Problem 1. Solow model: balanced growth path and convergence Consider the standard Solow model, where the production function is the Cobb-Douglas function Y = K (AL)*, where Yis GDP, K is capital. L is labour and A is labour effectiveness (also called labour augmenting technological progress). As usual, the population grows at the constant rate of n and labour effectiveness at the rate of g, while the capital depreciation rate is 8. Constant fraction s of output Y is saved to be invested into capital. a) Write the production function in its intensive form, i.e. in terms of y = Y/AL and k = K/ AL. b) For the general case of any production function the differential equation for k looked as follows: k = sf(k)-(n+8+). Derive for this case with a specific Cobb-Douglas production function, the capital accumulation equation of the Solow model expressing k as a function of k. c) Find the expressions for k* y* and, c* as functions of savings rates, population growth rate n, technological progress growth rate g, rate of depreciation 8. d) Let us assume that the share of income paid to capital (or the elasticity of output w.I.t. capital) is equal to 1/4. For the structural parameters of the economy let us assume that savings rate is s = 25%, the rate of population growth is n = 2%, rate of depreciation 8 = 5%, rate of knowledge growth g = 3%. What are the resulting equilibrium values of k* y* ,c*? Suppose now that the savings rate will decrease from s = 25% to s = 20%. The next questions ask how economy responds to that change in the short run and in the long run. e) What is going to be the new long-run balanced growth path values of the capital and output per unit of effective labour at the new lower savings rate? f) Show on the graphs approximately the (1) dynamics, (11) rate of change and (111) growth rate of the k, y and c after the decrease of s. This means you should sketch 9 graphs with time on horizontal axis and k, k, etc (similar variables for y and c) on vertical axis. In particular will the growth rate of the variables decrease or increase over the time as the economy approaches the new balanced growth path? NB! The paphs should be based on the respective equations derived in part c (additionally, you should derive formulas for the rate of change and growth rate of k, y and c). g) In that particular economy characterized by the given functions and parameter values, which value of the savings rate would maximize the balanced growth path consumption c*? Hint. Note that because log is a monotonic transformation, maximizing the log of some variable is equivalent to maximizing that variable itself, i.e. formally naxc* max hc* That is mentioned because taking the derivative from the log function is often easier than taking derivative from the original function Problem 2. Growth accounting exercise: empirical study for your country of origin This exercise allows you to do something beyond pure theory with your own country data. Please look in the Intemet at least one study about the growth accounting exercise about your country. If there is really a problem to find one (which is actually hard to believe), please choose something for one of the neighbouring countries. Please provide correct bibliographical references to the study, e.g. the reference to the report. Describe the main results of that study in approximately 200 words. In particular your descriptions should cover the following points: i What have been the relative contributions of different production factors - capital, labour and TFP - to the growth; ii. Present and describe the used growth accounting formula - in what respect it was similar to the one discussed in the classroom/text-books/slides and what were the differences iii. Discuss what could be the potential weaknesses of the approach used, e.g. any problems related to the measurement of capital and labour. Hint for searching. As the possible sources for the literature, you can search via Google Google Scholar, Sciencedirect (www.sciencedirect.com), EBSCO (log in from UT library homepage), JSTOR (login from UT library homepage). The paid databases can be accessed from the UT library homepage (https://utlib.ut.ee/en): you can use these also outside the university premises (eg. dormitory) by logging in using your Study Information System usemame and password. Problem 3. Comparing the Growth of two countries (selected by yourself) The pupose of the current exercise is to show how one can carry through the analysis of practical growth performance issues using Excel and the discrete time version of the Solow growth model in. First, you need to find GDP data for 2 countries you are expecting to converge (eng. your home country and some richer country, or eg. comparing some poorer European country with EU average GDP is also OK). Preferably, use time series of GDP per worker measured in purchasing power parity for at least 20 years. Longer time series including most recent years is the best choice. You can use e.g. Eurostat (https://ec.europa.eu/eurostat/data/database ), World Bank data (http://databank.worldbank.org/data/home.aspx ) or some other reliable source. Let's denote the values for the first year in your time series e.g. Yo and the last year as yr. Also let's use notation (poor) for poorer country and (rich) for richer country. a) To eliminate the effects of technological progress from the data we need to transform these initial data series by dividing these by the richer country's GDP per worker (year-by year) and multiplying the results by the 100. It means that we will normalize for each year the GPD per worker in richer country to 100 and (ii) compute for poorer country the GDP per worker as the percentage of that of the richer country. Compute these transformed data series and put them on the figure. Give short qualitative description of the poorer country's growth rate. Hint: The resulting series for richer country is just the list of 100-s. For poorer country, in case of observing some European country's convergence with EU average, you can find required series "ready" in Eurostat We next investigate, how well the Solow model could explain the catching up process of your selected countries. For that purpose we will next consider the Solow model in the discrete time. That is very similar to the Solow model in continuous time; all the notations are as before, but time is now in the lower index, e.g. Y, is the value of GDP at time t and Y4 is the value of GDP one period earlier Y = K (14) S = 51, K. = 5,-K. L. = (1+n) 4.,-1+g4, It can be shown that equation describing the evolution of the capital per effective labour is as follows: 1 k. (sk% +(1-5).). (1+nX1+g) If there is no technological progress, it simplifies to k (sk* +(1-5%). Instead assume, (1+n) that the technological level is characterized by the constant parameter B in the production function, 1 3 so as 1 k-pe. S S = +8 n+S that Y, =BKL), and then the dynamics for the kul becomes ku SBk +(1-5),) and the steady-state capital and GDP become respectively (1+n) *= and y* = B(k*)* = BIS Next we will look whether for both countries one could apply the Solow model with country- specific parameters. Let's make following assumptions: Richer country has been for the whole studied period on balanced growth path or steady - In case of both countries it holds that a= 1/4 and 8 = 0.04. Two countries have similar technological levels, i.e. the same value of B. For the savings rate and population growth rate in both countries, please find the real data- (preferably from the same source where you took GDP data). Hint: calculate the average values of s and n over the observed period. state. b) What should be the value of B for richer country in the steady state, if y* (rich) = 100? c) What are the values of k* and y* in steady state in poor country for the same value of B? What must have been in that case k(poor) in the beginning of the observed period (t=0)? Describe, how far away would the poor country be from its' steady state in t=0 according to these calculations, i.e. what were the percentages of ko and yo, respectively, to their steady state levels? d) Starting from the calculated values of the series ko, and by using the value of B calculated above, simulate the model, i.e. the values of k,, for poor country for the period of EO ... ET, and thereby create the model-based time-series. Plot the actual series and the model based series on the figure that also shows the steady state values of k*(rich) and y* (rich) as horizontal lines. How fast has poor country actually caught up to the rich country compared to how fast it should have caught up according to the Solow model under the assumption of the common technological level B? Hint - All the calculations can be easily done in Excel! Problem 1. Solow model: balanced growth path and convergence Consider the standard Solow model, where the production function is the Cobb-Douglas function Y = K (AL)*, where Yis GDP, K is capital. L is labour and A is labour effectiveness (also called labour augmenting technological progress). As usual, the population grows at the constant rate of n and labour effectiveness at the rate of g, while the capital depreciation rate is 8. Constant fraction s of output Y is saved to be invested into capital. a) Write the production function in its intensive form, i.e. in terms of y = Y/AL and k = K/ AL. b) For the general case of any production function the differential equation for k looked as follows: k = sf(k)-(n+8+). Derive for this case with a specific Cobb-Douglas production function, the capital accumulation equation of the Solow model expressing k as a function of k. c) Find the expressions for k* y* and, c* as functions of savings rates, population growth rate n, technological progress growth rate g, rate of depreciation 8. d) Let us assume that the share of income paid to capital (or the elasticity of output w.I.t. capital) is equal to 1/4. For the structural parameters of the economy let us assume that savings rate is s = 25%, the rate of population growth is n = 2%, rate of depreciation 8 = 5%, rate of knowledge growth g = 3%. What are the resulting equilibrium values of k* y* ,c*? Suppose now that the savings rate will decrease from s = 25% to s = 20%. The next questions ask how economy responds to that change in the short run and in the long run. e) What is going to be the new long-run balanced growth path values of the capital and output per unit of effective labour at the new lower savings rate? f) Show on the graphs approximately the (1) dynamics, (11) rate of change and (111) growth rate of the k, y and c after the decrease of s. This means you should sketch 9 graphs with time on horizontal axis and k, k, etc (similar variables for y and c) on vertical axis. In particular will the growth rate of the variables decrease or increase over the time as the economy approaches the new balanced growth path? NB! The paphs should be based on the respective equations derived in part c (additionally, you should derive formulas for the rate of change and growth rate of k, y and c). g) In that particular economy characterized by the given functions and parameter values, which value of the savings rate would maximize the balanced growth path consumption c*? Hint. Note that because log is a monotonic transformation, maximizing the log of some variable is equivalent to maximizing that variable itself, i.e. formally naxc* max hc* That is mentioned because taking the derivative from the log function is often easier than taking derivative from the original function Problem 2. Growth accounting exercise: empirical study for your country of origin This exercise allows you to do something beyond pure theory with your own country data. Please look in the Intemet at least one study about the growth accounting exercise about your country. If there is really a problem to find one (which is actually hard to believe), please choose something for one of the neighbouring countries. Please provide correct bibliographical references to the study, e.g. the reference to the report. Describe the main results of that study in approximately 200 words. In particular your descriptions should cover the following points: i What have been the relative contributions of different production factors - capital, labour and TFP - to the growth; ii. Present and describe the used growth accounting formula - in what respect it was similar to the one discussed in the classroom/text-books/slides and what were the differences iii. Discuss what could be the potential weaknesses of the approach used, e.g. any problems related to the measurement of capital and labour. Hint for searching. As the possible sources for the literature, you can search via Google Google Scholar, Sciencedirect (www.sciencedirect.com), EBSCO (log in from UT library homepage), JSTOR (login from UT library homepage). The paid databases can be accessed from the UT library homepage (https://utlib.ut.ee/en): you can use these also outside the university premises (eg. dormitory) by logging in using your Study Information System usemame and password. Problem 3. Comparing the Growth of two countries (selected by yourself) The pupose of the current exercise is to show how one can carry through the analysis of practical growth performance issues using Excel and the discrete time version of the Solow growth model in. First, you need to find GDP data for 2 countries you are expecting to converge (eng. your home country and some richer country, or eg. comparing some poorer European country with EU average GDP is also OK). Preferably, use time series of GDP per worker measured in purchasing power parity for at least 20 years. Longer time series including most recent years is the best choice. You can use e.g. Eurostat (https://ec.europa.eu/eurostat/data/database ), World Bank data (http://databank.worldbank.org/data/home.aspx ) or some other reliable source. Let's denote the values for the first year in your time series e.g. Yo and the last year as yr. Also let's use notation (poor) for poorer country and (rich) for richer country. a) To eliminate the effects of technological progress from the data we need to transform these initial data series by dividing these by the richer country's GDP per worker (year-by year) and multiplying the results by the 100. It means that we will normalize for each year the GPD per worker in richer country to 100 and (ii) compute for poorer country the GDP per worker as the percentage of that of the richer country. Compute these transformed data series and put them on the figure. Give short qualitative description of the poorer country's growth rate. Hint: The resulting series for richer country is just the list of 100-s. For poorer country, in case of observing some European country's convergence with EU average, you can find required series "ready" in Eurostat We next investigate, how well the Solow model could explain the catching up process of your selected countries. For that purpose we will next consider the Solow model in the discrete time. That is very similar to the Solow model in continuous time; all the notations are as before, but time is now in the lower index, e.g. Y, is the value of GDP at time t and Y4 is the value of GDP one period earlier Y = K (14) S = 51, K. = 5,-K. L. = (1+n) 4.,-1+g4, It can be shown that equation describing the evolution of the capital per effective labour is as follows: 1 k. (sk% +(1-5).). (1+nX1+g) If there is no technological progress, it simplifies to k (sk* +(1-5%). Instead assume, (1+n) that the technological level is characterized by the constant parameter B in the production function, 1 3 so as 1 k-pe. S S = +8 n+S that Y, =BKL), and then the dynamics for the kul becomes ku SBk +(1-5),) and the steady-state capital and GDP become respectively (1+n) *= and y* = B(k*)* = BIS Next we will look whether for both countries one could apply the Solow model with country- specific parameters. Let's make following assumptions: Richer country has been for the whole studied period on balanced growth path or steady - In case of both countries it holds that a= 1/4 and 8 = 0.04. Two countries have similar technological levels, i.e. the same value of B. For the savings rate and population growth rate in both countries, please find the real data- (preferably from the same source where you took GDP data). Hint: calculate the average values of s and n over the observed period. state. b) What should be the value of B for richer country in the steady state, if y* (rich) = 100? c) What are the values of k* and y* in steady state in poor country for the same value of B? What must have been in that case k(poor) in the beginning of the observed period (t=0)? Describe, how far away would the poor country be from its' steady state in t=0 according to these calculations, i.e. what were the percentages of ko and yo, respectively, to their steady state levels? d) Starting from the calculated values of the series ko, and by using the value of B calculated above, simulate the model, i.e. the values of k,, for poor country for the period of EO ... ET, and thereby create the model-based time-series. Plot the actual series and the model based series on the figure that also shows the steady state values of k*(rich) and y* (rich) as horizontal lines. How fast has poor country actually caught up to the rich country compared to how fast it should have caught up according to the Solow model under the assumption of the common technological level B? Hint - All the calculations can be easily done in Excel