$$ u(c)=\\frac{c^{1-\\sigma}-1}{1-\\sigma}, \\quad \\sigma \\in [0,1) $$<\/p>\n\n\n\n
$$ u'(c)=c^{-\\sigma} $$<\/p>\n\n\n\n
$$ u”(c)=-\\sigma c^{-\\sigma -1} $$<\/p>\n\n\n\n
Risk Aversion is \\( – \\frac{u”}{u’}\\). So,<\/p>\n\n\n\n
$$ – \\frac{u”}{u’}=\\frac{ -\\sigma c^{-\\sigma -1} }{ u'(c)=c^{-\\sigma} }=\\frac{\\sigma}{c_t} \\leftarrow CRRA$$<\/p>\n\n\n\n
How does the isoelastic utility function work?<\/p>\n\n\n\n
Recall a Euler equation \\(u'(c_t)=\\beta (1+r)u'(c_{t+1})\\).<\/p>\n\n\n\n
$$ c_t^{-\\sigma}=\\beta (1+r) c_{t+1}^{-\\sigma} $$<\/p>\n\n\n\n
$$ \\frac{c_t}{c_{t+1}}=( \\beta (1+r) )^{-\\frac{1}{\\sigma}} =e^{ -\\frac{1}{\\sigma} ln(\\beta(1+r)) }$$<\/p>\n\n\n\n
That implies the consumption as a ratio over time is a constant, depending on \\(\\beta, r, \\sigma\\). Also, as \\(\\beta (1+r)\\) is a very small number, \\(ln(\\beta (1+r))\\approx \\beta(1+r)\\). Thus, \\(\\frac{c_t}{c_{t+1}}<0\\).<\/p>\n\n\n\n
In microeconomics, we always think of factors that grow constantly over time, e.g. constant saving rate.<\/p>\n\n\n\n
Further study.<\/p>\n","protected":false},"excerpt":{"rendered":"
$$ u(c)=\\frac{c^{1-\\sigma}-1}{1-\\sigma}, \\quad \\sigma \\in [0,1) $$ $$ u'(c)=c^{-\\sigma} $$ $$ u”(c)=-\\sigma c^{-\\sigma -1} $$ Risk Aversion is \\( – \\frac{u”}{u’}\\). So, $$ – \\frac{u”}{u’}=\\frac{ -\\sigma c^{-\\sigma -1} }{ u'(c)=c^{-\\sigma} }=\\frac{\\sigma}{c_t} \\leftarrow CRRA$$ How does the isoelastic utility function work? Recall a Euler equation \\(u'(c_t)=\\beta (1+r)u'(c_{t+1})\\). $$ c_t^{-\\sigma}=\\beta (1+r) c_{t+1}^{-\\sigma} $$ $$ \\frac{c_t}{c_{t+1}}=( \\beta (1+r) … Continue reading Isoelastic Utility Function<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,6],"tags":[],"_links":{"self":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/1572"}],"collection":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1572"}],"version-history":[{"count":30,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/1572\/revisions"}],"predecessor-version":[{"id":1602,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/1572\/revisions\/1602"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1572"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1572"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1572"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}