Both problems 1 and 2 or at least 1 please

Economics Suppose a nation's economy is divided into n sectors that produce goods. Define the production vector x ER" to be the vector giving the output of each sector over the course of a year. Suppose there is an additional sector (the open sector) that does not produce goods, but only consumes them; we define the final demand vector de R" to be the list of the amount demanded from each of the productive sectors by the open sector (this represents consumer demand, government consumption, exports, etc.). Finally, in order to meet the consumer demand, the producers themselves require materials from the other sectors and possibly their own) in order to produce their goods, which we call intermediate demand. As you might imagine, the interactions between all of the sectors are quite complex. However, Harvard professor Wassily Leontief (a later Nobel Prize winner)posited a model in which the amount produced exactly matches the total demand (final plus intermediate). The basic assumption of this model is the existence of a unit consumption vector in IR" for each sector that gives the inputs needed for that particular sector to produce one unit of its goods. As a concrete example, suppose our economy has three (productive) sectors: manufac- turing, agriculture, and services. Each sector has an associated unit consumption vector. For example, if 1 unit of output of manufacturing requires 0.5 units from manufacturing, 0.2 units from agriculture, and 0.1 units from services, the unit consumption vector cm for manufacturing is 0.5 0.2 0.1 If the final demand on the manufacturing sector is a units, then the intermediate demand of manufacturing is com, since this represents the goods needed by manufacturing in order to produce enough goods to meet their demand. If the final demand on agriculture is 12 units and the final demand on services is 1 units, the total intermediate demand is C.. + x26, +33c, = Cx, where C is the consumption matrix whose columns are the unit consumption vectors of the sectors: C-1 Com Coc,] This yields the Leontief production equation (1) x=Cx+d which can be rewritten as (2) (1 - C)x = d. According to Theorem 2.5, this equation can be solved by finding (1 - C)-d (assuming 1- C is invertible). Problem 1. (8 points) Suppose we have a three-sector economy as described above, where for each unit of output manufacturing requires 0.2 units from itself, 0.3 units from agriculture, and 0.1 units from services; agriculture requires 0.2 units from manufacturing and 0.1 units from itself; and services requires 0.3 units from agriculture and 0.2 units from itselt. (a) Find the consumption matrix C for this economy. (b) Write the matrix I-C. and then se row reduction to compute (1-C)-(you may use a calculator, and round to two decimal places). (c) Find the required production vector if the final demand vector is d= 60 (round to one decimal place) 40 80 For an alternative view, imagine the sectors at first set their production levels to exactly meet demand, so that x =d. They then send out orders for their required inputs which creates an intermediate demand Cd. To meet this additional demand, the sectors will need the additional inputs C(Cd) = Cd. This in turn creates another intermediate demand, and so on and so forth. In a real-world scenario, this sequence will eventually terminate, but it leads to the equation x= d+ Cd +C%d + Cd + ... = (1 +C+ C+C +...d We can derive the following identity from a polynomial identity in calculus (4) (1-C)(1+C++C+...+ C) = 1 - C+1 One feature of our consumption matrix C is that com a m + , so that the right-hand side of (4) approaches I. This implies the following approximation for the inverse: (1-0)- 21+ C + + + + Problem 2. (5 points) Let C be the consumption matrix you found in Problem 1(a). Approximate (I C)- by the quantity I+C+C2 +C3 +C4 +C5 (use your calculator, rounding to two decimal places), and use this approximation to recalculate the production vector (rounding to one decimal place) needed to meet the final demand given in Problem 1(c). Compare your results to those in Problem 1. Economics Suppose a nation's economy is divided into n sectors that produce goods. Define the production vector x ER" to be the vector giving the output of each sector over the course of a year. Suppose there is an additional sector (the open sector) that does not produce goods, but only consumes them; we define the final demand vector de R" to be the list of the amount demanded from each of the productive sectors by the open sector (this represents consumer demand, government consumption, exports, etc.). Finally, in order to meet the consumer demand, the producers themselves require materials from the other sectors and possibly their own) in order to produce their goods, which we call intermediate demand. As you might imagine, the interactions between all of the sectors are quite complex. However, Harvard professor Wassily Leontief (a later Nobel Prize winner)posited a model in which the amount produced exactly matches the total demand (final plus intermediate). The basic assumption of this model is the existence of a unit consumption vector in IR" for each sector that gives the inputs needed for that particular sector to produce one unit of its goods. As a concrete example, suppose our economy has three (productive) sectors: manufac- turing, agriculture, and services. Each sector has an associated unit consumption vector. For example, if 1 unit of output of manufacturing requires 0.5 units from manufacturing, 0.2 units from agriculture, and 0.1 units from services, the unit consumption vector cm for manufacturing is 0.5 0.2 0.1 If the final demand on the manufacturing sector is a units, then the intermediate demand of manufacturing is com, since this represents the goods needed by manufacturing in order to produce enough goods to meet their demand. If the final demand on agriculture is 12 units and the final demand on services is 1 units, the total intermediate demand is C.. + x26, +33c, = Cx, where C is the consumption matrix whose columns are the unit consumption vectors of the sectors: C-1 Com Coc,] This yields the Leontief production equation (1) x=Cx+d which can be rewritten as (2) (1 - C)x = d. According to Theorem 2.5, this equation can be solved by finding (1 - C)-d (assuming 1- C is invertible). Problem 1. (8 points) Suppose we have a three-sector economy as described above, where for each unit of output manufacturing requires 0.2 units from itself, 0.3 units from agriculture, and 0.1 units from services; agriculture requires 0.2 units from manufacturing and 0.1 units from itself; and services requires 0.3 units from agriculture and 0.2 units from itselt. (a) Find the consumption matrix C for this economy. (b) Write the matrix I-C. and then se row reduction to compute (1-C)-(you may use a calculator, and round to two decimal places). (c) Find the required production vector if the final demand vector is d= 60 (round to one decimal place) 40 80 For an alternative view, imagine the sectors at first set their production levels to exactly meet demand, so that x =d. They then send out orders for their required inputs which creates an intermediate demand Cd. To meet this additional demand, the sectors will need the additional inputs C(Cd) = Cd. This in turn creates another intermediate demand, and so on and so forth. In a real-world scenario, this sequence will eventually terminate, but it leads to the equation x= d+ Cd +C%d + Cd + ... = (1 +C+ C+C +...d We can derive the following identity from a polynomial identity in calculus (4) (1-C)(1+C++C+...+ C) = 1 - C+1 One feature of our consumption matrix C is that com a m + , so that the right-hand side of (4) approaches I. This implies the following approximation for the inverse: (1-0)- 21+ C + + + + Problem 2. (5 points) Let C be the consumption matrix you found in Problem 1(a). Approximate (I C)- by the quantity I+C+C2 +C3 +C4 +C5 (use your calculator, rounding to two decimal places), and use this approximation to recalculate the production vector (rounding to one decimal place) needed to meet the final demand given in Problem 1(c). Compare your results to those in Problem 1