Recall the government’s budget constraint again,<\/p>\n\n\n\n
$$ p_tg_t+d_t=\\frac{d_{t+1}}{1+i_{t+1}}+T_t+(m_t-m_{t-1}) $$<\/p>\n\n\n\n
Divide by \\(p_t\\), and assume for simplicity that \\(m_t=m_{t+1}=…=\\bar{m}\\)<\/p>\n\n\n\n
$$ g_t+\\frac{d_t}{p_t}=\\frac{d_{t+1}}{p_t (1+i_{t+1})}+\\tau_t $$<\/p>\n\n\n\n
denote \\(\\tilde{d_t}=d_t\/p_t\\), and recall that \\((1+i_{t+1})\\frac{p_t}{p_{t+1}}=1+r_{t+1}\\).<\/p>\n\n\n\n
Thus,<\/p>\n\n\n\n
$$ g_t+\\tilde{d_t}=\\frac{\\tilde{d}_{t+1}}{1+r_{t+1}}+\\tau_t $$<\/p>\n\n\n\n
Iterate forward, and impose the “no-Ponzi condition”, \\( \\lim_{s\\rightarrow \\infty} \\frac{\\tilde{d_{t+s}}}{\\prod_{j=1}^s 1+r_{t+j}}=0\\) to get,<\/p>\n\n\n\n
$$ \\tilde{d_t}=\\sum_{s=0}^{\\infty} \\beta^s (\\tau_{t+s}-g_{t+s}) $$<\/p>\n\n\n\n
, where the equilibrium condition \\( \\beta=\\frac{1}{1+r}\\) has been imposed.<\/p>\n\n\n\n
Implication:<\/strong><\/p>\n\n\n\n $$ \\frac{d_t}{p_t}=\\sum_{s=0}^{\\infty} \\beta^s (\\tau_{t+s}-g_{t+s}) $$<\/p>\n\n\n\n The “unpleasant arithmetic” stated that if the government has leadership, it can coerce expansions in money.<\/p>\n\n\n\n In contrast, FTPL says that the above restrictions are not a constraint to the CB or government. Instead, it is an equilibrium relation.<\/p>\n\n\n\n As a consequence, the CB and the government may choose policies independent of the above constraint. In the end, the price level \\(p_t\\) must then adjust such that the equation holds.<\/p>\n","protected":false},"excerpt":{"rendered":" Recall the government’s budget constraint again, $$ p_tg_t+d_t=\\frac{d_{t+1}}{1+i_{t+1}}+T_t+(m_t-m_{t-1}) $$ Divide by \\(p_t\\), and assume for simplicity that \\(m_t=m_{t+1}=…=\\bar{m}\\) $$ g_t+\\frac{d_t}{p_t}=\\frac{d_{t+1}}{p_t (1+i_{t+1})}+\\tau_t $$ denote \\(\\tilde{d_t}=d_t\/p_t\\), and recall that \\((1+i_{t+1})\\frac{p_t}{p_{t+1}}=1+r_{t+1}\\). Thus, $$ g_t+\\tilde{d_t}=\\frac{\\tilde{d}_{t+1}}{1+r_{t+1}}+\\tau_t $$ Iterate forward, and impose the “no-Ponzi condition”, \\( \\lim_{s\\rightarrow \\infty} \\frac{\\tilde{d_{t+s}}}{\\prod_{j=1}^s 1+r_{t+j}}=0\\) to get, $$ \\tilde{d_t}=\\sum_{s=0}^{\\infty} \\beta^s (\\tau_{t+s}-g_{t+s}) $$ , where the equilibrium … Continue reading The Fiscal Theory of the Price Level<\/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":[9,6,13],"tags":[],"_links":{"self":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/1452"}],"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=1452"}],"version-history":[{"count":17,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/1452\/revisions"}],"predecessor-version":[{"id":2668,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/1452\/revisions\/2668"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1452"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1452"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1452"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}