ABC Consider an economy with two industries, using a common resource to produce two different types...

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ABC Consider an economy with two industries, using a common resource to produce two different types of goods and to achieve a certain level of technology. Given a fixed technology level K (considered a parameter), and with an allocation of x resource units to the first industry and y units to the second, the units produced by each industry can be calculated by the formulas f₁(x, K) = 16x¹/2 K¹/4 y f2(y, K) = 8y¹/2 K¹/4 Suppose further that to reach a technology level K it is necessary to use H(K) = 2K² units of the common resource, and that the total available resource is L = 160 units. In this context, the central planner's problem consists of maximizing total revenue in the two industries subject to the resource constraint, that is: Maximize I(x,y)=P₁ •f₁(x,K)+P2 f₂(y,K) restricted to x+y=L-H(K). (a) State the Lagrange conditions (first-order conditions) to solve the problem. (b) Solve the problem using Lagrange's method, and assuming that the unit prices of the goods produced by each industry are pl = 1 and p2 = 2. (c) Explain why the necessary and sufficient conditions are fulfilled in this case. (d) By how much (approximately) does maximum revenue change if the level of technology goes from K to K + 1? ABC Consider an economy with two industries, using a common resource to produce two different types of goods and to achieve a certain level of technology. Given a fixed technology level K (considered a parameter), and with an allocation of x resource units to the first industry and y units to the second, the units produced by each industry can be calculated by the formulas f₁(x, K) = 16x¹/2 K¹/4 y f2(y, K) = 8y¹/2 K¹/4 Suppose further that to reach a technology level K it is necessary to use H(K) = 2K² units of the common resource, and that the total available resource is L = 160 units. In this context, the central planner's problem consists of maximizing total revenue in the two industries subject to the resource constraint, that is: Maximize I(x,y)=P₁ •f₁(x,K)+P2 f₂(y,K) restricted to x+y=L-H(K). (a) State the Lagrange conditions (first-order conditions) to solve the problem. (b) Solve the problem using Lagrange's method, and assuming that the unit prices of the goods produced by each industry are pl = 1 and p2 = 2. (c) Explain why the necessary and sufficient conditions are fulfilled in this case. (d) By how much (approximately) does maximum revenue change if the level of technology goes from K to K + 1?

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a The Lagrange conditions firstorder conditions for solving the problem are 1 partialpartial x Ixy l... View the full answer