Macro Midterm #2
How to find the Golden Rule rate of savings
This is the rate of saving that leads to the highest stead-state level of consumption. The rate can be found by setting MPK=depreciation. Since here depreciation=0, MPK is never zero since you can always get more productive by adding more capital. Therefore, there is no Golden Rule level of savings.
Given a per capita production function of y=k^1/4, capital available in year 1 is 10, depreciation is 0, how do you find the amount of capital, production, and consumption available in its 5th year? Assume that s=11, and then repeat again assuming s=40.
-Must know the amount of capital available in the next period is equal to the amount of capital in period plus the amount that is saved, which mathematically is Kt+1 = Kt +savings(St); and that production not saved as capital is consumed in the current period which is Yt=Ct+It. -Start with the amount of capital in year 1, 10, and plug it into production function to get 10^(1/4). To get savings, multiply this # by .11 and consumption by .89. The resulting # from savings will be added to the capital in period 2, and then repeat the process until year 5. -To find percent growth between year 1 and year 5, you do (production Year 5-production Year1)/(production Year1) -Same process except utilize s=40 instead.
Go back to the case where y=k^(1/4), depreciation =0.05, and the savings rate is .11%. Now suppose that the population is growing at 4.1% per year. Solve for the new steady state level of capital, output, and consumption Solve for the new golden rule level of savings, capital, output, and consumption. According to this model, what does population growth do to economic growth?
-The new equation we must utilize in this case is (depreciation+population growth rate)k=sy=sk^(1/4). This comes out to 0.091k=.11k^(1/4). In solving for k, we get k=1.288. Now, we can find y by plugging this value of k into the production function to get y=1.065. Now we can solve for s by plugging in the values and getting the equation .091*1.288=s*1.065. Here, we get s=11%. And therefore, c is 1.065-1.065*.11 = .948 -In order to find the golden rule, we have the equation MPK=(depreciation + Pop growth rate) , which comes out to (1/4)k^(-3/4)=(.091). In solving for k here, we get k=3.848. Now we can find y which is 3.484^(1/4) which comes out to 1.401. From here, we can find the value of s, which is found through the equation .091*3.848=s*3.848^(1/4) which comes out to s=.25. Once we have s=.25, we can find the value of c, which is 1.401-1.401*.25=1.05 According to this model, while the optimal savings rate is the same under population growth and no population growth, population growth reduces the golden rule level of output and consumption again by implicitly increasing the rate at which per capita capital disappears.
Assume s=11%, K in first year =10, and depreciation =.05. Solve for the steady state level of capital ,output, and consumption for the first country. Also, solve for the Golden Rule of savings, capital, output and consumption.
-since (depreciation)*k=sy=s*k^(1/4), we plug in the given values and come out to an equation of (0.05)k=.11(1.778). Solving for k here , we get k=2.861, and then from here we can plug this value of k back into the previous equation to solve for y, which results in y=1.301. Then, in order to find consumption, we can either do y*.89 or use the equation of c=y-sy in which we can plug in previously solved values. Calculating, we get c=1.157 -To find the golden rule of capital, we use the equation MPK=depreciation. So, (1/4)k^(-3/4)=.05. In solving for k, we get: k=8.55. Once we have k, we can use this to find the rest of the variables with, again, the equation (depreciation)*k=sy=sk^(1/4). Now we know what y is since we plug k=8.55 into the production function to get y=1.710. Now, to find s we have .05*8.55=s*1.710, and s comes out to .25. To find c, we can use the equation c=y-sy to come out to 1.282.
If a country has a higher level of consumption in the fifth year from the previous example, does this mean that this country is better off in 5th year?
Consumption is higher in year 5 with the country that has the lower savings rate, S=11%, but the higher savings country, s=40%, has higher production. While the country with the lower saving rate starts out with the highest consumption it will eventually lose out to the higher savings country in terms of production. The high savings country forgoes current consumption in exchange for future consumption. Its high savings rate is setting it up for high rates of consumption in the future.
"Devoting a larger share of national output to investment would help restore rapid productivity growth and rising living standards." Do you agree with this claim? Explain, using the Solow model.
A- suppose the economy begins in steady state with capital per worker below the Golden Rule. Then suppose the economy adopts the higher saving rate needed to achieve the Golden Rule steady state. Comparing the two steady states, output per worker and consumption per worker are both higher with the higher saving rate, so the standard of living increases in the long run. But in both steady states the growth rate of output per worker is zero, so the long-run rate of productivity growth is unaffected by the higher saving rate. We can also consider the transition from one steady state to the other. the higher saving rates to higher growth in the output per worker in the short run, but it also lowers consumption per worker in the short run, so there is a short-term sacrifice involved in achieving a higher standard of living in the long run.
Country A and B both have the production function Y=F(K,L)=K^(1/3)L^(2/3). a.) Does this production function have constant returns to scale? Explain. b.) what is the per worker production function y=f(k)? c.) Assume that neither country experiences population growth or technological progress and that 20% of capital depreciates each year. Assume that further country A saves 10 percent of output each year and country B saves 30 percent of output each year. Using your answer from part (b) and the steady-state condition that investment equals depreciation, find the steady state level of capital per worker for each country. then find the steady state levels of income per worker and consumption per worker.
a.) A production function has constant returns to scale if increasing all factors of production by an equal percentage causes output to increase by the same percentage. mathematically, if the function were multiplied by function z --> F(zK, zL)=zK^(1/3)L^(2/3) = zY. b.) The per worker function represents per-capita, and in this case is just y=k^(1/3). Mathematically, we can solve this from the original production function by dividing by L on both sides of the function, and remembering that y=K/L, and k=K/L. c.) The steady state condition on capital per worker is sf(k) = dk. Since we know f(k) =k^(1/3), we can plug this value into the steady-state condition and solve for k where we get k=(s/d)^(3/2). Once we have this, then we plug in the given values for each country and solve. we know c=(1-s)y or c=y-sy, so we can find this for each country
Suppose now that the GDP growth equation is Y=K/3. Depreciation=0, assume k1=10 a.) if the savings rate is 11% what is the annual growth rate in the economy? b.) if the savings rate is 40% what is the annual growth rate in the economy? C.) what fundamental properties of production function does this model satisfy and not satisfy? d.) If this model is accurate, what is the best way for countries to achieve economic growth? e.) If this model is accurate, how could rich countries help poor countries?
a.) In order to find the annual growth rate, we set up a chart with year, capital, production, saving, and growth rate. since capital is 10 in the first year, we plug into production function K/3 to get production which is 3.333. Since the savings rate is 11%, we do 3.333*.11 which is .367 and add it to 10 in order to get the amount of capital available in year two, which is now 10.367. Then we plug this value into the production function and get a production value of 3.457. this value then is multiplied by .11 and added to 10, which comes out to 10.747. Once we have this capital available in year 3, we can find the percent growth between year 2 and 3, which is done through the formula (10.747-10.367)/(10.367) and we get a percentage growth of 3.67%. b.) Conduct and identical process, instead replace .11 with .40. c.) The model satisfies constant returns to scale because if you double K, you double Y. The model satisfies positive constant marginal product since if K=10 and you add one more unit of K, Y increases by 1/3, and the same can be said if K=100. Does not satisfy diminishing marginal product since dY/dK =1/3 (which is a constant) and not negative as it requires. d.) What is the best way for countries to achieve economic growth? -Growth is a direct function of the savings rate. Faster growth is accomplished only through higher savings. e.) What is the best way for rich countries to help poor countries? - Any transfer of capital will increase output. Any steps taken to help countries increase their savings rate will increase growth.
In the discussion of German and Japanese postwar growth, the text describes what happens when part of the capital stock is destroyed in a war. By contrast, suppose that a war does not directly affect the capital stock, but that casualties reduce the labor force. Assume the economy was in a steady state before the war, the saving rate is unchanged, and the rate of population growth after the war is the same as it was before. a. What is the immediate impact of the war on total output and on output per person? b. What happens subsequently to output per worker in the postwar economy? Is the growth rate of output per worker after the war smaller or greater than it was before the war?
a.) the production function in the solow model is Y=F(K,L) or expressed in per worker terms as y=f(k). If a war reduces the labor force through casualties, then L falls, but k=K/L rises. Thus, total output falls because there are fewer workers, but output per worker rises because each worker has more capital. b.) The economy is initially in a steady state with capital per worker k*. After the war, the reduction in the labor force increases capital per worker to k1. With capital per worker of k1, investment is lower than break-even level, so capital per worker, and therefore output per worker, will fall over time as the economy converges to the steady state. The economy will experience an economic contraction.
Consider how unemployment would affect the Solow growth model. Suppose that output is produced according to the production function Y = K^α[(1 − u)L]^1−α, where K is capital, L is the labor force, and u is the natural rate of unemployment. The national saving rate is s, the labor force grows at rate n, and capital depreciates at rate δ. a. Express output per worker (y = Y/L) as a function of capital per worker (k = K/L) and the natural rate of unemployment (u). b. Write an equation that describes the steady state of this economy. Illustrate the steady state graphically, as we did in this chapter for the standard Solow model. c. Suppose that some change in government policy reduces the natural rate of unemployment. Using the graph you drew in part (b), describe how this change affects output both immediately and over time. Is the steady-state effect on output larger or smaller than the immediate effect? Explain.
a.) to find output per worker y, divide the given solow expression by L on both sides in order to get y=k^g
consider an economy described by the production function Y=F(K,L)=K^0.4L^0.6. a.) what is the per-worker production function? b.) Assuming that no population growth or technological progress, find the steady-state capital per worker, output per worker, and consumption per worker as a function of the saving and depreciation rates.
a.) to find the per worker production function, we have to divide both sides by L. Here we utilize the fact that y=Y/L and k=K/L in order to get y=k^0.4. b.) Recall that ∆k=sf(k)-dk. Steady state capital per worker k* must satisfy the condition that sf(k)=dk, or equivalently, (k/f(k))=(s/d). again, remember that consumption per worker is c=(1-s)y
