The word “real” means the real term in contract with the word “monetary”. Therefore, the real business cycle is not about the monetary policy, but about the negative supply shock. The RBC theory explains most of the business cycle in human history.
Examples
For example. in the early agriculture society, agriculture consists most of GDP. If extreme weather condition happens (the real shock), then there are bad harvests and bad outputs for almost all economy. People have less to eat, and an economic recession emerges. In the modern economy, outputs are more diversified. Another is that in the 1973 oil crisis, the OPEC oil embargo induced the oil price increase. The increase in oil prices made production costs increase for other goods and services, and led to an overall recession. A recent example is a crisis in Brazil. A decrease in commodity prices hugely reduced incomes (net export). Also, the Brazilian government became erratic and unpredictable (no clear target and no credible), bringing further risks to the Brazilian economy.
Shocks
Examples of shocks are,
Technology schoks
Policy shocks: Fiscal policy & Monetary shocks
Political shocks: changes in polical party
Expectations shocks: animial spirits
Natural disaster
Propagation mechanisms
Two Propagation mechanisms are here, the labour propagation mechanism and the intertemporal one. See notes.
Potential Solutions
Try to avoid the problem in the first place. For example, if the oil price is expected to increae, then invest in other alternative energy to decreae the effects of oil price increase on production costs. In other word, diversity the production costs and make the production process not rely too much on oil.
Make the economy more flexible and can be adjustable to negative supply shocks quickly.
Problems
It do not explain all business cycles, which are not caused by supply shocks. For example, a lot busienss cycles are about monetary polcy, banking, and credit.
It does not explain why unemployment rate is so high in labour economics.
In short, before the RBC model, macroeconomic studies mainly focus on the IS-LM and AD-AS. The building up of RBC solves the problem that macroeconomic study did not have a solid microeconomic background.
RBC model is like a new classical model with shocks, based on the key assumption that markets are perfectly competitive. Then, market players maximise their utility subject to certain constraints. Through the RBC model, we can get the co-movement of outputs, labours and capitals. Market fluctuations are caused by shocks. Without shocks, the markets are in equilibrium condition over time, because markets are competitive. Meanwhile, money is not included in the RBC model, so all factors are in real terms.
The first oil crisis started with the oil embargo proclaimed by OPEC.
OPEC: Oil exporting nations accumulated vast wealth due to the price increase. US: the oil price increase induced the recession, inflation, reduced productivity, and low economic growth.
Whyt did Keynesian economics fail in the 1970s?
According to Keynesians, the growth in the money supply can increase employment and promote economic growth. Keynesian economists believe in the Philips relationship between unemployment (economic growth) and inflation. However, both of them hiked in the 1970s.
Why did stagflation occur?
The prevailing belief has been that high levels of inflation were the result of an oil supply shock and the resulting increase in the price of gasoline, which drove the prices of everything else higher (cost-push inflation).
A now well-founded principle of economics is that excess liquidity in the money supply can lead to price inflation. Monetary policy was expansive during the 1970s, which could help explain the rampant inflation at the time.
How did Friedman work?
“Inflation is always and everywhere a monetary phenomenon.”
Milton Friedman
During the energy crisis of the 1970s, while everyone was blaming OPEC in the early part of the 70s, or the Iranian revolution in 1979, Friedman recognized who the real culprits were — Richard Nixon, who in 1973 instituted wage-price controls and, following Nixon, Gerald Ford and Jimmy Carter who continued these price controls on oil, gasoline, and natural gas.
“The present oil crisis has not been produced by the oil companies. It is a result of government mismanagement exacerbated by the Mideast war.”
– Milton Friedman, “Why Some Prices Should Rise,” Newsweek, November 19, 1973.
Friedman believed prices could not increase without an increase in the money supply. The Fed followed a constrictive monetary policy that helped drive interest rates to double-digit levels, reduce inflation.
P.S. Fed’s credibility and inflation expectation (inflation targets) also play roles in resulting in stagflation.
Inspiration
Inflation (or hyperinflation) is a monetary phenomenon by Friedman and some economists. In China’s case, stagflation seems unable to happen if there are no vast increase in money supply and loss of credibility of the central bank.
Starting with The Wealth of Nations, 1776, by Adam Smith. The central idea is that the market can be self-correcting. The central assumption implied is that all individuals choose to maximise their utility.
Neoclassical Economics
Neoclassical economics is formalised by Alfred Marshall (Marshallian demand, and Cambridge quantitative theory of money). The school is based on the mathematical formulation of the general equilibrium by Léon Walras (Walras’ Law).
Neoclassical economics states that the production, consumption and valuation (pricing) of goods and services are driven by the supply and demand model. Value is determined by maximising utility s.t. constraints.
Assumptions: 1. people have rational preferences (complete and transitive, see R100 at the Cambridge uni); 2. individuals maximise their utility and firms maximise profits; 3. people act independently on the basis of full and relevant information.
Neoclassical schools dominated until the Great Depression during the 1930s. However, John Maynard Keynes led with the publishment of The General Theory of Employment, Interest and Money. Keynesian dominated until 1973-1975 recession triggered by the 1973 oil crisis (stagflation crisis resulted from oil price increase) that Keynesian policy failed to reduce unemployment and also lead to hyperinflation. Phillips curve also failed because high unemployment and inflation came together. Then, new classical took the dominant.
New Neoclassical Economics
The new classical school works on real business cycle (Real Business Cycle model) theory that used fully specified general equilibrium models and used changes in technology to explain fluctuations in economic output.
M&M theorem (Modigliani and Miller, 1958) is used to value a firm. It states that a firm’s value is based on its ability to earn revenue plus its risk of underlying assets. The way a firm finances its operations should not affect its value.
At its most basic level, the theorem argues that, with certain assumptions in place, it is irrelevant whether a company finances its growth by borrowing, by issuing stock shares, or by reinvesting its profits.
Assumptions are 1. the markets are completely efficient; 2. there are no costs of bankruptcy or agency dynamics and no taxes.
However, there are of course taxes and costs in the reality, and the assumptions do not hold. Therefore, the M&M theorem implies that firms are more valuable if financed by debts than financed by equities. The reason is the tax shield effects of debts.
Mathematic Example
Consider two companies, same risks, same expected cash flow before interest, \( Y\).
Co1, has debt with market value of \(D_1\). Total market value \(V_1=E_1+D_1\).
Co2, has no debt. \(V_2=E_2\), market value of equity.
We can invest,
Investment A: We own a fraction \(a\) (e.g. 6%) of shares in Co1z. They worth \(aE_1\) (e.g. $6,000). Expected cash flow from investment A, \(y_A\), is:
Co1 needs to pay an interest rate of its debt, \(R_D D_1\). Purchasing Co1 means only purchasing the equity of Co1, which is EV-Debts.
Investment B: We sell the shares of Co1. We receive amount \(aE_1\) ($6,000). Then, we use this to buy shares in Co2, which is ungeared.
To Produce the same gearing and risk as investment A, we borrow a further amount worth \(aD_1\) (‘home-made gearing’ or ‘artificially gearing’), at interest rate \(R_D\), and buy more shares in Co2.
E.G. if gearing of Co1 is \( \frac{D_1}{D_1+E_1}=0.25\), then to get same gear for our investment B, we borrow \(6,000\times \frac{0.25}{0.75}=2,000\).
Expected cash flow from investment B, \( y_B\), is:
In equilibrium, \( y_A=y_B\). Otherwise, arbitrage opportunity emerges. Therefore, we get \(V_1=V_2\).
In conclusion, gearing does not affect value (Equity plus Debt).
Implication
Since expected net cash flow (\(Y\)) and company value are the same for each company, the cost of equity for ungeared Co2 and WACC for geared Co1 must be equal. So WACC must be constant with respect to gearing.
Let the cost of equity for a company with no debt be \(R_{ungeared}\).
So we get a linear relation between cost of equity and D/E, assuming a constant cost of debt, assuming a constant cost of debt (and assuming \(R_{ungeared}>R_D\)). Implicly, changes in gearing structure (or how to finance the business) do not affect the WACC for a company.
The relation is shown in this figure
Violation of the constant debt assumption of course would make WACC unconstant.
MM Theory & Beta
In the CAPM, expected returns on assets differ because their betas differ.
, where \(\beta_{ungeared}\) denotes asset beta, and \( \beta_{geared}\) denotes actual beta of shares of a geared company (estimated from the market data).
A Modigliani-Miller Theorem for Open-Market Operations
Monetary policy determines the composition of the government’s portfolio. Fiscal policy (the size of the deficit on the current account) determines the path of net government indebtedness. Wallace showed that alternative paths of the government’s portfolio consistent with a single path of fiscal policy can be irrelevant. The irrelevance means that both the equilibrium consumption allocation and the path of the price level are independent of the path of the government’s portfolio.
Typos are there. See the original paper issued by Fed in 1979.
Reference
Modigliani, F. and Miller, M.H., 1958. The cost of capital, corporation finance and the theory of investment. The American economic review, 48(3), pp.261-297.
Wallace, N., 1981. A Modigliani-Miller theorem for open-market operations. The American Economic Review, 71(3), pp.267-274.
In statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory.
Wikipedia
During my study in Cambridge, Professor Oliver Linton introduced the Kalman Filter in Time Series analysis, but I did not get it at that time. So, here is a revisit.
My Thinking of Kalman Filter
Kalman Filter is an algorithm that estimates optimal results from uncertain observation (e.g. Time Series Data. We know only the sample, but never know the true distribution of data or never know the true value when there are no errors).
Consider the case, I need to know my weight, but the bodyweight scale cannot give me the true value. How can I know my true weight?
Assume the bodyweight scale gives me error of 2, and my own estimate gives me error of 1. Or in another word, a weight scale is 1/3 accurate, and my own estimation is 2/3 accurate. Then, the optimal weight should be,
$$ Optimal Result = \frac{1}{3}\times Measurement + \frac{2}{3}\times Estimate $$
, where \( Measurement\) means the measurement value, and \(Estimate\) means the estimated value. We conduct the following transformation.
$$ Optimal Result = \frac{1}{3}\times Measurement +Esimate- \frac{1}{3}\times Estimate $$
Optimal Result = Esimate+\frac{1}{3}\times Measurement – \frac{1}{3}\times Estimate
Optimal Result = Esimate+\frac{1}{3}\times (Measurement – Estimate)
Therefore, we can get
Optimal Result = Esimate+\frac{p}{p+r}\times (Measurement – Estimate)
, where \(p\) is the estimation error and \(r\) is the measurement error.
For example, if the estimation error is zero, then the fraction is equal to zero. Thus, the optimal result is just the estimate.
, where \(p_{n,n-1}\) is Uncertainty in Estimate, \(r_n\) is Uncertainty in Measurement, \(\hat{x}_{n,n}\) is the Optimal Estimate at \(n\), and \(z_n\) is the Measurement Value at \(n\).
The Optimal Estimate is updated by the estimate uncertainty through a Covariance Update Equation,
Intuitively, I need \( \hat{x}_{k-1}\) (, which is the weight last week) to calculate the optimal estimate weight this week \(\hat{x}_k\). Firstly, I estimate the weights this week \(\hat{x}_k^-\) and measure the weight this week \(y_k\). Then, combine them to get the optimal estimate weights this week.
Reading
The application of the Kalman Filter could be found in the following reading. Also, I will continue in my further study.
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.
Recall
Suppose that \( y_t=u_{t+1}=…=y\), then
$$\quad 1=\beta(1+i_{t+1})\frac{p_t}{p_{t+1}}$$
Now if guess both \(x_{t+1}=x_{t+2}=0\), then the velocity of money \(v_t=1\).
\( \quad p_t=p_{t+1}=\frac{m}{y}, \quad \) and \(\quad i_{t+1}=\frac{1}{\beta}-1\geq0\)
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.
Here, by complementary slackness, \( x_{t+2}\times i_{t+1}=0\).
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,
We, in the following, assume \(x u'(x)\) is decreasing in x.
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?
We write it in this way for simplification.
u'(y)=\beta(1+i_t)\frac{p_t }{m}y’u'(y’)
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.
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.
From the figure, we can find that once the effective federal fund rate (The effective federal funds rate (EFFR) is calculated as a volume-weighted median of overnight federal funds transactions) hits zero, excess reserves increases. Injecting more money would only cause excess money reserves in the liquidity trap.
If future outputs decrease and price is sticky,
An extension. If the price is “sticky” in the short run. In other words, \( \bar{p}_t=\frac{m}{y}\), price cannot fall below a certain threshold. Then, a decrease in \(y_{t+1}\) would end up with decrease in current output \(y_t\). As shown in the following equation,
Future output decrease, then RHS increases, and so LHS has to increase as well. \(\frac{\partial u'(y)}{\partial y}=u”(y)\) is negative. For example, in the isoelasticity form \( u(c)=\frac{c^{1-\sigma}}{1-\sigma} \), and \(0\leq \sigma \leq1\).
In summary, recession in \(t+1\) would bring down \(y_{t+1}\). Then, firstly, decrease \(i_{t+1}\) to 0; secondly, reduce \(p_t\) to \(\bar{y}\) if price is stikcy; and thirdly, drive \(y_t\) decrease in the end. (All those are based on the guess of \(x_{t+1}=x_{t+2}=0\))
In a liquidity trap with sticky prices, outputs become “demand-driven”. The reason is that the Euler Equation is derived from the private sector, and thus \(u'(y_t)=u'(c_t)\) if not replaced with the markets clearing condition in equilibrium. The equation would then show that the increase in the LHS is driven by a decrease in consumption. A disequilibrium starts. Finally, a recession begins if nothing happened to productive capacity.
Intuition
Private sectors initially earn income, say 100, and buy goods for100 as well (Normal situation).
When they receive a “news” that income will decrease in the future from \(y_{t+1}\) to \(y’\), then they all wish to save.
However, in the aggregate, nobody can save, because noboday want to borrow or invest.
So the interest rate, as the benefits of saving, decrease to eventally zero, and private sectors start to hoard cash.
Thus, instead of spending 100, they spend80 and save $20. The demand drives down current outputs.
Role of price stickiness
Initally, current and future outputs (endownments) are all $100. \(y_{t}=y_{t+1}=100\).
A news tells us future output decrease to 80. In the current period, we save20 and spend $80. Same as the above process.
So, current spending is 80 and future spending is100.
If the price is sticky, consume $80 today and price decreases 20% at the same time. Ending up with the same amount of current consumption, \(y_t\). No recession.
If the price is sticky, then agents spend $20 fewer goods in the current. Worse off. And recession.
Here, we would consider the Planner’s Problem that makes social optimal.
Planner’s Problem
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.
Here let’s compare the planner’s Euler equation with the private sector one.
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 \( \pi^*=\frac{-r}{1+r}\).
The implication is that the optimal inflation rate is negative and close to the negative real interest rate.
Cash Credit Good Model
Stokey and Lucas (1987) included the cash-credit good into the cash in advance model.
In equilibrium, markets clear and resources constraints,
$$ y_{t-1}=c_{t-1}^1+c_{t-1}^2 $$
y_{t}=c_{t}^1+c_{t}^2
F.O.C.
$$ u'(c_t^1)=\lambda_t p_t $$
$$ u'(c_t^2)=\beta\lambda_{t+1}p_t $$
$$\lambda_t=\beta \lambda_{t+1}(1+i_{t+1})$$
Combining them we can get
$$ \frac{u'(c_t^1)}{ u'(c_t^2) }=1+i_{t+1}$$
The ratio of marginal utility is equal to one plus the nominal interest rate.
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.
However, the planner problem is that
$$ \frac{u'(c_t^1)}{ u'(c_t^2) }=1 $$
Thus, the optimal rule is to set \(i_{t+1}=0\).
The Euler equation in the steady state (\( c_t^i=c_{t+1}^i=…=c^i \)) is that,
$$ \beta \frac{1+i_{t+1}}{1+\pi_t}=1 $$
By plugging in \(i_{t+1}=0\), \(\pi^*=\beta -1 \), the Friedman rule also holds.
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.
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).
v'(y)=\beta u'(y)\frac{1}{\pi}
Therefore, we can find that,
Money is netural: if change \( m \) (stock of money, or money supply), then output is not affected. For example, if money doubles in all time, the fraction \( \frac{p_t}{p_{t+1}}\) keeps constant. No affecting the real term of output \( y\).
Moeny is not super netural: if change \(\pi\) (inflation rate), then output would change. (y decrases if \(\pi\) increases. That can be analysed by the curvture of \( v\) and \( u\) functions).
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?
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.
The question and answer also state that the neutrality of money bases the key cash-in-advance assumption.
Empirical study examples are as McCandless and Weber (1995)
Reference
McCandless, G.T. and Weber, W.E., 1995. Some monetary facts. Federal Reserve Bank of Minneapolis Quarterly Review, 19(3), pp.2-11.