I am not sure how to use the substitution and lagrange methods. Thanks for help

2. Consumption-Savings Decision: The Household's decision problem is: 17 17 max { c1 + c2 } 61,62; 1 '7 1 '7 st. c1 +3 = (5)3;1 C2 = (S)(1 + Tls + :92 1The original source of the data are the Penn World Tables v9.0, FRED is a database that we are using to access them. 2You will need to download GDP, population, 'nurnber of engaged persons', 1average annual hours per engaged person' and then calculate, e.g., GDP per worker as GDP divided by number of engaged persons. 3You can actually get FRED to do this for you before you download the data. Or alternatively you can just download the data in levels and calculate the growth rates yourself using a spreadsheet or any other software you like. where cl and C2 are consumption in periods 1 and 2 respectively; yl and y2 are income in periods 1 and 2 respectively; .9 is savings; 5" is the interest rate on savings. 7 is a parameter controlling the concavity of the utility function, and will determine intertemporal substitution of consumption.4 We assume that 7 > 1; so utility is increasing and concave in consumption. (a (b (c (d Use the method of substitution to solve for optimal savings 3*. Use the method of the Lagrangian to solve for optimal savings 3*. ) ) ) Find the solutions for optimal consumption in periods 1 and 2, ie. of and c3. ) For 7 = 1 look at how consumption in period 1 depends on the interest rate 'r' (you can just copy the expression for a; from the lecture slides). Describe how monetary stimulus (a decrease in the interest rate) works to increase demand (consumption) today. (Whether you do this by taking derivatives of the optimal choices with respect to the interest rate, or just by describing it's effect, is up to you.)