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Consider a household with the following utility function representing their preferences over consumption: U = u(C) + Bu(C++1) with u(C) = exp(-aC), BE (0,1), a > 0 == where C and C++1 represent consumption in the current and future periods, respectively. The household faces a two-period decision problem. They receive endowments of Y, and Yt+1 in the current and future periods, respectively. The real interest rate is denoted by rt. Notice: The utility function u(C) takes on negative values for all positive consumption levels. However, in economic models, the absolute value of utility is less important than how utility changes with consumption. A higher level of utility represents a more preferred outcome for the household. Question: Formulate the household's budget constraints for the current and future periods. Com- bine them to derive the household's intertemporal budget constraint. Write down the household's optimization problem (objective function) that they seek to maximize. Derive the first-order conditions (Euler Equation) that characterize the optimal con- sumption plan. Provide an economic interpretation of these conditions (hint: you need to take logs at some point to make the expressions linear

Consider a household with the following utility function representing their preferences over consumption: U = u(C) + Bu(C++1) with u(C) = exp(-aC), BE (0,1), a > 0 == where C and C++1 represent consumption in the current and future periods, respectively. The household faces a two-period decision problem. They receive endowments of Y, and Yt+1 in the current and future periods, respectively. The real interest rate is denoted by rt. Notice: The utility function u(C) takes on negative values for all positive consumption levels. However, in economic models, the absolute value of utility is less important than how utility changes with consumption. A higher level of utility represents a more preferred outcome for the household. Question: Formulate the household's budget constraints for the current and future periods. Com- bine them to derive the household's intertemporal budget constraint. Write down the household's optimization problem (objective function) that they seek to maximize. Derive the first-order conditions (Euler Equation) that characterize the optimal con- sumption plan. Provide an economic interpretation of these conditions (hint: you need to take logs at some point to make the expressions linear

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Step 1: Describe the Problem We are given a two-period model with the utility function: U = u(Ct) + ßu(Ct+1) where u(C) = exp(−aC). Step 2: Derive Intertemporal Budget Constraint The budget constraints for the two periods are: 1. Y₁ = Ct + S 2. Yt+1 = C++1 + S(1 + rt) Combining these gives us the intertemporal budget constraint: YE+1 C₁ + Cell = Y₁ + Y Step 3: Write Down Optimization Problem The optimization problem is to maximize utility subject to the intertemporal budget constraint: Maximize: U = u(C₁) + Bu(C++1) Subject to: C₁+1 C₁ + C++ = Y₁ + Y 1+re 1+re Step 4: Derive First-Order Conditions (Euler Equation) The Lagrangian for this problem is: C1+1 L = u(Ct) + ßu(C++1) − A (C₁ + C++ - Y₁ – Yt Y₁+1 ( The first-order conditions are derived by taking the partial derivatives of the Lagrangian: 1. For C+: = u' (Ct) - =0 2. For C++1 ас C = Bu' (C++1)=0 DC++1 - 3. For (the budget constraint): ас = = C++ C₁+1 - Yt- Y₁+1 = 0 1+r To linearize the Euler equation, we take the logarithm of the first-order conditions derived from the marginal utilities: u' (Ct) = a exp(-aC₁) u' (C++1) = a exp(-aC++1) The Euler equation is: a exp(−aCt) = 8(1 + r;)a exp(−aC++1) Taking the natural logarithm of both sides gives us: -aC₁ = log(ß) + log(1 + †) − aC++1 The three first-order conditions, including the Lagrange multiplier and considering the logarithmic transformation to linearize the expressions, will allow us to solve for the optimal consumption values C and C++1 as well a✓ Lagrange multiplier A. This revised method addresses the hint from the exercise and correctly includes all first-order conditions.
Transcribed Image Text:Step 1: Describe the Problem We are given a two-period model with the utility function: U = u(Ct) + ßu(Ct+1) where u(C) = exp(−aC). Step 2: Derive Intertemporal Budget Constraint The budget constraints for the two periods are: 1. Y₁ = Ct + S 2. Yt+1 = C++1 + S(1 + rt) Combining these gives us the intertemporal budget constraint: YE+1 C₁ + Cell = Y₁ + Y Step 3: Write Down Optimization Problem The optimization problem is to maximize utility subject to the intertemporal budget constraint: Maximize: U = u(C₁) + Bu(C++1) Subject to: C₁+1 C₁ + C++ = Y₁ + Y 1+re 1+re Step 4: Derive First-Order Conditions (Euler Equation) The Lagrangian for this problem is: C1+1 L = u(Ct) + ßu(C++1) − A (C₁ + C++ - Y₁ – Yt Y₁+1 ( The first-order conditions are derived by taking the partial derivatives of the Lagrangian: 1. For C+: = u' (Ct) - =0 2. For C++1 ас C = Bu' (C++1)=0 DC++1 - 3. For (the budget constraint): ас = = C++ C₁+1 - Yt- Y₁+1 = 0 1+r To linearize the Euler equation, we take the logarithm of the first-order conditions derived from the marginal utilities: u' (Ct) = a exp(-aC₁) u' (C++1) = a exp(-aC++1) The Euler equation is: a exp(−aCt) = 8(1 + r;)a exp(−aC++1) Taking the natural logarithm of both sides gives us: -aC₁ = log(ß) + log(1 + †) − aC++1 The three first-order conditions, including the Lagrange multiplier and considering the logarithmic transformation to linearize the expressions, will allow us to solve for the optimal consumption values C and C++1 as well a✓ Lagrange multiplier A. This revised method addresses the hint from the exercise and correctly includes all first-order conditions.
Consider a household with the following utility function representing their preferences over consumption: U = u(Ct) + Bu(C++1) with =- u(C) = exp(-aC), BE (0,1), a > 0 where C and C++1 represent consumption in the current and future periods, respectively. The household faces a two-period decision problem. They receive endowments of Y, and Yt+1 in the current and future periods, respectively. The real interest rate is denoted by rt. Notice: The utility function u(C) takes on negative values for all positive consumption levels. However, in economic models, the absolute value of utility is less important than how utility changes with consumption. A higher level of utility represents a more preferred outcome for the household. Question: Formulate the household's budget constraints for the current and future periods. Com- bine them to derive the household's intertemporal budget constraint. Write down the household's optimization problem (objective function) that they seek to maximize. Derive the first-order conditions (Euler Equation) that characterize the optimal con- sumption plan. Provide an economic interpretation of these conditions (hint: you need to take logs at some point to make the expressions linear
Transcribed Image Text:Consider a household with the following utility function representing their preferences over consumption: U = u(Ct) + Bu(C++1) with =- u(C) = exp(-aC), BE (0,1), a > 0 where C and C++1 represent consumption in the current and future periods, respectively. The household faces a two-period decision problem. They receive endowments of Y, and Yt+1 in the current and future periods, respectively. The real interest rate is denoted by rt. Notice: The utility function u(C) takes on negative values for all positive consumption levels. However, in economic models, the absolute value of utility is less important than how utility changes with consumption. A higher level of utility represents a more preferred outcome for the household. Question: Formulate the household's budget constraints for the current and future periods. Com- bine them to derive the household's intertemporal budget constraint. Write down the household's optimization problem (objective function) that they seek to maximize. Derive the first-order conditions (Euler Equation) that characterize the optimal con- sumption plan. Provide an economic interpretation of these conditions (hint: you need to take logs at some point to make the expressions linear
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