I have uploaded the question and my answer can you check if it's correct? Step 1: Describe the ProblemWe 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 ConstraintThe budget constraints for the two periods are:1. Y₁ = Ct + S2. Yt+1 = C++1 + S(1 + rt)Combining these gives us the intertemporal budget constraint:YE+1C₁ + Cell = Y₁ + YStep 3: Write Down Optimization ProblemThe optimization problem is to maximize utility subject to the intertemporal budget constraint:Maximize:U = u(C₁) + Bu(C++1)Subject to:C₁+1C₁ + C++ = Y₁ + Y1+re1+reStep 4: Derive First-Order Conditions (Euler Equation)The Lagrangian for this problem is:C1+1L = u(Ct) + ßu(C++1) − A (C₁ + C++ - Y₁ –YtY₁+1(The first-order conditions are derived by taking the partial derivatives of the Lagrangian:1. For C+:= u' (Ct) - =02. For C++1асC = Bu' (C++1)=0DC++1-3. For (the budget constraint):ас== C++C₁+1 - Yt-Y₁+1 = 01+rTo linearize the Euler equation, we take the logarithm of the first-order conditions derived from themarginal 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++1The three first-order conditions, including the Lagrange multiplier and considering thelogarithmic transformation to linearize the expressions, will allow us to solve for the optimalconsumption values C and C++1 as well a✓ Lagrange multiplier A. This revised methodaddresses the hint from the exercise and correctly includes all first-order conditions. Consider a household with the following utility function representing their preferencesover consumption:U = u(Ct) + Bu(C++1)with=-u(C) = exp(-aC), BE (0,1), a > 0where 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, andYt+1 in the current and future periods, respectively. The real interest rate is denoted byrt.Notice: The utility function u(C) takes on negative values for all positive consumptionlevels. However, in economic models, the absolute value of utility is less important thanhow utility changes with consumption. A higher level of utility represents a morepreferred 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 tomaximize.Derive the first-order conditions (Euler Equation) that characterize the optimal con-sumption plan. Provide an economic interpretation of these conditions (hint: you needto take logs at some point to make the expressions linear

Given,The utility function Where,However,

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