Assume the cash-in-advance constraint always binds \\((x_{t+1}=0)\\).<\/p>\n\n\n\n
Still, private sectors maximise their utility s.t. budget constraint and cash-in-advance constraint. Let’s also include labor as a disutility and assume output is produced by labour.<\/p>\n\n\n\n
$$ \\max_{c_t, b_{t+1}, x_{t+1}} \\sum_{t=0}^{\\infty}\\ \\beta^{t} [u(c_t)-v(l_t)] $$<\/p>\n\n\n\n
$$ p_{t-1}y_{t-1}+b_t(1+i_t)+x_{t}-T_t=x_{t+1}+p_t c_t+b_{t+1} $$ <\/p>\n\n\n\n
0 \\leq x_{t+1}<\/span><\/p>\n\n\n\n $$ 0 \\leq l_t \\leq 1 $$ <\/p>\n\n\n\n , with \\( y_{t-1}=l_{t-1}\\)<\/p>\n\n\n\n Now the output is not exogenous anymore but depends on an agent’s willingness to work.<\/p>\n\n\n\n F.O.C.<\/p>\n\n\n\n w.r.t. \\(c_t: \\quad u'(c_t)=\\beta (1+i_{t+1})\\frac{p_t}{p_{t+1}}u'(c_{t+1}) \\)<\/p>\n\n\n\n w.r.t. \\(l_t: \\quad v'(l_t)=\\beta u'(c_{t+1})\\frac{p_t}{p_{t+1}} \\)<\/p>\n\n\n\n At the steady state, \\( \\frac{p_t}{p_{t+1}}=\\frac{1}{1+\\pi}\\) and \\(y_t=l_t=c_t=y\\) (output is equal to labour’s production in the long run). The output could be calculated as the following equation. (at the steady state means in the long run).<\/p>\n\n\n\n v'(y)=\\beta u'(y)\\frac{1}{\\pi}<\/span><\/p>\n\n\n\n Therefore, we can find that,<\/p>\n\n\n\n Question: First we assume cash-in-advance constraint binds. The QTM states that \\( growth rate of money\\) and \\(inflation \\) is one-to-one correlated only if assuming \/(y\/) is stable (\\(m_t=p_t y\\)). However, we find the relationship between inflation and output here. There seems a contradiction of whether fixes \\(y\\) or not. So, how to bridge the connection between inflation and money growth?<\/p>\n\n\n\n Answer: From the demand point of view or the Cash-in-Advance constraint ( \\(p_t c_t=M_t, or x_t=0\\) by our previous assumption ). At the steady state, consumption is stationary, so \\( \\frac{M_t}{p_t}=\\frac{M_{t+1}}{p_{t+1}}= \\frac{M_{t+2}}{p_{t+2}} =…\\) imply the stock of money and price level are connected, and so the connection between growth rate of money and inlfation works. The inspiration is the cash-in-advance constraint binds, and we consider the problem by fixing consumption in the long-run stationary condition.<\/p>\n\n\n\n The question and answer also state that the neutrality of money bases the key cash-in-advance assumption.<\/p>\n\n\n\n Empirical study examples are as McCandless and Weber (1995)<\/p>\n\n\n\n McCandless, G.T. and Weber, W.E., 1995. Some monetary facts. Federal Reserve Bank of Minneapolis Quarterly Review<\/em>, 19<\/em>(3), pp.2-11.<\/p>\n","protected":false},"excerpt":{"rendered":" Assume the cash-in-advance constraint always binds \\((x_{t+1}=0)\\). Still, private sectors maximise their utility s.t. budget constraint and cash-in-advance constraint. Let’s also include labor as a disutility and assume output is produced by labour. $$ \\max_{c_t, b_{t+1}, x_{t+1}} \\sum_{t=0}^{\\infty}\\ \\beta^{t} [u(c_t)-v(l_t)] $$ $$ p_{t-1}y_{t-1}+b_t(1+i_t)+x_{t}-T_t=x_{t+1}+p_t c_t+b_{t+1} $$ 0 \\leq x_{t+1} $$ 0 \\leq l_t \\leq 1 $$ … Continue reading The Neutrality of Money<\/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\/155"}],"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=155"}],"version-history":[{"count":37,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/155\/revisions"}],"predecessor-version":[{"id":208,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/155\/revisions\/208"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=155"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=155"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=155"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Reference<\/h4>\n\n\n\n