Question 2 Solow grouth model. In this question, you will explore how changes in the saving rate and the rate of technological progress affect an economy's growth. In addition, you will examine how the golden rule saving rate depends on the production function. Consider the Solow (neoclassical) growth model with aggregate production function Y = K (AN)'. Each period lasts a year. (1) Using the parameter values in Table 1, calculate the steady-state values of capital per effective worker k = K/AN, output per effective worker y = Y/AN and consumption per effective worker C = C/AN. Calculate the golden rule saving rate. Also calculate the growth rates of output per worker and consumption per worker along the balanced grouth path. (2 points) 1/3 9A SN 30% 10% Table 1: Benchmark Parameter Values-Solow model (2) Suppose that the economy is initially in steady-state. In year t=0 the saving rate increases from s = 0.16 to 8 = 0.25 (i.e., from 16% to 25%) while all other parameters have their benchmark values () Calculate the new steady-state levels of capital per effective worker, output per effective worker and consumption per effective worker. Does long-run consumption per effective worker increase? Also calculate the long run growth rates of output per worker and con sumption per worker? Do the long-run grouth rates of output per worker and consumption per worker increase? Explain. (3 points) (11) Calculate and plot the time-paths of (a) capital per effective worker, output per effective worker and consumption per effective worker for 100 years (t = 0,1...... 100) and of (b) log output per worker and log consumption per worker. Describe and explain the short-run effect of the change in the saving rate on these variables. (3 points) Note: Here log means the natural logarithm, in(-) = log.(). To answer part (il) of this question we need to know the level of productivity at year t = 0. Assume this initial level of productivity is Ao = 1. You will probably want to use a spreadsheet program to implement these calculations. (3) We will now contrast these results with a change in the rate of technological progress. Suppose instead that at time t = 0, the rate of technological progress increases to 4% per year. With all other parameters as in Table 1 (in particular, with the saving rate back at its benchmark value of 16%), calculate and plot the time-path of log output per worker for 100 years after the change in the rate of the technological progress (for t = 1,2,...,100), again assuming A = 1. Compare the time path of log output per worker from the increase in the saving rate in part (2) to the time path with the increase in the rate of technological progress. How many years pass before output per worker surpasses the level that would be obtained from the increase in the saving rate? What does this suggest about the relative importance of level versus growth rate effects? Explain. (3 points) (4) In the Solow model, the golden rule saving rate is defined as the saving rate that maximize the steady state consumption per effective worker. Using the expression of the steady state consumption per effective worker to derive how the steady state consumption per effective worker depends on the saving rate. With all other parameters as in Table 1, plot how the steady state consumption per effective worker depends on the saving rate for s=0,001,0.02, 0.03 ...,1. In your plot, the horizontal axis should be the saving rate ranging from 0 to 1 and the vertical axis is the steady state consumption per effective worker. What is the golden rule saving rate? (3 points) Question 2 Solow grouth model. In this question, you will explore how changes in the saving rate and the rate of technological progress affect an economy's growth. In addition, you will examine how the golden rule saving rate depends on the production function. Consider the Solow (neoclassical) growth model with aggregate production function Y = K (AN)'. Each period lasts a year. (1) Using the parameter values in Table 1, calculate the steady-state values of capital per effective worker k = K/AN, output per effective worker y = Y/AN and consumption per effective worker C = C/AN. Calculate the golden rule saving rate. Also calculate the growth rates of output per worker and consumption per worker along the balanced grouth path. (2 points) 1/3 9A SN 30% 10% Table 1: Benchmark Parameter Values-Solow model (2) Suppose that the economy is initially in steady-state. In year t=0 the saving rate increases from s = 0.16 to 8 = 0.25 (i.e., from 16% to 25%) while all other parameters have their benchmark values () Calculate the new steady-state levels of capital per effective worker, output per effective worker and consumption per effective worker. Does long-run consumption per effective worker increase? Also calculate the long run growth rates of output per worker and con sumption per worker? Do the long-run grouth rates of output per worker and consumption per worker increase? Explain. (3 points) (11) Calculate and plot the time-paths of (a) capital per effective worker, output per effective worker and consumption per effective worker for 100 years (t = 0,1...... 100) and of (b) log output per worker and log consumption per worker. Describe and explain the short-run effect of the change in the saving rate on these variables. (3 points) Note: Here log means the natural logarithm, in(-) = log.(). To answer part (il) of this question we need to know the level of productivity at year t = 0. Assume this initial level of productivity is Ao = 1. You will probably want to use a spreadsheet program to implement these calculations. (3) We will now contrast these results with a change in the rate of technological progress. Suppose instead that at time t = 0, the rate of technological progress increases to 4% per year. With all other parameters as in Table 1 (in particular, with the saving rate back at its benchmark value of 16%), calculate and plot the time-path of log output per worker for 100 years after the change in the rate of the technological progress (for t = 1,2,...,100), again assuming A = 1. Compare the time path of log output per worker from the increase in the saving rate in part (2) to the time path with the increase in the rate of technological progress. How many years pass before output per worker surpasses the level that would be obtained from the increase in the saving rate? What does this suggest about the relative importance of level versus growth rate effects? Explain. (3 points) (4) In the Solow model, the golden rule saving rate is defined as the saving rate that maximize the steady state consumption per effective worker. Using the expression of the steady state consumption per effective worker to derive how the steady state consumption per effective worker depends on the saving rate. With all other parameters as in Table 1, plot how the steady state consumption per effective worker depends on the saving rate for s=0,001,0.02, 0.03 ...,1. In your plot, the horizontal axis should be the saving rate ranging from 0 to 1 and the vertical axis is the steady state consumption per effective worker. What is the golden rule saving rate? (3 points)