Econonic Growth, Cycles and Policy

The following notes come from the course Economic Growth, Cycles and Policy, held by Prof. Dr. Hans Gersbach at ETH Zurich.

(I spent one semester working for Hans Gersbach at KOF and I couldn't be more enthusiastic: he’s clever, prepared, and very inspiring.)

Please refer to his list of publications to deep dive in the subject.

Index

• Introduction
• Macroeconomic Schools
• IS-LM Model in Closed Economy
• Aggregate Demand and Aggregate Supply Curves
• Control of Interest Rates and Taylor Rule
• Schools of Thought
• Consumption and investment
• The Solow Growth Model
• Money holding: Inflation and Monetary Policy
• Crises in Market Economnies
• IS-LM Model and Open Economy
• Theories of exchange rate determination
• Technical Appendix

1.1 Introduction

Macroeconomics tries to answer some key questions:

• What determines key macroeconomic variables? _(GDP, unemployment, inflation, credit, consumption, interest rate...)
• How can economic policy influence these variables? _(government spending, tax policies, monetary policy, "wage policy"...)
• How can macroeconomic policy and market regulation operate together?
• Do macroeconomic processes show inherent dynamics? (cycles, trend and instabilities)
• Should economic policy focus mainly on long-run growth or on stabilization of the business cycle?

To start answering them, we firstly start with some stylized facts of Macroeconomics:
Note that, if a variable is procyclical, it moves in the same direction as the overall economy.

Some additional common rules of macroeconomic policy, based on consensus, argue that:

• in case of liquidity problems, the central bank should provide liquidity by raising money supply.
• Government spending and investments of infrastructure should not be lowered during recession periods.
• In downturns automatic stabilizer (first responders" to an economic crisis) should work.
• In the long run monetary policy has to focus on price stability (inflation around 2% a year).
• Real wage increases should not exceed productivity growth whem at the sime time the number of employees remain constant.

1.2 Macroeconomic Schools

The following graph shows the evolution of the macroeconomic thought from 1950 till today.

• Keynesians: focus on the short-run. They argue that markets don't always clear and that government intervention is necessary to manage the business cycle.
• New Kenynesians: they introduce microfoundations _(explaining how people behanve and spend) and nominal rigidities (prices and wages don't adjust instantly, which justifies government intervention).

• Classic: they argue that markets are self-correcting and the invisible hand ensures optimal employment in the long run.
• New Classic: Emerged in 1970, they introduced rational expectations, suggesting that because people anticipate government policy, those policies often become ineffective.

• Monetarists: led by Milton Friedman, they argued that changes in the money supply are the main driver of economic fluctuations.
• RBC Economics (Real Business Cycle): they are a subset of New Classical thought. They argue that "cycles" aren't failures of the market, but rather efficient responses to technological shocks or changes in productivity.

2.1. IS-LM Model in Closed Economy

In the long run:

• prices are flexible
• output is ddetermined by factors of production and technology
• unemployment equals its natural rate

In the short run (IS-LM setting):

• prices are fixed (or change slowly)
• oputput is determined by aggregate demand

In equilibrium, money supply equals money demand:

where $L(i,Y)$ is a function representing the liquitidy preferences, $i$ is the interest rate and $p$ the price level.

If we consider:

where:

• $\alpha$: Sensitivity of money demand to changes in income ($\frac{\partial L}{\partial Y}>0$).
• $\beta$: Sensitivity of money demand to changes in the interest rate ($\frac{\partial L}{\partial i}<0$).

If we write $M^s=p(\alpha Y-\beta i)$ and solve for $i$:

This yields an upward-sloping LM curve, which represents the positive relation between the interest rate and the output.

In this model:

• the Central Bank (CB) controls money supply
• if output increases, the demand for money increases at any $i$.
• for a fixed supply of money, $i$ must increase to lower the demand for money and maintain equilibrium.

It is also important to distinguish between:

• Movements along the curve: caused by a change in output ($Y$).
• Shifts of the curve: caused by changes in Monetary Policy (the Money Supply).

Indeed, increasing money suppòly leads to:

• increase in demand for money to maintain equilibrium $\to i$ must decrease
• for any level of output $Y$, the corresponding $i$ is lower.

The result is that an increase in the money supply leads the LM curve to shift down:

• $\Delta M^s\to \Delta i=-\frac{\Delta M^s}{p\beta }$ or, equivalently: $\frac{\partial i}{\partial M^s}=-\frac{1}{\beta p}$
• $\Delta i\to \Delta M^s=-p\beta \Delta i$ or, equivalently: $\frac{\partial M^s}{\partial i}=-\beta p$.

Assumptions:

• there is only one good that can be used for consumption or investment
• all demanded goods will be supplied (no rationing of demand).

We then model planned expenditures $E$:

where $C$ is the consumption, $I$ physical investments (not financials!) and $G$ the public spending.

When $E=Y$ planned and effective expenditures are equal and we obtain the IS-curve.

If we consider: $E=Y=cY+I_0-\gamma i+G$, we obtain the following:

Even though the IS-curve is often referred to as the goods market equilibrium condition, this is misleading.
In microeconomics, a market equilibrium balances demand and supply via price adjustments (market clearing condition).
The IS-curve does not reflect endogenous price levels.

Intiuitively, if we consider increase in public spending $G$:

• increase in public spending moves the IS-curve to the right, $Y$ increases.
• the LM-curve remains unchanged
• in the short run, interest rate and outputincrease.

In formulas, if we consider $\Delta G$ and we solve both IS and LM equations simultaneously, from the LM-curve we get:

if we substitute this into the IS-expression:

If we consider $\Delta G$ or, equivalently, we take its partial derivative:

Dividing both sides gives us the Fiscal Policy Multiplier:

• $(1-c)$: represents the how much goes into savings (smaller this is, the bigger the multiplier).
• $\alpha \gamma /\beta$: is the monetary drag.
  • If $\alpha$ is high, income growth creates massive money demand.
  • If $\gamma$ is high, investment is very sensitive to interest rates.
  • If $\beta$ is low, it takes a larger increase in $i$ to balance the money market.

• Central Bank increases money supply by setting lower interest rate at given $Y$.
• The LM-curve shifts downward
• The IS-curve remains unchanged
• $Y$ increases (with $G$ unchanged because exogenous, $I$ increases and $C$ increases) and $i$ declines

• It may be caused by financial crisis, wars, terror attacks or whatever
• Households consume less and save more of given income
• the IS-curve odes leftward
• LM-curve remains unchanged assuming, for simplicity, that the demand for money does not change
• $Y$ and $i$ decline.

Possible counter-measures to stabilize $Y$:

• Expansive fiscal policy can shift the IS-curve to initial position
• Easing of monetary policy implies downard shift of LM-curve

2.2. Aggregate Demand and Aggregate Supply Curves

So far we've been using the IS-LM model to analyze the short-run dynamics. However, a change in price level would shift the LM-curve and therefore affects the output.
The (aggregate demand) AD-curve describes the relationship between price and output.

On the other hand, we can model the short-run aggregate supply curve with an upward slope (given by sticky wages and sticky costs).

We now give an example of complete adjustment and we see what happend with a reduction of the government spending.

Firstly, a reduction in $G$ shifts both IS and AD-curves left.

In the short-equilibrium, $Y$ is lower than the initial-natural value $\bar{Y}$.

Thefore, aggregate supply moves down and we see a gradual adjustment of pcises, leading to a downward-shift of the LM-curve that, finally, brings the system back to $\bar{Y}$.

In reality, downward price and wage adjustments are difficult (see for example, the Greek crisis in 2010-2016):

• very long wage contract make them hard to re-negotiate
• resistance of particular groups of employees.

In the case of Greece:

• costs were also partly determined by international markets due to the import
• governments has to rely too much on increasing valued added-takex that leads to an increase in prices.

2.3. Control of Interest Rates and Taylor Rule

Models with LM-curve assume that the CB uses the money supply as an instrument, therefore $i$ will determined endogenously by the equilibrium.
In reality, CB uses short-term interest rates as their policy instruments (rather than the money supply).

The main advantage of targeting interest rating versus money supply is that, in the second case, the difference between fluctuating demand and stable siupply would lead to strongly functuating interest rates.
Nowadays CB counterbalance fluctuations in money demand by adapting the money supply to maintain the short-term interest rate at the targeted level.

CB influences short-terminterest rates $i_0,i_1,i_2...$.

A risk-neutral investors must be indifferent between:

• make a long-term investment with interest rate $i_L$
• make short-term investments in each period $t$ at interest rate $i_t$ (with $i_t^e$ denoting the expected interest rate)

But, in practice:

where ${\tau }_{n-1,n}^P$ denotes the positive term premium.

In this way, by influencing the short-term nominal interest rates, the CB can also influence the long-term ones.

3. Schools of Thought

The following basic RBC-model starts with the following assumptions:

• an infinitely long-living household/agent maximizes the expected value of

with respect to the intertemporal budget restriction, where $u_t$ represents the utility in period $t$ and $0<\delta <1$ the discount factor.

• a representative firm maximizes profits, given its production function.
• there exist no frictions (no price rigidities, no externalities, no asymmetric info, no public goods...)

The aggregate production function is then:

• $Y_t$ aggregate output
• $A_t$ measure for the efficeny of labor (stochastic), example: $\ln A_t=\bar{A}+\rho {ϵ}_{t-1}+{ϵ}_t$, with $ϵ\sim N(0,{\sigma }^2)$, where $\rho$ describes the persistence of the shock.
• $K_t$ capital
• $L_t$ labor input
• $\alpha$ share of physical capital

Intuition: the stochastic component $A_t$ induce business cycles.

$A_t$ is then a measure fo the efficiency of labor over time and relative to capital. It's a catch-all variable that includes levels of health, skill, education and technological knoweldge.

Under technical assumptions - (i) local non-satiation of preferences, (ii) compete markets with no frictions and no externalities, (iii) price taking behaviors - any allocation that forms a competitive equilibrium is Pareto-efficient.

However, the RBC model received several critiques:

• there's no broad evidence that strong technology shocs always affect the economy
• there's no adeguate explanation of lasting recessions
• in typical RBC models, employment fluctuactions comesfrom the household'ìs intertemporal substitution of labor, empirical studies, in reality, show little evidence about that.
• RBC models do not consider monetary shocks, but only technology shocks.
• the assumption of a unique representative agent is not accurate, small heterogenity in utilities can have a strong influence on the model.

Early Monetarism started around 1920 when Irving Fisher applied the quantity theory of money as a tool for quantitative analysis of prices, inflation and interest rates. The key concept behind is that the amount of money circulating in an economy is the primary driver of its health, inflation, and growth.

The quantity equation is the central and formal definition of this model:

• $M$: stock of money
• $V$: velocity of money
• $P$: price level
• $Y$: real GDP

The Quantity Theory (coming from the Early Monetarism) is an interpretation of the previous equation that assumes that $V$ is costant: if people’s spending habits don't change, then any change in $M$ (Money) must lead to a proportional change in $P$ In its extreme form, it predicts the neutrality of money: an exogenous increase of the stock of money is followed by a proportional increase of the price level, without any effect on real variables like consumption, output and investment.

Recall that, the velocity of money is defined as:

In the 50ies, Milton Friedman and Anna Schwartz argued that money might be neutral in the long run, but it has real effects in the short run.
They linked the previous crisis with their theories: before Friedman, most people thought the Great Depression was a failure of capitalism. Friedman and Schwartz proposed that it was actually a failure of the Federal Reserve because Fed didn't put money back into the system in moments of need, resulting in a strong credit crunch and deflation.

If compared with Keynesians, Monetarists show a different approach:

Friedman was also skeptical about discretionary policy, he proposed:

• constant-money growth rule: central banks should increase annually 3% the amount of money supplied
• avoid interest rate targeting and too low (below the natural level)
• distrust of fiscal policy: governemtn spending "crowds out" (reduce) investment and has little impact in the long run.

However, in 1980, monetarism lost his appeal for different reasons:

• the assumption on $V$ was not realistic
• it was hard to understand the definition of money stock (cash, liquid investments, cash equivalents...?)
• money growth was not a good indicator of demand

• M1 (Narrow Money): includes currency, i.e. banknotes, coins and balances that can be immediately converted into currency or used for payments.
• M2 (Intermediate Money): includes M1 + deposits with maturity up to 2 years and deposits redeemable with notice peiriod up to 3 months.
• M3 (Broad money) inclues M2 + marketable instruments issued by Monetary Financial Institutions.

Today's money architecture is structured as follows:

The model, based on Lucas (1972) and Phelps (1970), starts from the following assumptions:

• the economy is populated by a laege number of farmers indexed by $i$
• each farmer produces a specific good
• each farmer selles its good in a competitive market at individual price $P_i$ and market price $P$
• revenues allow each farmer to consume a bundle of goods
• there are two 2 possible shocks:
  • prefrerence shock
  • monetary shock

For simplicity, we also add the assumption that there's perfect information. We then model each farmer's production as:

where $Y_i$ is the output and $N_i$ his amount of labour.

The budget constraint for each individual with consumption $C_i$ is:

The utility function of each individual is then:

Since there's perfect information, each individual knows the aggregate price level $P$ and can solve the optimization problem:

that leads to:

if we consider $n_i=\ln N_i$ and $y_i=\ln Y_i$, we can rewrite the first-order condition as:

Intuition: a farmer increases his production whenever the relative price of his product increases.

If we then assume that the demand function depends on the good's relative price , on real income and on a real preference shock $z_i$, we can write the log-demand as:

where $\eta$ is the elasticity of demand between goods.

Therefore, the aggregate output is given by $y={\int }_0^1{y}_i{d}_i$ and the aggregate price level by $p={\int }_0^1{p}_i{d}_i$.
(Note that the preference shock is assumed to have zero mean: ${\int }_0^1{z}_i{d}_i=0$)

We then consider aggregate demand, given by: $M=PY$ _(intuition: similar to the quantity equation with $V=1$).
In log-terms, this rewrites as:

with $m$ assumed to be stochastic $m\sim N(0,{\sigma }^2)$, representing the monetary shock (central bank's policy is unpredictable, creating random fluctuations in demand).

In equilibrium, the market has to clear, i.g. $y_i=y_i^d$:

solving for $p_i$ yields to:

if we then take the integral over all goods, we get:

Hence, we obtain the equilibrium as: $y=0$ (remember that $y=\ln Y\to Y=1$) and $m=p$.
For the indidual good, this means:

Intution:

• money is neutral and can only affect the price level, output is fixed at the full employment rate $Y=1$. (Classical dichotomy: monetary policy cuase nominal prices and wages to change, leaving the real equilibrium unchanged).
• the equilibrium $y_i$ depends only on the relative price $p_i$ which depends on $z_i$.

We now consider the same model with imperfect information, so each farmer only observes the price $p_i$ of his produced good but not the market price/price index $p$.

In this way, a monetary shock has real effects on output, raising aggregate demand. As the unpredicted increase of money supply generates price movements, individuals $i$ may misinterpret this partially as an increase of relative price index of a good.

In this model, output is given by the Lucas Supply Function:

Intuition:

• deviation of $y$ from its natural level is an increasing function of the surprise in the price level
• if we make use of the aggregate demand function $y=m-p$, we can solve Lucas Supply Function for $p$ and $y$:

This model argues that there's a statistical output-inflation relation: however, if policy markers attempt to take advantage of this (raising output by raising the money growth rate above the trend level), they may cause this relationsbip to break down.

However, models based on Neoclassical economy (like Lucas Supply function) are appropriate to study:

• if monetary policy has large effect if it surprises the public
• if trying to create repeatedly surprises will be futile

The main assumption behind is that prices are not completely flexible (Some models also consider wage rigidities).

Firm's profit maximization yields to the following New Keynesian Phillips Curve:

where ${\pi }_t$ represents the inflation, $y_t$ the output, $\kappa >0$ and $0<\beta <1$.

• an increasing output level leads to higher inflation
• high expected inflation tomorrow yields to higher inflation today. (If firms expect higher prices, they set higher prices today).

On the other hand, households's intertemporal consumption optimization probel yields to the following IS-cuve:

The IS-curve states that output $y_t$ depends negatively on the real interest rate and positively on the expected future output.
Also, if the real interest rate is expected to rise in the future, savings become more attractive and consumption declines.

A forward iteration of the IS-curve and the Philipps curve brings to:

Intuition: firms set their nominal prices based on the expectations of future output:

• output depends on all expected future real interest rates
• infation depends on all expected output levels.

For the analysis of the optimal monetary policy, we also add supply shocks ${\xi }_t$ and demand shocks ${\sigma }_t$, both described by an AR(1) process.

with ${\rho }_1,{\rho }_2\in (0,1)$ measure the persistence of the shocks.

The forward iteration of the IS-curve and the Phillips curve with shocks leads to:

Since output depends now also on expected future demand shocks, shocks on the IS-curve can be completely accomodated. Indeed, at the optimum, the policy maker ajusts the interest rate such that the shocks are offset.
On the other hand, cost-push shocks ${\xi }_t$ (supply shocks) generate a trade-off between output and inflation ($\to$ efficient frontier).

Criticism:

• there's no delay in the response of inflatio to shocks
• inflation is just forward-looking, past inflantion is irrilevant
• in the cases of demand shocks, disinflation can be achieved costlessly and may even generate a boom if anticipated. However, disinflations have historically entailed signficant output losses.

4. Consumption and Investment

The Keynesian consumption function has the form:

We immediately notice that consumption depends only on current income.

We then define the Marginal. Propensity of Consumption (MPC), that lies between $0$ and $1$ as:

Furthermore, the Average Propensity of consumption (APC), that dereases with an increasing income, is defined as:

Intuition: savings are luxuries and can only be afforted with higher income.

However, empirial observation don't proove this result: APC decreases with increasing income in the short term but, as the income increases over a long-term horizon, the APC remains constant.

The Life Cycle and Permanent Income hypoteses try to solve the previous inconsistency.
The Life Cycle Hypotesis was proposed by Ando, Brumberg and Mofigliani in the 1950s

The core idea is that households pursue consumption-smoothing, i.e. they want to spend eqial amount of money in each period of their life (so they borrow when they earn less and save in the opposite situation).

As a result, wealth is increasing until retirement, and is gradually used for consumption afterwards.
Savings, on the other hand, do not depend nuch on current income, but on total lifetime income.

The Permanent Income Hypotesis, proposed by Milton Friedman in 1971, states that income $Y$ is a sum of permament income $Y_p$ and transitory insome $Y_t$:

• Permanent income is the long-term/average income

• Transitory income is the one coming from surprises and fluctuates, with $\mathbb{E}[Y_t]=0$.

Consider the permanent income as your salary for your work contract, hopefully increasing with seniority; and the transitory income as some unexpected bonus, lottery wins, monetary heritage etc....

The theory suggests that, since people want to avoid fluctuactions in consumption, current consumption will mainly depend on permanent income. Moreover, people will use borrowings and savings to smooth their consumption over time.

At the limit, transitory income becomes irrilevant: $C=\alpha Y_p$ with $\alpha \in (0,1)$.

In this model, the APC can also be expressed as:

As a result, for a given level of permament income, people with high levels of transitory income will have APCs lower than average.
Also, over the long-period, the APC should be independent on the transitory income, since $\mathbb{E}[Y_t]=0$ (shocks and fluctuactions cancel each other).

According to this theory then, economic policy can affect consumption only if it affects permanent income (example: tax cuts can increase consumption only if they are not perceived as transitory).

We now consider a simple formal model of optimal consumption/saving.

• Consider an agent that lives for $2$ periods.
• The agent has a period utility $u(C_t)$ with a discount factor $\delta$ with $0<\delta <1$.
  We also assume $u^{\prime }(.)>0$ and $u^{\prime \prime }(.)<0$.
• The agents faces the following budget constraints:
  1. Period: $S=Y_1-C$
  2. Period: $C_2=S(1+r)+Y_2$ where $r$ denotes the real interest rate.

So, if we plug toghether, we end up with the following intertemporal-budget constraint:

The agent then faces the following optimization problem:

At the optimum, the Euler/Tancency condition (the point where a consumer has perfectly balanced their budget to get the maximum possible satisfaction) must hold:

Intuition:

• if $u^{\prime }(C_1)>\delta (1+r)u^{\prime }(C_2)$: the agent should consume more on the first period and save less.
• if $u^{\prime }(C_1)>\delta (1+r)u^{\prime }(C_2)$: the agent should shift consumption to the second period and save more. The model has then these implications:
• Consumption depends on the (expected) discounted income over both periods (in contrast with the Keynesian approach when consumption depends only on current income)
• Consumption typically depend on the real interest rate (changes in $r$ lead to income and substitution effect).

In a 2-period model, the government levies taxes $T_t$ and spends $G_t$ in public goods (non rival and non-excludable).

• In the first period, the government can end up in a fiscal defitic (or surplus if $D<0$):

• in the second period, the government has to repay the deficit plus the interests:

If we combine the two equationsm,we obtain the Intertemporal Budget Constraint:

Intuition: the present value of governemtn spending equals the present values of tax incomes.

As a consequence:

• if the governement decides to lower taxes in the first period by $\Delta T$ keeping spending constant
• then, in the second period, it has to raise taxes by $\Delta T (1 + r)
• the tax cut increases the income of the houdeholds by $\Delta T$ in period 1 and decreases it in period 2 by $\Delta T(1+r)$
• lifetime income is given by $Y_1+Y_2/(1+r)$, so, after the changes in taxation, it becomes:

• according to the Fisher model, in each period, the households consume the same amount as before the tax policy, since lifetime income was NOT affected.

This is called Ricardian Equivalence Theorem.

Intuition: househols have more money in the first period, but they know they will have less in the second, so they save to pay higher taxes in the future.

Despite this theorem, governments often try to simulate consumption by reducing taxes and keeping spending costant. These policies are NOT completely futile, since, in practice, the Ricardian Equivalent Theorem doesnt hold perfectly. (The increase in households' current income increases their consumptoion by some positive fraction of the additional income).
This comes from the fact that, in most cases, household are myope: they notice an increase in income, but they are unable to predict a future increase in taxation (households are not 100% rational and follow simple rules).

However, even in theory, there are some possible deviations from the Ricardian Equivalence:

• Limited lifetime: the government can take long periods of times to repay its debt, shifting the tax burden to future generation and redistributing consumption from a generation to another.
  • For this reason, if households took the utility of future generations into account, they should save m,ore and NOT expand consumption for the expected future tax burden.
• Distortionary taxation: not always who benefit the tax reduction will also repay the tax burden (the governement can levy taxes from different parts of the population)

5. The Solow Growth Model

Real GDP per capita is often considered the most important indicator for average standard of living.

(Is this really the case? Luciano Canova in this book answers this: is the PIL the right measure to measure happiness?)

More in detail:

• Economic growth is the increase in value of all goods and services produced by an economy. It is conventionally measured as the percent rate of increase in real gross domestic product.
• GDP is defined as the total market valoue of all final goods and servied produced in a country in a gived time period. It's defined by the National Income Identity:

• A closed economy (no trade)
• One good that can be consumed or used as capital
• Houseolds save a costant exogenous fraction $s$ of their income (we are not intered in why they save $s$, we simply put that value into the model)
• Rate of technological progress is exogenous
• No taxes, no subsidies _($\approx$ no governement)
• Discrete time $t=0,1,\mathrm{2...}$

Output $Y_t$ is produced according to the aggregate production function:

• $A_t$: efficiency of labor
• $L_t$: size of the labor force $⟹A_t{L}_t$ is the effective labor
• $K_t$: stock of capital

Further assumptions:

• constant return to scale (CRS): $F(\lambda A_t{L}_t,K_t)=\lambda F(A_t{L}_t,K_t)$
• $F^{\prime }>0,F^{\prime \prime }<0$ for each argument

We model technological progress as $(A_{t+1}-A_t)/A_t$ and we consider as exogenous.

Most of the times, in the Solow Model, we assume a Cobb-Douglas production function:

(it can be easily checked that this function respectes all the assumptios).

Because of the costan return to scale, the productionm function can be written in intestive form:

if we now introduce:

• $f(x):=F(1,x)$
• $k:=K_t/(A_t{L}_t)$
• $y_t:=Y_t/(A_t{L}_t)$

we obtain:

(This makes easier to solve the model since we only have one argument).

We now introduce some assumptions on the population growth and the technological progress:

• population (coinciding with labor force) at $t=0$ is $L_0$
• population grows at a costant rate $n$:

• technological level at $t=0$ is $A_0$
• technological level grows at a costant rate $g$: $A_t=A_0(1+g{)}^t$

The capital stock increases by investment $I$ and decreases by depreciation at a costant rate $\delta$.

We then obtain the foundamental law of motion of the Solow Model:

In a closed economy without government spending:

If we also assume that $S_t=s{Y}_t,s\in (0,1)$ is costant over time, this implies:

We now take some time to describe the variable. Given the ones in absolute form ($K_t,L_t,A_t,Y_t,C_t)$, we have the Per Capita Values (simply dividing by population) as:

• ${\bar{k}}_t:=\frac{K_t}{L_t}$
• ${\bar{y}}_t:=\frac{Y_t}{L_t}$
• ${\bar{c}}_t:=\frac{C_t}{L_t}$

and the Intensive Form (simply dividing by effective labor) as:

• $k_t:=\frac{K_t}{L_t{A}_t}$
• $y_t:=\frac{K_t}{L_t{A}_t}$
• $c_t:=\frac{K_t}{L_t{A}_t}$

If we divide the foundamental law of motion by the effective labor too, we obtain the Capital Accumulation Equation in intensive form:

Where the last equality is true if we consider the Cobb-Douglas function and it represent the Foundamental Law of motion in intensive form.

$(\delta +n+g+ng)k_t$ is the break-even investment (the min investment needed to prevent the capital stock in intensive form from shrinking). If actual investment $s{k}_t^{\alpha }$ is equal to the break-even investment, $k_t$ is costant over time.

A diminishing marginal product of capital implies that $k_t$ converges to where actual investment equals break-even investment.
In the long run, $k_t$ is costant over time, reaching a steady state

The long-run equilibrium ($k_{t+1}=k_t=k^{\ast }$) implies the following mathematical solution:

that brings the following properties:

• $y^{\ast }$, the ratio of output fo effective labor, is costant:

• $c^{\ast }$ is costant

• $k^{\ast },c^{\ast },y^{\ast }$ do NOT depend on the initial capital $k_0$.
• Capital per capita $\bar{k}$, output per capita $\bar{y}$ and consumptio per capita $\bar{c}$ grow at rate $g$ of technological progress:

• $K,Y,C$ grow at rate $g+n+gn\approx g+n$ for $g,n<<1$:

We can also se that changes in $s,n,\delta ,\alpha$ have only effects on per-capita steady-state value.
A change in $g$ has growth effects on per-capital steady-state values.
A shock to the capital stock has temporary (but no long-run) effects.

Example: the Second World brought a shock to the capital stock:

We now ask which saving rate $s^{\ast }$ maximizes steady-state consumption $c^{\ast }$:

whoose solution is characterized by:

which is called the Golden Rule of Capital Accumulation.

Note that, for the Cobb-Douglas production function, $s_{gold}=\alpha$ and that an economy with $s>s_{gold}$ is dynamically inefficient.

Implications of the Solow Model:

• long-run growth per-capita is only driven by technological progress
• in the long-run, capital accumulation is a consequence of growth in technology
• in the short run, growth is faster for countries far away that their steady state.

6. Money holding: Inflation and Monetary Policy

In barter, each side must want exactly what the other side offers. Money solvey this by provding a generally accepted medium of exchange.

Fiat money has value because people expect it to be acceptedby future exchange. This expectation is self-fulfilling as long as confidence is maintained.

In economics and monetary economics, money is typically defined by its functions:

• unit of account
• medium of exchange
• store of value

A practical definition: money is the set of instruments that are widely accepted at par (at par means: $1$ dollar is $1$ dollar, no due diligence or questions asked) to settle obligations and make payments.

Note that, according the economics and monetary definition, Bitcoin (and other criptocurrencies) are not a form of money: they usually miss te unit-of-measure function and the medium one.

In the current economic system, cash doesn't play a major role anymore

• In Switzerland _(december 2025): banknotes and coir Sns represented the $9.7\%$ of aggregate money M1.

A bank trasfer is essentialy: a message $+$ a balance sheet update
It can be broken down in two steps:

• Clearing: who owes what to whom?
• Settlement: how are obligations discharged (with what asset)?

In the two-tier system, interbank settlement uses central bank reserves.

Let's condier the following transaction: Alice trasfers $10$ to Bob

1. Alice instructs Bank A to transfer 10 to Bob.
2. Bank A debits Alice’s deposit account by 10.
3. Bank A sends the interbank payment instruction into the payment system.
4. At the central bank layer, Bank A’s reserve account is debited by 10 and Bank B’s reserve account is credited by 10.
5. Bank B credits Bob’s deposit account by 10.
6. The payment is complete once the interbank obligation has been settled in central bank money.

• Incoming payments from other banks
• Borrowing from other banks or the central bank (a bank borrows from another in the interbank market, a bank borrows from the central bank through central bank lending operations against collateral).
• Selling assets.

Banks can redistribute reserves among themselves, but only the central bank can create reserves for the system as a whole.

• Central bank money
  • cash is created physically through the issuance of banknotes and coins
  • reserves are created electronically through central bank operations (lending, asset purchases)
• Commercial Bank Money (deposits)
  • Created electronically when banks expand their balance sheets and credit deposit accounts (main case: banks make loans, some other options include when banks buy assets from non-banks).

The Bank balance sheet can be simplified in:

• assets: loands, securities, reserves
• liabilities: depotis, wholesale funding, equity.

Example: a new loan of $x$ to a customer increases assets and liabilities by $x$. Iin this case $x$ units of money (deposits) are created.
The bank doesn't need to "find depotis first" to make the loan.

Bank can create deposits ($\approx$ give loans) and expand their balance sheet, but there're some limits:

• capital and leverage constraints
• liquidity constraints
• profitability and risk: credit risk, funding cost, competition...
• borrower demand
• monetary policy

In securities trade, the buyer owes cash and te seller owes securities. If te two egs are exchange sequentially, one side may deliver while te oter side fails. This creates the principal risk problem in securities trade_, since there's exposure to the full value of te trade, not just the "replacement" cost.

Delivery versus Payment (DvP) was designed to solve this problem.

Current frictions in payments and settlmenet:

• cross-border payments can be slow, costly and opaque (multiple intermediaries, different time zones etc...)
• settlmenent is often fragmented across different ledgers and insititutions (example: in DvP cas leg and asset leg may sit on different systems)
• real-tie 24/7 settlment is not universal (many systems are still limited to operating hours)
• programmability is limited (conditional payments, automated settlmenent and integrated DvP are difficult to implement)
• access to central bank money is restricted (households can old cash, not reserves, for digital settlment, most users rely on commercial bank money)

$⟹$ The current system works very well in many domestic settings, but it is not frictionless, especially across borders.

Crypto-assets _(like Bitcoin) are not typically a claim on a issuer: no underlying liability or redemption at par.
They are native digital assets: trasfer and settlment take placde on its own ledger and its value is market-determined and often highly volatile.
Generally speaking, it often functions more like a speculative asset tan as money.

A stablecoin is a digital token designed to maintain a stable value relative to a reference asset (usually a fiat currency).
The biggest ones are currently (march 2026) Theter (USDT) and USDC, both tied one to one to the US dollar.

They are typically issued by a private equity and backed by reserve assets, together with a redemption premise. In the past, there were also some algorith-backed stablecoins _(see for example TerraUSD).

Unlike unbacked-crypto assets, stablecoins aim at price stability in the unit of account but, unlinke deposits:

• they operate outside the banking framework
• they are tradable on the secondary market
• transfers can settle without interbank settlement in central bank money

Currently (march 2026) the total stable coins market is about $320$ Bn USD. The two biggest ones USDC and USDT account for more than 90% of te market cap.

They have been "relatively" stable in the past:

But stress episodes happened. In March 2023 for example, USDC came under stress during the Silicon Valley Bank failure and a portiio of Circle's reserves (USDC's issuer) became temporarily uncertain.

What are stablecoins currently used for? (March 2026)

• Within the crypto-markets:
  • as a relatively stable trading and settlment asset
  • as a collateral and liquidy in DeFi (decentralized finance) -as a place to "park" value between more volatile positions
• Beyond crypto-markets:
  • cross-borders trasfers
  • dollar access in some countries
  • digital payments on a token-based system
  • to bypass restrictions in traditional financial systems (example: capital controls, anti-money laundering (AML) or combating the financing of terrorism (CFT) safeguards)

$⟹$ Still, there are several core risks: reserve quality, redemption/run dynaimcs, governance and integrity, AML issues and bank disintermediation.

Note that a stablecoin-dominated system would be a distinc monetary architecture from today's one, with privately issued settlmenet assets potentially operating outside the traditional two-tier system.
An alternative is CDBC + tokenized deposits that is mainly a modernization of existing forms of money within the current architecture.

A tokenized deposit is still a bank deposit (a calim on regulated commercial bank) but it's represented as a digital token on a programmable platform, often using DLT. It preserves both the two-tier monetary system while still changing the transfer and settlmenet infrastructure, promising for greater programmability and potentially more integraded settlement in tokenized markets.

This would be part of a broader trend: not only money, but also bonds and contracts can be brought onto programmable digital rails.
Indeed a toekn can bundle both information about an asset or claim and the rules governing how it can be used.

If the target is not reached $⟹$ the funds are automatically returned, reducing the risk of fraude or delay.

wCBDC are Digital Central Bank Money for banks and financial institutions only, they are economically close to the central bank reserves,but potentially with new functionalities enabled by the tokenization.

Retail CBDC are a digital form of central bank money for the general public, often with a token-based design, an intermediate model and more privacy and offline capabilities.

Often described as "digital cash", but its design can differ substiantally depending on policy goals and they would be more than a new payment technology _(a now monetary architecture choice!).

Currently (april 2026) the world is still not ready for this system:

• avanced pilots in wholesale market infrastructure for wCBDC are ongoing
• The SNB Prokect Helvetia on SDX and the Eurosystem DLT work
• In China, laege-scale pilots (e-CNY) for rCBDC are in place
• The US is still hostile to rCBDC.

7. Crises in Market Economies

8. IS-LM Model and Open Economy

The monetary policy goals can be summarized, as in the standard US textbook by Frederic Mishkin, in:

• low unemployment (high GDP)
• economic growth
• price stability
• interest rate stability
• financial market stability

The problem is that this multitude of goals often anno be reached simultaneously, requiring some trade-offs.

We speak by of discretionary policy when the central bank can choose its policy period by period.
We speak of rule-based monetary policy when policy is constrained by a previously announced rule or commitment.

At first look, a discretionary policy seems superior to a less-flexible rule based, however, when current policy affects private expectatios for the future, discretion may lead to inferior outcomes.

(Since the early 1990s, many countries have moved towards a more rule-based monetary system, largely independent by the government).

On the other side, a dicretionary-policy has some major drawbacks and the most important is the so-called Time Inconsistency Problem.

The classic analysis of this problmes comes from Kydlan and Prescott, that, for this achievement, received the Nobel Prize.

A plan is time inconsistent if it is judjed optimal at $t_0$, but at $t_1$ the same decision maker prefers to revise the plan, even though no new relevant information is available and the current enviroment has not yet changed.

(This problem is also linked with credibility: announcements are no longer credible if one knows that they will later changed).

When a central bank knows that it cannot influencer real variables in the long run but it can influence inflation rate, it will ex-ante formulate its poity in a such a way that it results in a optimal inflation.

However, if the central bank can influence real-variables in the short run, it will be tempted to deviate from the previous policy.

Therefore, it is no longer time-consistent.

The equilibrium is thus an inflation rate which is higher than socially desired, but without changing output in the long run.

(The solution would be a independent CB with rule-based monetary policy).

The model a CB that bases in actions on a social welfare function (discrete-monetary policy).

The CB can directly coose an infòation rate $\pi$ for the short-run.

The private sector is the second player and it is described by rational expectations, we then model a strategic game with simultaneous decisions.

The social welfare funtion can be defined as:

or as a loss function as:

with:

• $\alpha >0$: costant
• $\pi$: inflation rate
• $y$: output
• $\tilde{\pi }$: target inflation _(we assume here $\tilde{\pi }=0$)
• $\tilde{y}$: target production.

The Philipps curve summarizes the interplay between real and nominal variables:

with:

• $y^N<\tilde{y}$: natural output (\tilde y = \lambda y^N$with$\lambda > 1$)
• $\beta >0$: costant
• ${\pi }_E$: expected inflcation

The prive sector has rational expectations, that is:

This immediately im,plies that (via the Philipps Curve) in equilibrium $y$ must satisfy:

Unfortunately, with discrete-monetary policyt, this is not what we get in equilibrium!

So let's go back to the loss function and let's substitute the Phillips curve:

We now minimize this function with respect to $\pi$:

that leads to the Monetary policy reaction function:

Together with the reaction function (expecations) of the private sector, ${\pi }_E=\pi$, the equilibrium is:

The equilibrium ouput is then: $y=y^N$.

Therefore, the optimal scenario ($\pi =0,y=y^N$) is not reached in equilibrium, but rahter a solution that is inferior.

• The inferior solution is time-consistent, while announging $\pi =0$ is cleary not.
• There's a suboptimal high inflation is the equilibrium, called inflation-bias
• Intuition: a discretionary policy provides an inferior result and a monetary policy dependent on ad-hoc governement decisions is therefore not optimal.

For these reasons, sine discretionary policy does not provide the desired results, it is socially-optimal to delegate monetary policy to an independent institution, that it then credibily announce $\pi =0$.

Along these lines the constitutions of the ECB was drafted.

Possible extensions of the Barro-Gordon Model:

• Look at repeated games instead of a one-shot game
• Include information imperfecions
• Take supply-shocks into account
• Model wage settings

$⟹$ also with these changes, the results presented here remain qualitatively very robust.

9. Theories of exchange rate determination

10. Technical Appendix