In the Northeast United States food can be produced on either good (G) land or bad (B) land. The (Leontief) production function for land of type i,i=G,B is: Qi=min[Ti,Li/ai] where Qi is food output from land of type i,Ti is hectares of land of type i, and Li is the number of workers working on land of type i. Parameters aG=1 and aB=3, meaning that bad land requires 3 workers per hectare and good land only 1 worker per hectare. Using ai workers a hectare (either G or B ) produces 1 unit of food. The wage of a worker is w whether she works on good land or on bad land. The price of renting a hectare of land of type i is ri. Food produced from both types of land is the same (i.e., consumers regard the two outputs as perfect substitutes) so has the same price. 1. What is the total cost function, unit cost, and marginal cost of producing food on each type of land? ?1 2. How much lower must rB be from rG for bad land to offer the same unit cost as good land? 3. Say that the price of food is p. Landowners rent out their land to competitive farmers who must hire workers at wage w to produce and who sell food at price p and make no profit. What will the rent be on each type of land as a function of p and w ? 4. Preferences are Cobb-Douglas over food and manufactures with equal shares so that the utility of a representative individual is: U=(CM)1/2(CF)1/2 where CM is consumption of manufactures and CF is consumption of food (regardless of the type of land its produced on). Food has price p and manufactures have price 1. Consider an individual with a budget Y. Set up her utility maximization problem as a Lagrangian maximization and take the first-order conditions. 5. Using two first-order conditions above show that she will choose CM/CF in proportion to p. 6. Manufactures require 1 worker to make 1 unit of output. (QM=LM, where QM is output of manufactures and LM are workers in manufacturing.) Production is competitive, so that the wage is also 1 as a workers can move between farming and manufacturing. The Northeast has 40 workers who can make manufactures, work on good land, or work on bad land (40=LM+LG+LB). It has 4 hectares of good land and 2 hectares of bad land. If all the land is used in food production how much food is produced and how many workers are needed? How many workers are left to produce manufactures? 1 Remember that total cost functions have as arguments factor prices (here w and rG or rB ) and output Q. The unit cost function is just the total cost function evaluated at Q=1. 1 7. Combining results in 5 and 6 , what is p if all land is used to make food and the remaining workers make manufactures? Using your answer to 3 , what are rG and rB ? Was it correct to assume that all land was used? 8. The Northeast expands by incorporating an area called Midwest which has no workers or bad land but 8 hectares of good land. (so that the total economy now has 40 workers, 12 hectares of good land, and 2 hectares of bad land). Revisit your answers to 6 and 7 . What will be the resulting p,rG, and rB?2 In the Northeast United States food can be produced on either good (G) land or bad (B) land. The (Leontief) production function for land of type i,i=G,B is: Qi=min[Ti,Li/ai] where Qi is food output from land of type i,Ti is hectares of land of type i, and Li is the number of workers working on land of type i. Parameters aG=1 and aB=3, meaning that bad land requires 3 workers per hectare and good land only 1 worker per hectare. Using ai workers a hectare (either G or B ) produces 1 unit of food. The wage of a worker is w whether she works on good land or on bad land. The price of renting a hectare of land of type i is ri. Food produced from both types of land is the same (i.e., consumers regard the two outputs as perfect substitutes) so has the same price. 1. What is the total cost function, unit cost, and marginal cost of producing food on each type of land? ?1 2. How much lower must rB be from rG for bad land to offer the same unit cost as good land? 3. Say that the price of food is p. Landowners rent out their land to competitive farmers who must hire workers at wage w to produce and who sell food at price p and make no profit. What will the rent be on each type of land as a function of p and w ? 4. Preferences are Cobb-Douglas over food and manufactures with equal shares so that the utility of a representative individual is: U=(CM)1/2(CF)1/2 where CM is consumption of manufactures and CF is consumption of food (regardless of the type of land its produced on). Food has price p and manufactures have price 1. Consider an individual with a budget Y. Set up her utility maximization problem as a Lagrangian maximization and take the first-order conditions. 5. Using two first-order conditions above show that she will choose CM/CF in proportion to p. 6. Manufactures require 1 worker to make 1 unit of output. (QM=LM, where QM is output of manufactures and LM are workers in manufacturing.) Production is competitive, so that the wage is also 1 as a workers can move between farming and manufacturing. The Northeast has 40 workers who can make manufactures, work on good land, or work on bad land (40=LM+LG+LB). It has 4 hectares of good land and 2 hectares of bad land. If all the land is used in food production how much food is produced and how many workers are needed? How many workers are left to produce manufactures? 1 Remember that total cost functions have as arguments factor prices (here w and rG or rB ) and output Q. The unit cost function is just the total cost function evaluated at Q=1. 1 7. Combining results in 5 and 6 , what is p if all land is used to make food and the remaining workers make manufactures? Using your answer to 3 , what are rG and rB ? Was it correct to assume that all land was used? 8. The Northeast expands by incorporating an area called Midwest which has no workers or bad land but 8 hectares of good land. (so that the total economy now has 40 workers, 12 hectares of good land, and 2 hectares of bad land). Revisit your answers to 6 and 7 . What will be the resulting p,rG, and rB?2