3. Suppose we have the Neoclassical Growth model that we have developed in class. Suppose At grows at the rate g. Assume a general period utility function, u(a). a. Write down the Social Planners problem in terms of per capita consumption and capital stock. Set up the Lagrangian, solve, and give the equations that for every time period are necessary for a solution (Euler equation and resource constraint). b. Using the resource constraint, show why along a BGP per capita consumption and capital stock are growing at the rate of technological progress, g. c. Using the Euler Equation explain what restriction we must make on our period utility function in order for a BGP to exist. Write the Euler equation with this restriction, i.e. so it is independent of time. d. Solve for the initial capital stock per capita, k, and output per capita, y. Write down the expressions for the capital stock per capita at time t, kt, and output per capita at time t, y e. Suppose capital fully depreciates each period. What is the savings rate, , along a BGP? Discuss how the savings rate changes with a and B, explain the intuition. How does the savings rate change with g? Explain why the effect of g on s differs as o changes. 3. Suppose we have the Neoclassical Growth model that we have developed in class. Suppose At grows at the rate g. Assume a general period utility function, u(a). a. Write down the Social Planners problem in terms of per capita consumption and capital stock. Set up the Lagrangian, solve, and give the equations that for every time period are necessary for a solution (Euler equation and resource constraint). b. Using the resource constraint, show why along a BGP per capita consumption and capital stock are growing at the rate of technological progress, g. c. Using the Euler Equation explain what restriction we must make on our period utility function in order for a BGP to exist. Write the Euler equation with this restriction, i.e. so it is independent of time. d. Solve for the initial capital stock per capita, k, and output per capita, y. Write down the expressions for the capital stock per capita at time t, kt, and output per capita at time t, y e. Suppose capital fully depreciates each period. What is the savings rate, , along a BGP? Discuss how the savings rate changes with a and B, explain the intuition. How does the savings rate change with g? Explain why the effect of g on s differs as o changes