{"id":202,"date":"2021-11-12T15:57:03","date_gmt":"2021-11-12T07:57:03","guid":{"rendered":"https:\/\/fanyuzhao.com\/?p=202"},"modified":"2022-03-02T16:52:44","modified_gmt":"2022-03-02T08:52:44","slug":"friedman-rule","status":"publish","type":"post","link":"https:\/\/fanyuzhao.com\/?p=202","title":{"rendered":"Friedman Rule"},"content":{"rendered":"\n

Let’s continue with the previous blog post The Neutrality of Money<\/a><\/em>.<\/p>\n\n\n\n

In the previous model, consumers maximise their utility subject to contraints.<\/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

We have solved it and get the Euler condition,<\/p>\n\n\n\n

v'(y)=\\beta u'(y)\\frac{1}{\\pi}<\/span> <\/p>\n\n\n\n

Here, we would consider the Planner’s Problem that makes social optimal.<\/p>\n\n\n\n

Planner’s Problem<\/h4>\n\n\n\n

In the planner’s problem, we would release the budget constraints and cash-in-advance constraints, because the planner only needs to achieve social optimal. The planner’s problem is as the following.<\/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)]<\/span> <\/p>\n\n\n\n

$$ s.t. \\quad c_t=l_t $$<\/p>\n\n\n\n

F.O.C.<\/p>\n\n\n\n

$$ u'(c_t)=v'(l_t) $$<\/p>\n\n\n\n

Here let’s compare the planner’s Euler equation with the private sector one.<\/p>\n\n\n\n

To make them equal, the only thing we need to adjust is to let \\( \\beta\\times\\frac{1}{1+\\pi}=1\\). The implication is that we need \\( \\pi =\\beta -1\\). As in the steady state, the discount factor \\( \\beta = \\frac{1}{1+r}\\), so the optimal inflation rate should be \\( \\pi^*=\\frac{-r}{1+r}\\). <\/p>\n\n\n\n

The implication is that the optimal inflation rate is negative and close to the negative real interest rate.<\/p>\n\n\n\n

Cash Credit Good Model<\/h4>\n\n\n\n

Stokey and Lucas (1987) included the cash-credit good into the cash in advance model.<\/p>\n\n\n\n

\\max_{ \\{ c_t,b_{t+1} \\}_{t=0}^{\\infty} } \\sum_{t=0}^{\\infty}\\ \\beta^{t} [u(c_t^1)+u(c_t^2)]<\/span> <\/p>\n\n\n\n

$$ s.t. \\quad b_{t+1}+p_t c_t^1+p_{t-1} c_{t-1}^2 =(1+i_t)b_t+p_{t-1}y_{t-1}$$<\/p>\n\n\n\n

In equilibrium, markets clear and resources constraints,<\/p>\n\n\n\n

$$ y_{t-1}=c_{t-1}^1+c_{t-1}^2 $$<\/p>\n\n\n\n

y_{t}=c_{t}^1+c_{t}^2<\/span> <\/p>\n\n\n\n

F.O.C.<\/p>\n\n\n\n

$$ u'(c_t^1)=\\lambda_t p_t $$<\/p>\n\n\n\n

$$ u'(c_t^2)=\\beta\\lambda_{t+1}p_t $$<\/p>\n\n\n\n

$$\\lambda_t=\\beta \\lambda_{t+1}(1+i_{t+1})$$<\/p>\n\n\n\n

Combining them we can get<\/p>\n\n\n\n

$$ \\frac{u'(c_t^1)}{ u'(c_t^2) }=1+i_{t+1}$$<\/p>\n\n\n\n

The ratio of marginal utility is equal to one plus the nominal interest rate. <\/p>\n\n\n\n

The implication is that people want to consume \\(c_t^2\\) instead of \\(c_t^1\\), pay money at the time at \\(t\\), and hold some bonds and earn the nominal interest rate.<\/p>\n\n\n\n

However, the planner problem is that<\/p>\n\n\n\n

$$ \\frac{u'(c_t^1)}{ u'(c_t^2) }=1 $$<\/p>\n\n\n\n

Thus, the optimal rule is to set \\(i_{t+1}=0\\).<\/p>\n\n\n\n

The Euler equation in the steady state (\\( c_t^i=c_{t+1}^i=…=c^i \\)) is that,<\/p>\n\n\n\n

$$ \\beta \\frac{1+i_{t+1}}{1+\\pi_t}=1 $$<\/p>\n\n\n\n

By plugging in \\(i_{t+1}=0\\), \\(\\pi^*=\\beta -1 \\), the Friedman rule also holds.<\/p>\n","protected":false},"excerpt":{"rendered":"

Let’s continue with the previous blog post The Neutrality of Money. In the previous model, consumers maximise their utility subject to contraints. $$ \\max_{c_t, b_{t+1}, x_{t+1}} \\sum_{t=0}^{\\infty}\\ \\beta^{t} [u(c_t)-v(l_t)] $$ $$ p_{t-1}y_{t-1}+b_t(1+i_t)+x_{t}-T_t=x_{t+1}+p_t c_t+b_{t+1} $$ 0 \\leq x_{t+1} $$ 0 \\leq l_t \\leq 1 $$ We have solved it and get the Euler condition, v'(y)=\\beta u'(y)\\frac{1}{\\pi} … Continue reading Friedman Rule<\/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\/202"}],"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=202"}],"version-history":[{"count":55,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/202\/revisions"}],"predecessor-version":[{"id":729,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/202\/revisions\/729"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=202"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=202"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=202"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}