{"id":321,"date":"2021-11-12T23:19:59","date_gmt":"2021-11-12T15:19:59","guid":{"rendered":"https:\/\/fanyuzhao.com\/?p=321"},"modified":"2022-03-02T16:52:44","modified_gmt":"2022-03-02T08:52:44","slug":"liquidity-trap","status":"publish","type":"post","link":"https:\/\/fanyuzhao.com\/?p=321","title":{"rendered":"Liquidity Trap"},"content":{"rendered":"\n

Recall the Euler condition in the previous blog post A Cash-in-Advance Model<\/em>. <\/p>\n\n\n\n

$$ u'(y_t)=\\beta(1+i_t)\\frac{p_t}{p_{t+1}}u'(y_{t+1}) $$ <\/p>\n\n\n\n

Assumption<\/h4>\n\n\n\n

For simplification, we assume no government spending, \\(g_t\\), government debt, \\(d_t\\), and taxes, \\(T_t\\). Also, we assume money is stable such that \\(m_t=m_{t+1}=m\\) (so there is not seignorage). We here consider \\(y_t\\) is exogenous.<\/p>\n\n\n\n

Recall<\/strong><\/p>\n\n\n\n

Suppose that \\( y_t=u_{t+1}=…=y\\), then<\/p>\n\n\n\n

$$\\quad 1=\\beta(1+i_{t+1})\\frac{p_t}{p_{t+1}}$$<\/p>\n\n\n\n

Now if guess both \\(x_{t+1}=x_{t+2}=0\\), then the velocity of money \\(v_t=1\\).<\/p>\n\n\n\n

\\( \\quad p_t=p_{t+1}=\\frac{m}{y}, \\quad \\) and \\(\\quad i_{t+1}=\\frac{1}{\\beta}-1\\geq0\\)<\/p>\n\n\n\n

P.S. if violate the guess \\(x_{t+1}=x_{t+2}=0\\), then the euler equation shows \\(1+\\beta (1+i_{t+1})\\frac{p_t}{p_{t+1}}\\) would be \\( p_{t+1}=\\beta p_{t}\\). So, \\( p_{t+1}<p_t\\). By QTM \\(m \\cdot v_t= p_t \\cdot y\\) (\\(m, y\\) are constant), \\( v_{t+1}<v_T\\) must be true to make next-period price level be low than the current price level. Lower velocity means \\( x_{t+2}>x_{t+1}\\) (people would hoard more money on hand in the next period). The loop begins, and price level would decline in the following periods.<\/p>\n\n\n\n

If future outputs decrease,<\/h4>\n\n\n\n

u'(y_t)=\\beta(1+i_t)\\frac{p_t}{p_{t+1}}u'(y_{t+1})<\/span> <\/p>\n\n\n\n

If replace \\(p_{t+1}=\\frac{m_{t+1} v_{t+1}}{y_{t+1}}\\),<\/p>\n\n\n\n

u'(y_t)=\\beta(1+i_t)\\frac{p_t \\times y_{t+1}}{m_{t+1}v_{t+1}}u'(y_{t+1})<\/span> <\/p>\n\n\n\n

u'(y_t)=\\beta(1+i_t)\\frac{p_t \\times y_{t+1}}{m_{t+1}-x_{t+2}}u'(y_{t+1})<\/span> <\/p>\n\n\n\n

Here, by complementary slackness, \\( x_{t+2}\\times i_{t+1}=0\\).<\/p>\n\n\n\n

If replace \\(p_{t+1}=\\frac{m_{t+1} v_{t+1}}{y_{t+1}}\\), =\\frac{m_{t+1}}{y_{t+1}}\\) by assume not in liquidity trap in the first so \\(v_{t+1}=1\\). Then we get,<\/p>\n\n\n\n

u'(y_t)=\\beta(1+i_t)\\frac{p_t \\times y_{t+1}}{m_{t+1}}u'(y_{t+1})<\/span> <\/p>\n\n\n\n

We, in the following, assume \\(x u'(x)\\) is decreasing in x.<\/p>\n\n\n\n

If the economy experiences a fall in period \\( t+1\\) output from \\(y_{t+1}\\) to \\(y’_{t+1}\\). What happens to the nominal interest rate?<\/strong><\/p>\n\n\n\n

We write it in this way for simplification.<\/p>\n\n\n\n

u'(y)=\\beta(1+i_t)\\frac{p_t }{m}y’u'(y’)<\/span> <\/p>\n\n\n\n

As \\(y_{t+1}\\) decrease, \\(y’u'(y’)\\) increase as our assumption. The LHS keeps stable, so the interest rate has to decrease to keep the equality holding. Therefore, \\(i_{t+1}\\) we’ll eventually hit zero.<\/p>\n\n\n\n

As \\( i_{t+1}=0\\), the economy enters into the liquidity trap, and people start to hoard money ,\\(x_{t+1}>0\\). Recall the QTM equation, \\(p_t=\\frac{ m_t-x_{t+1} }{y_t}=\\frac{mv_t}{y} \\), \\(p_t\\) would decrease. So, the price level at time \\(t\\) finally decreases as well.<\/p>\n\n\n\n