Taking logarithms and rearranging implies the following demand relationship for good k: The aggregate index can be derived using (5) and (7). Specifically, (5) implies: while equation (7) implies: Substituting (9) into (10) implies the aggregate consumption price index is an aggregation of individual goods prices: Once the consumer has made their decision at the top level, they move to the next level, which involves the choice between domestically produced and imported varieties of the goods subject to the value of aggregate expenditure decided at the previous level. Assuming that Dixit-Stiglitz preferences also apply at this level, the consumer has the following nested maximisation problem: subject to: where is the elasticity of substitution between domestic and imported varieties of is a k k k, weighting parameter which implies home bias, is the volume of domestically produced good ditk k, is the price of domestically produced is the volume of imported good and is the pditk k, mitk k pmitk price of imported k. This implies the following demand relationships for domestically produced and imported k: and index for good k: Finally, consumers in country must choose between imported varieties of good sourced from i k different trading partners. Assuming Dixit-Stiglitz preferences also apply at this level yields the following nested maximisation problem: subject to:
Taking logarithms and rearranging implies the following demand relationship for good k: The aggregate index can be derived using (5) and (7). Specifically, (5) implies: while equation (7) implies: Substituting (9) into (10) implies the aggregate consumption price index is an aggregation of individual goods prices: Once the consumer has made their decision at the top level, they move to the next level, which involves the choice between domestically produced and imported varieties of the goods subject to the value of aggregate expenditure decided at the previous level. Assuming that Dixit-Stiglitz preferences also apply at this level, the consumer has the following nested maximisation problem: subject to: where k is the elasticity of substitution between domestic and imported varieties of k, k is a weighting parameter which implies home bias, ditk is the volume of domestically produced good k, pditk is the price of domestically produced k, mitk is the volume of imported good k and pmitk is the price of imported k. This implies the following demand relationships for domestically produced and imported k: and index for good k: Finally, consumers in country i must choose between imported varieties of good k sourced from different trading partners. Assuming Dixit-Stiglitz preferences also apply at this level yields the following nested maximisation problem: subject to:
P2017-T160720-working_paper_2017-01.docx