1- 1- + 1-O 1-O 1. (Parts a to f 10 points, g 20 points) Consider an endowment economy with two goods (A and B) and with the periods. Households earn Y. and Y:+1 as endowment income. They can save, S, with an interest rate of (1 + r) and consume goods A and B. Consumption levels for goods A and B at time t are given by CA and C#, respectively. The lifetime utility of the household is given by: (c)! + BC 1-O 1-0 (a) Write the flow budget constraints of the household. Give economic intuition. (b) Using the flow budget constraints, derive the household's intertemporal budget constraint. What is the economic meaning of this constraint? (e) Write down the household's problem and derive optimality conditions. Interpret each condition economically Derive the relationship between current consumption levels A and B and their relationship with their respective future consumption levels. (d) Derive the intertemporal elasticity of substitution (IES) for goods A and B. Give economic intuition for the IES. Now define, aggregate consumption level in the economy, Cs, as the sum of consumption levels for goods A and B at time t (similarly for t + 1). Derive the IES for aggregate consumption level. (e) Derive the consumption function for goods A and B. (1) Now assume that o = 1 and show the effect of an increase in r, on current consumption. Give proper economic intuition. (8) Define competitive equilibrium for this economy. Solve for the equilibrium real interest rates. How would equilib- rium interest rate would change if y, changes? (h) This is a numerical exercise that can be done in Excel. Here we will visualize the effect of different levels of IES on consumption growth. But, as the Euler equation suggests, we have know the values of real interest rates. Here are the steps for this exercise: 1. Go to FRED database from St. Louis Fed (https://fred.stlouisfed.org/). ii. Search for "Effective Fed Funds Rate" data. This is the nominal interest rate set by the Federal Reserve Board (central bank of United States). By clicking on the link, you will be able to visualize the data. Then click on "Max" tab given above the graph to have the maximum sample of observations. Finally click on "Edit Graph" tab and "Modify frequency monthly and "Aggregation Method" to average. iii. Next, by clicking "Add Line", search for "Consumer Price Index for All Urban Consumers: All Items in U.S. City Average." Then click "Add Data Series." Note that this gives you the CPI index. Using this index, we will calculate inflation. To do so, we change "Units" to "Percentage Change from Year Ago." The reason we do this is to convert inflation to annualized numbers like the interest rates. iv. Close the Edit Line" tab and "Download" data to Excel. To calculate real interest rates, subtract CPI inflation from the Effective Fed Funds Rate. This gives you ex-post real interest rate. v. Now set 8 = 0.99. Using the natural log version of the Euler equation (the equation you used for calculating IES) and data for real interest rates, calculate aggregate consumption growth for each value of real interest rate by setting a = 2, 3, 4, respectively. Plot implied consumption growth levels on the same graph. What is the effect of higher a on aggregate consumption growth? Does the effect depend on whether real interest rates are negative or positive? Give proper economic intuition. In your answer, make sure that you submit the Excel graph. 1- 1- + 1-O 1-O 1. (Parts a to f 10 points, g 20 points) Consider an endowment economy with two goods (A and B) and with the periods. Households earn Y. and Y:+1 as endowment income. They can save, S, with an interest rate of (1 + r) and consume goods A and B. Consumption levels for goods A and B at time t are given by CA and C#, respectively. The lifetime utility of the household is given by: (c)! + BC 1-O 1-0 (a) Write the flow budget constraints of the household. Give economic intuition. (b) Using the flow budget constraints, derive the household's intertemporal budget constraint. What is the economic meaning of this constraint? (e) Write down the household's problem and derive optimality conditions. Interpret each condition economically Derive the relationship between current consumption levels A and B and their relationship with their respective future consumption levels. (d) Derive the intertemporal elasticity of substitution (IES) for goods A and B. Give economic intuition for the IES. Now define, aggregate consumption level in the economy, Cs, as the sum of consumption levels for goods A and B at time t (similarly for t + 1). Derive the IES for aggregate consumption level. (e) Derive the consumption function for goods A and B. (1) Now assume that o = 1 and show the effect of an increase in r, on current consumption. Give proper economic intuition. (8) Define competitive equilibrium for this economy. Solve for the equilibrium real interest rates. How would equilib- rium interest rate would change if y, changes? (h) This is a numerical exercise that can be done in Excel. Here we will visualize the effect of different levels of IES on consumption growth. But, as the Euler equation suggests, we have know the values of real interest rates. Here are the steps for this exercise: 1. Go to FRED database from St. Louis Fed (https://fred.stlouisfed.org/). ii. Search for "Effective Fed Funds Rate" data. This is the nominal interest rate set by the Federal Reserve Board (central bank of United States). By clicking on the link, you will be able to visualize the data. Then click on "Max" tab given above the graph to have the maximum sample of observations. Finally click on "Edit Graph" tab and "Modify frequency monthly and "Aggregation Method" to average. iii. Next, by clicking "Add Line", search for "Consumer Price Index for All Urban Consumers: All Items in U.S. City Average." Then click "Add Data Series." Note that this gives you the CPI index. Using this index, we will calculate inflation. To do so, we change "Units" to "Percentage Change from Year Ago." The reason we do this is to convert inflation to annualized numbers like the interest rates. iv. Close the Edit Line" tab and "Download" data to Excel. To calculate real interest rates, subtract CPI inflation from the Effective Fed Funds Rate. This gives you ex-post real interest rate. v. Now set 8 = 0.99. Using the natural log version of the Euler equation (the equation you used for calculating IES) and data for real interest rates, calculate aggregate consumption growth for each value of real interest rate by setting a = 2, 3, 4, respectively. Plot implied consumption growth levels on the same graph. What is the effect of higher a on aggregate consumption growth? Does the effect depend on whether real interest rates are negative or positive? Give proper economic intuition. In your answer, make sure that you submit the Excel graph