on incomes and real-estate prices. To start, let the consumer utility function be given by q1/2c1/2a1/2, where c is consumption of "bread" (a catch-all commodity), q is real es- tate (housing), and a is amenities, which are valued by the consumer given that a' s ex- ponent is positive. Letting y denote income, it can be shown that the consumer demand functions for bread and housing are given by C = y/2 and q = y/(2p), where p is the price per unit of real estate. (a) Substitute the above demand functions into the utility function to get what is known as the "indirect" utility function, which gives utility as a function of income, prices, and amenities. (b) Using your answer from (a), how does utility change when income y rises? When the real-estate price p rises? How does utility change when amenities increase? With free mobility, everyone must enjoy the same utility level regardless of where they live. Let this constant utility level be de- noted by u. (c) Set the utility expression from (a) equal to . The resulting equation shows how y and p must vary with amenities a in order for Page 270 of 286 95% everyone to enjoy utility . To see one impli- cation of the equation, solve it to yield p as a function of the other variables. According to your solution, how must p change when amenities rise, with y held constant? Give an intuitive explanation of your answer. How must p change if y were to rise, with amen- ities held constant? Again, explain your an- swer. As was explained in the chapter, an- other condition is needed to pin down an explicit solution that tells how y and p vary as amenities change. That condition comes from requiring that the production cost of firms be constant across locations. To gener- ate this condition, let the production func- tion for bread be given by D42where a now represents the firm's real-estate input, L is labor input and a again is amenities (D is a constant). The exponent could be either positive or negative, indicating that an in- crease in a could either raise or lower output. Recalling that p is the price of real estate and y is the price of labor, it can be shown that the cost per unit of bread output is equal to (d) This function shows that an increase in p or y raises unit cost, but that an increase Page 271 of 286 95% in a could either raise or lower costs. Give an example for each possibility, identify- ing an amenity a valued by consumers that could alternatively raise, or lower, produc- tion costs for particular goods. (e) The condition ensuring that costs are constant across locations can be written as plyt24 =1.. Suppose that O>0, so that higher amenities reduce costs. What must happen to p as a increases to keep costs constant, with y held fixed? Give an intuitive explan- ation for your answer. (f) To generate an explicit solution for y in terms of amenities, take the p solution from (c) and substitute it into the constant-cost condition from (e). The resulting equation just involves y, and use it to solve for y as a function of a. (g) Suppose that is negative. Using your so- lution from (f), how does y change when a increases? Suppose instead that is positive but that its magnitude is unknown. Can you say how y responds to an increase in a ? How about if is positive and small? How about if O is positive and large? How about ife is zero? (h) Now take the y solution from (f) and use it to eliminate y from the p solution in (c). Solve the resulting equation for p as Page 271 of 286 95% a function of a. Suppose that is positive. How does p change when a increases? Sup- pose instead that is negative but that its magnitude is unknown. Can you say how p responds to an increase in a ? How about if o is negative but close to zero? How about if is negative and far from zero? How about if equals zero? (i) Summarize your conclusions about how amenities affect incomes and real-estate prices. Although some conclusions are am- biguous, it is possible to offer a clear- cut statement when the effect of amenities on production is "small," either positive or negative (with close to zero). What conclu- sion can be stated in this case? (j) Relate your answer from (i) to the dia- grammatic analysis from the chapter. Page 271 of 286 95% on incomes and real-estate prices. To start, let the consumer utility function be given by q1/2c1/2a1/2, where c is consumption of "bread" (a catch-all commodity), q is real es- tate (housing), and a is amenities, which are valued by the consumer given that a' s ex- ponent is positive. Letting y denote income, it can be shown that the consumer demand functions for bread and housing are given by C = y/2 and q = y/(2p), where p is the price per unit of real estate. (a) Substitute the above demand functions into the utility function to get what is known as the "indirect" utility function, which gives utility as a function of income, prices, and amenities. (b) Using your answer from (a), how does utility change when income y rises? When the real-estate price p rises? How does utility change when amenities increase? With free mobility, everyone must enjoy the same utility level regardless of where they live. Let this constant utility level be de- noted by u. (c) Set the utility expression from (a) equal to . The resulting equation shows how y and p must vary with amenities a in order for Page 270 of 286 95% everyone to enjoy utility . To see one impli- cation of the equation, solve it to yield p as a function of the other variables. According to your solution, how must p change when amenities rise, with y held constant? Give an intuitive explanation of your answer. How must p change if y were to rise, with amen- ities held constant? Again, explain your an- swer. As was explained in the chapter, an- other condition is needed to pin down an explicit solution that tells how y and p vary as amenities change. That condition comes from requiring that the production cost of firms be constant across locations. To gener- ate this condition, let the production func- tion for bread be given by D42where a now represents the firm's real-estate input, L is labor input and a again is amenities (D is a constant). The exponent could be either positive or negative, indicating that an in- crease in a could either raise or lower output. Recalling that p is the price of real estate and y is the price of labor, it can be shown that the cost per unit of bread output is equal to (d) This function shows that an increase in p or y raises unit cost, but that an increase Page 271 of 286 95% in a could either raise or lower costs. Give an example for each possibility, identify- ing an amenity a valued by consumers that could alternatively raise, or lower, produc- tion costs for particular goods. (e) The condition ensuring that costs are constant across locations can be written as plyt24 =1.. Suppose that O>0, so that higher amenities reduce costs. What must happen to p as a increases to keep costs constant, with y held fixed? Give an intuitive explan- ation for your answer. (f) To generate an explicit solution for y in terms of amenities, take the p solution from (c) and substitute it into the constant-cost condition from (e). The resulting equation just involves y, and use it to solve for y as a function of a. (g) Suppose that is negative. Using your so- lution from (f), how does y change when a increases? Suppose instead that is positive but that its magnitude is unknown. Can you say how y responds to an increase in a ? How about if is positive and small? How about if O is positive and large? How about ife is zero? (h) Now take the y solution from (f) and use it to eliminate y from the p solution in (c). Solve the resulting equation for p as Page 271 of 286 95% a function of a. Suppose that is positive. How does p change when a increases? Sup- pose instead that is negative but that its magnitude is unknown. Can you say how p responds to an increase in a ? How about if o is negative but close to zero? How about if is negative and far from zero? How about if equals zero? (i) Summarize your conclusions about how amenities affect incomes and real-estate prices. Although some conclusions are am- biguous, it is possible to offer a clear- cut statement when the effect of amenities on production is "small," either positive or negative (with close to zero). What conclu- sion can be stated in this case? (j) Relate your answer from (i) to the dia- grammatic analysis from the chapter. Page 271 of 286 95%