ECON 5118 Assignment 3 The Optimal Growth Model with Labor and Technology Growth Consider an optimal growth model with a production function Yt = Ktα(AtLt)1−α , where Kt is the aggregate capital and Lt is the labor. Capital depreciates at rate 1 and the representative consumer’s preference is given by u(ct) = log ct , where ct is the consumption per capita. The model also includes National income identity: Yt = Ct + It , where It is the aggregate investment. Capital accumulation: Kt+1 = It + (1 − δ)Yt , where the depreciation depends on total production now. Labor dynamics: Lt − Lt−1 Lt−1 = n, where n is the growth rate of population. Technology dynamics: At − At−1 At−1 = na, where na is the technology growth rate. 1 (1) Please find the expression for the per effective unit of labor form of the production function. (6 points) (2) Please derive the budget constraint for the representative consumer at time t. (6 points) (3) Please write down the maximization problem for the representative consumer. (6 points) (4) Please write down the Lagrangian and derive the first-order conditions. (6 points) (5) Please derive the Euler’s equation and discuss its implication. (6 points) (6) Please find the modified Golden rule. (6 points) (7) Please find the closed form solutions for ˜ct and ˜kt+1. (You may define the relevant parameters by yourself.) (6 points) (8) Please find the endogenous saving rate based on (7). (6 points) (9) Suppose the consumer discounts the future utility at rate 0.74, α is 0.3, and δ = 0.4. The growth rate of technology is 0.02 and the growth rate of labor is 0.02 as well. The economy starts with ˜k0 = 0.001×the last digit of your student ID, A0 = 10×the first digit of your student ID, and L0 = 4. Please show the response function of aggregate output when the consumer gets impatient. You may discuss what impatience means in terms of the changes in the value of the relevant parameter. Please submit your excel file along with your answers. (20 points) (10) Does being impatient affect the endogenous saving rate? Please show it using math or the excel and talk about its economics intuition. (6 points) (11) Does being efficient (a lower δ) affect the endogenous saving rate? Please show it using math or the excel and talk about its economics intuition. (6 points) The Growth Accounting (19 points) Please choose one country and use data to analyze the contribution of different factors (capital, labor, human capital, and one more factor that is relevant in your opinion) to economic growth. Please discuss your assumptions and results. Please submit your excel file along with your answers. [Hint: the recommended the database is the PWT 10.01] 2
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ECON 5118 Assignment 3
The Optimal Growth Model with Labor and Technology Growth Consider an optimal growth model with a production function Yt = Ktα(AtLt)1−α , where Kt is the aggregate capital and Lt is the labor. Capital depreciates at rate 1 and the representative consumer’s preference is given by u(ct) = log ct , where ct is the consumption per capita. The model also includes National income identity: Yt = Ct + It , where It is the aggregate investment. Capital accumulation: Kt+1 = It + (1 − δ)Yt , where the depreciation depends on total production now. Labor dynamics: Lt − Lt−1 Lt−1 = n, where n is the growth rate of population. Technology dynamics: At − At−1 At−1 = na, where na is the technology growth rate. 1 (1) Please find the expression for the per effective unit of labor form of the production function. (6 points) (2) Please derive the budget constraint for the representative consumer at time t. (6 points) (3) Please write down the maximization problem for the representative consumer. (6 points) (4) Please write down the Lagrangian and derive the first-order conditions. (6 points) (5) Please derive the Euler’s equation and discuss its implication. (6 points) (6) Please find the modified Golden rule. (6 points) (7) Please find the closed form solutions for ˜ct and ˜kt+1. (You may define the relevant parameters by yourself.) (6 points) (8) Please find the endogenous saving rate based on (7). (6 points) (9) Suppose the consumer discounts the future utility at rate 0.74, α is 0.3, and δ = 0.4. The growth rate of technology is 0.02 and the growth rate of labor is 0.02 as well. The economy starts with ˜k0 = 0.001×the last digit of your student ID, A0 = 10×the first digit of your student ID, and L0 = 4. Please show the response function of aggregate output when the consumer gets impatient. You may discuss what impatience means in terms of the changes in the value of the relevant parameter. Please submit your excel file along with your answers. (20 points) (10) Does being impatient affect the endogenous saving rate? Please show it using math or the excel and talk about its economics intuition. (6 points) (11) Does being efficient (a lower δ) affect the endogenous saving rate? Please show it using math or the excel and talk about its economics intuition. (6 points) The Growth Accounting (19 points) Please choose one country and use data to analyze the contribution of different factors (capital, labor, human capital, and one more factor that is relevant in your opinion) to economic growth. Please discuss your assumptions and results. Please submit your excel file along with your answers. [Hint: the recommended the database is the PWT 10.01] 2
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