of capital in eachcountry in steady state?Additional Assumptions:Yt= F (Kt,Lt)=(Kt).3(Lt).7, Kt+1= Kt(1 − δ)+ItIt= sF (Kt,Lt), Lt+1= Lt(1 + n)δ = .06,n=0,g= .0ands = .1,.2,.3Hint: Find the capital-labor ratio k that makes investment i = sF (k, 1) equalto depreciation δk. It is easy to see that y = F (k, 1) = k.3for the productionfunction above.4. Calculate the Golden Rule capital-labor ratio. Do this for the modeleconomy considered in problem 3 above. To answer this question it is helpfulto know that the marginal product of capital is Fk(k, 1) = .3k−.7.5. Theory tells us that the real interest rate equals the marginal productof capital less the depreciation rate. Theory also tells us that the marginalproduct of capital decreases when the capital-labor ratio increases. We knowfrom observation that the capital-labor ratio has been rising steadily for over acentury, but that the real interest rate and the depreciation rate have remainedrelatively constant. How can all of these statements be correct? Explain.Hint: One possible answer uses technological change in an important way.Graphing the production function for different technology levels may be helpful.BONUS QUESTION: How Fast Will Mexico Converge to the US?6. Suppose that two contries both share a common technology and that bothcountries have the same saving rate. Suppose further that one economy (US) isin steady state, whereas the other (Mexico) has a per capita output level thatis one fourth the US level.1Under the additional assumptions listed below, how many model periods willit take Mexico to have a per capita output level that is one half the US level?Additional Assumptions:F (Kt,LtAt)=(Kt).3(LtAt).7, At+1= At(1 + g)Kt+1= Kt(1 − δ)+It, It= sF (Kt,LtAt), Lt+1= Lt(1 + n)δ = .06 - depreciation raten = 0 - population growth rateg = .02 - technology growth rates =