研一上以前高級宏觀slide the solow model and its extensions i

1、The Solow Model and Its Extensions ISolow ModelAssumes constant returns to scale production function with labor augmenting technical progress:Y = f(K,AL)Y: outputK: capital A: technologyL: laborSolow ModelAlso assume: A grows at rate g per yearL “ “ “ n “ “Constant savings rate: s “ depreciation rat

2、e: dSavings = Investment (S=I)Easier to work with y = Y/AL: output per effective worker k = K/AL: capital “ “ “ Solow ModelAnd the number of effective workers in the economy can grow either because we have more workers, or because each worker is more productive thanks to better technology. So we can

3、 rewrite our production function: y=f(k)Solow ModelNote: Diminishing returns to more capital. Then we can add the actual amount of savings per effective worker:S per effective worker = sy = sf(k)I “ “ “ “ “ “But that is equal to investment per effective worker, since the total of amount of investmen

4、t equals the total amount of savings. Since the savings rate is a constant, like 20% or .2, then savings per effective worker will be a constant fraction of output per effective worker, so its just going to be a smaller version of it:Solow ModelSo the more capital there is in the economy, the more o

5、utput and e there, so savings and investment are also higher. But again you have diminishing returns. The model is completed by adding the breakeven investment line:Breakeven investment line: (n+d+g)k. Amount of I (per effective worker) needed to maintain constant k. Solow ModelQ: Without investment

6、 little k=K/AL would fall over time for three reasons. What are they? Depreciation, the rise in A, and the rise in L. Note that this equation is a straight line: its linear in k. So well graph it that way. Solow ModelThey cross at a unique level of capital. Well call that k*. At that value of capita

7、l, the actual amount of investment in the economy is exactly equal to the amount required to stay at that value of k. So we stay there. Its an equilibrium. And its the only equilibrium in the entire model. Solow ModelIf k is below k*, what is the relationship between actual I and breakeven I? Actual

8、 I is above breakeven I, so we have more than enough I to stay at that value of k, so k grows. And if k is above k*, you have less I than you need to maintain k so k falls. So the model converges to this unique equilibrium. That gives us a constant long-run value for k and a constant long-run value

9、for y. EquilibriumIn equilibrium, k and y constantk=K/AL and y=Y/ALBalanced growth pathBut what we really care about is output and e per worker not effective worker. That is a real measure of living standards. Y/L = Y/AL x A = y x A Y/L grows at rate g (so does K/L)These variables are growing at con

10、stant rates. Thats why this steady state is known as a Balanced growth path.And we can graph this growth. Y/L is growing exponentially, so its easier to graph the log of it, which is linear:The Engine of GrowthOver time here living standards are improving at a constant rate. And that rate is determi

11、ned by one thing only: the rate of technical progress. Faster technical progress would give us a steeper growth path and e per worker would grow faster. So the key result in the Solow Model is that only technical progress can generate sustainable long-run growth. Savings and Investment?The surprisin

12、g aspect of this is what does not seem to matter for long-run growth. What is that? Savings and investment. Steady-state growth does not depend on the savings rate, s, at least in the long-run. So a country that saves and invests a lot here wont grow any faster than a country that saves and invests

13、a little. Amazing! Effects of Higher Savings Rate It does boost capital and output per effective worker. But once it reaches a new steady-state, k and y are constant again, just like before. So output per worker grows at the same rate it did before: rate g. Effects of Higher Savings Rate So there is

14、 a level effect on living standards, but no growth rate effect. Eventually growth will return to the rate of technical progress, independent of the savings rate. That is a strong result. It comes from the fact that more savings and investment boost the amount of capital per worker so much that event

15、ually it takes all that extra investment just to offset annual depreciation and to create enough new capital to give to new entrants to the labor force every year. Effects of Higher population growthHere is another important result: higher population growth reduces e per worker. It rotates upward th

16、e breakeven investment line and we settle at a lower level of output per effective worker and lower e per worker. So it is unambiguously bad. Why? Effects of Higher population growth e per worker depends in part on capital per worker, and with more entrants to the labor force, the capital stock gets

17、 spread out over more workers, so each worker has less capital to work with. No good! So to boost e per worker, have low population growth and a high savings rate. But only technical progress determines long-run growth. No Technical Progress?Q: Without technical progress what would we get? Stagnatio

18、n! No matter what the rate of I in the economy, eventually living standards would stop rising. Golden RuleOne last important result. If we care about living standards, what is more important than maximizing e per worker? Consumption per worker. Consumer welfare depends on consumption, not e. Where i

19、s consumption maximized? Golden RuleGolden Rule: consumption per worker is maximized where f(k) = n+g+df(k) = df(k)/dkThe derivative of f(k), our production function, is just the slope of our f(k) line. And the Golden Rule is satisfied where that is parallel to our breakeven investment line. Golden

20、RuleLets Think CriticallyBut is there any guarantee in this model that the equilibrium value for k will coincide with the golden rule value? No! So there is nothing that guarantees that this economy will be efficient. Weve already seen that according to this model the rate of investment cannot expla

21、in long-run economic growth. That comes from technical progress. But it turns out the rate of I also cant explain much of the difference in e per worker across countries.A Few FactsIf you assume a Cobb-Douglas production function:Y = K(AL)1-Most countries 1/3 (share of e going to capital)But then ea

22、ch 1% rise in the capital stock only boosts output by 1/3 of a %:Elasticity of Y w.r.t. K 1/3 (Low)A Few FactsNow: K/L 3 times higher in rich countries boost Y/L by 31/3 = 1.44 or 44% higherSo differences in the amount of capital per worker would predict that workers are only about 50% more producti

23、ve in rich countries. But in fact they are about 10 times more productive and they earn 10 times the e. But Y/L 10 times higher in rich countriesA Few FactsSo variations in the amount of physical capital in countries can only explain about 5% of cross-country e differences. They are not the main rea

24、son some countries are rich and others are poor. There must be some other reason. Another result that Romer works out is the speed of convergence to a new equilibrium:Slow convergence to a new equilibrium: 17 year half-lifeSo not only do changes in the savings rate have a weak long-run effect on out

25、put, they happen very slowly too. Savings and investment just dont matter that much in this model. Evaluating the Solow ModelWhat are its strengths and weaknesses? Strengths: suggests that countries that dont save much or have high population growth have lower levels of e. True. Helps us to understa

26、nd differences in living standards across the globe. Predicts rising wages and constant MPK and a constant K/Y ratio. All true. Criticisms of Solow ModelDoesnt explain rate of technical progressAssumes all nations have access to same technology (A). True? Doesnt explain savings rate Cant explain why

27、 Y/L varies so much in the world (should differ by factor of 4 vs. 32). Cant explain why little convergence in living standards (MPK should be huge in low capital countries attract I). Extensions of Solow ModelRamsey-Cass-Koopmans Model (1965): endogenizes savings rateDiamond overlapping-generations

28、 model (1965): endogenous saving rate and finite-lived householdsMankiw, Romer, and Weil (1992): added human capitalAdding finite resourcesRamsey-Cass-Koopmans ModelThe first major extension of the Solow ModelRamsey-Cass-Koopmans Model (1928/1965)Named after Frank Ramsey (who started work on this in

29、 1928), David Cass, and Tjalling Koopmans (who shared a Nobel Prize in 1975). Ramsey-Cass-Koopmans ModelEndogenized the savings rate (s)In the Solow model it was just given and there was nothing that guaranteed that it would satisfy the Golden Rule. But what if rational utility-maximizing consumers

30、chose it? Would that make a difference? To answer that question this model made the following assumptions: Ramsey-Cass-Koopmans ModelAssumptions:People live forever (infinite horizon model)2. They discount future utility at rate per year (e.g. 2%)3. Perfectly competitive firms4. No depreciationRamse

31、y-Cass-Koopmans ModelResult 1: In steady-state the saving rate is constant and Y/L grows at rate gHow does this compare to the Solow Model? Exactly the same. So assuming a constant saving rate in that model was okay. Rational consumers will do the same thing. And his basic result that long-run growth comes from technical progress also holds up. But not the second result:Ramsey-Cass-Koopmans ModelResult 2: Steady state is Pareto optimal (dynamically efficient)People choose the optimal savings rate for the economy and it maximizes peoples utility over time. We end up on the bes