4 Comparative statics in the Solow growth model In this exercise, we want to assess the contribution of capital accumulation to our understanding of differences in output per capita across countries. We will distinguish between the results obtained from the static model of production and from the Solow growth model. Question 4.1 Start with the Cobb Douglas production function Y=AKL1. We want to compute what is the change in output after a parameter (like the TFP A) changes. We will proceed exactly in the same way as when we computed annual growth rates. Denote Yold the amount of capital before the change, and Ynew the amount of capital after the change in a parameter. Then we can define YoldYnew=1+gY where gY is the growth rate of output from before until after the change. Substitute in for Ynew/Yold from equation (4.1) and do the same for variables A,K,L. By using the logarithmic approximation, derive the same formula for gY as the one that we were using to compute annual growth rates (gY will be a function of gA,gK,gL and the parameter ). Question 4.2 We will now do the same with the steady state relationship in the Solow growth model. Take the formula for the steady-state capital-output ratio K/Y (it is a function of s and ), and substitute in for Y from the Cobb-Douglas production function. Express K as a function of A,s,,L and only. Once you are done, take again the production function and substitute in for K using the formula that you just obtained to get an expression for Y as a function of A,s,,L and 3 only. Hint: What you should ultimately get is Y=A1a1(s)111L but make sure you provide all the intermediate steps in your answer. Question 4.3 Take the expression for output as a function of parameters. We now want to express the cumulative growth rate of output between the old and the new steady state when one of the parameters ,s,A,L, changes (treat as constant). In order to do so, define again the cumulative growth rate of output from the old to the new steady state as YoldYtiew=1+gY and correspondingly the growth rates of ,s,A,L; for instance soldsnew=1+gs. Remember that gY is not an annual growth rate of output - it is the total percentage change in output between the old and the new steady state. Similarly, g5 is the percentage change in the saving rate parameter. Now find the growth rate of output gY as a function of gA,g5,g and gL. Hint: The coefficient on gS that you should obtain is 1aA. Important: It is important to understand what growth rate are we computing here. This is not an annual growth rate, it is the cumulative growth rate that expresses the percentage change in the steady state level of output after one of the variables of interest (, s,A,L ) changes by a given percentage. It may therefore be more appropriate to call gY a 'percentage change' rather than the growth rate but given the similarities with our previous computations, we will stick to the term growth rate, with the implicit understanding of what is going on. In what follows assume that the capital share is equal to 31, as we have usually assumed. Question 4.4 If the TFP parameter increases by 10%, how does total output change in the model of production and in the Solow growth model? Why is there a difference? Question 4.5 If population increases by 3%, how does the total output in the economy change in the model of production and in the Solow growth model? Why (or why not) is there a difference? Question 4.6 So far, we computed changes in total output. We will now focus on the growth rates of output per capita. Compute the growth rate of output per capita, LY, in the model of production and in the Solow growth model. Hint: Notice that you do not need to derive everything from scratch, just slightly reorganize the results for gY. Question 4.7 What happens to the level of output per capita when population increases by 10% in the model of production? What happens to the steady state level of output per capita when population increases by 10% in the Solow growth model? Provide an economic explanation for the results. Why do you observe an negative sign in your answer for the model of production? And why not in the Solow growth model? 4 Comparative statics in the Solow growth model In this exercise, we want to assess the contribution of capital accumulation to our understanding of differences in output per capita across countries. We will distinguish between the results obtained from the static model of production and from the Solow growth model. Question 4.1 Start with the Cobb Douglas production function Y=AKL1. We want to compute what is the change in output after a parameter (like the TFP A) changes. We will proceed exactly in the same way as when we computed annual growth rates. Denote Yold the amount of capital before the change, and Ynew the amount of capital after the change in a parameter. Then we can define YoldYnew=1+gY where gY is the growth rate of output from before until after the change. Substitute in for Ynew/Yold from equation (4.1) and do the same for variables A,K,L. By using the logarithmic approximation, derive the same formula for gY as the one that we were using to compute annual growth rates (gY will be a function of gA,gK,gL and the parameter ). Question 4.2 We will now do the same with the steady state relationship in the Solow growth model. Take the formula for the steady-state capital-output ratio K/Y (it is a function of s and ), and substitute in for Y from the Cobb-Douglas production function. Express K as a function of A,s,,L and only. Once you are done, take again the production function and substitute in for K using the formula that you just obtained to get an expression for Y as a function of A,s,,L and 3 only. Hint: What you should ultimately get is Y=A1a1(s)111L but make sure you provide all the intermediate steps in your answer. Question 4.3 Take the expression for output as a function of parameters. We now want to express the cumulative growth rate of output between the old and the new steady state when one of the parameters ,s,A,L, changes (treat as constant). In order to do so, define again the cumulative growth rate of output from the old to the new steady state as YoldYtiew=1+gY and correspondingly the growth rates of ,s,A,L; for instance soldsnew=1+gs. Remember that gY is not an annual growth rate of output - it is the total percentage change in output between the old and the new steady state. Similarly, g5 is the percentage change in the saving rate parameter. Now find the growth rate of output gY as a function of gA,g5,g and gL. Hint: The coefficient on gS that you should obtain is 1aA. Important: It is important to understand what growth rate are we computing here. This is not an annual growth rate, it is the cumulative growth rate that expresses the percentage change in the steady state level of output after one of the variables of interest (, s,A,L ) changes by a given percentage. It may therefore be more appropriate to call gY a 'percentage change' rather than the growth rate but given the similarities with our previous computations, we will stick to the term growth rate, with the implicit understanding of what is going on. In what follows assume that the capital share is equal to 31, as we have usually assumed. Question 4.4 If the TFP parameter increases by 10%, how does total output change in the model of production and in the Solow growth model? Why is there a difference? Question 4.5 If population increases by 3%, how does the total output in the economy change in the model of production and in the Solow growth model? Why (or why not) is there a difference? Question 4.6 So far, we computed changes in total output. We will now focus on the growth rates of output per capita. Compute the growth rate of output per capita, LY, in the model of production and in the Solow growth model. Hint: Notice that you do not need to derive everything from scratch, just slightly reorganize the results for gY. Question 4.7 What happens to the level of output per capita when population increases by 10% in the model of production? What happens to the steady state level of output per capita when population increases by 10% in the Solow growth model? Provide an economic explanation for the results. Why do you observe an negative sign in your answer for the model of production? And why not in the Solow growth model