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Table of Contents

Lecture 16Utility6Indifference Curve6Marginal Rate of Substitution6Notations6Consumer Budget Constraint7Government Budget Constraint7Government Savings8Economy-Wide Resource Frontier8Consumer Optimisation9Price Ratio9National Savings10Policy Experiment: Effect of change in tax on national savings.10Ricardian Equivalence11Effects of Tax Policy – In depth analysis11Lump Sum Tax11Proportional/Distortionary Tax11Lecture 212Inifinite Period Framework12Notations12Consumer Optimisation using Lagrangian13Case 1: case13Case 2: case13Consumer Optimisation using MRS15Steady State15Lecture 316Roles of Money16Money in Utility16Bonds16Notations16Money and Bond Market16Open Market Operations17Expansionary and Contractionary Monetary Policy17Money in Utility (MIU) Function18Notations18Consumer Optimisation19Budget Constraint19Lagrangian19Analysing Monetary Policy21Money Shock21Money and Inflation in the Long-Run22Lecture 423Monetary and Fiscal Policy Interactions23Fiscal Authority23Monetary Authority23Consolidated Government Budget24Active vs Passive Policies24Fiscal Policy24Monetary Policy24Definition24Present Value Analysis24Seignorage Revenue25Lifetime Consolidated Government Budget Constraint25Ricardian Policy26Definition26Cases of Polices26Case 1: Fiscal Theory of Inflation26Case 2: Fiscal Theory of Price Level26Case 3: Mix of FTI and FTPL27Inflationary Finance27Fiscal Theory of Inflation27Fiscal Theory of Price Level27Lecture 528Definitions28Exchange Rate28Appreciation and Depreciation28Foreign Reserves28Foreign Reserves Changes28Types of Exchange Rate28Floating28Fixed28The Argentine Currency Peg29Singapore Foreign Exchange Policy29Building Blocks of the Fiscal Theory of Exchange Rates301.Money Demand Function30Nominal and Real Exchange Rates302.Purchasing Power Parity303.Interest Rate Parity31Summary of Auxillary Assumptions31Government Budget Constraint31Assumptions of government324.Government Budget Constraint32Notations32Four Building Blocks of the Fiscal Theory of Exchange Rates33Definition of Balance of Payments33Changes in Exchange Rate33Case 1: Fixed Exchange Rate in Place and Individuals Expect it to Remain in Place34Case 2: Unanticipated, one-time devaluation35Balance of Payment Crisis36Definition36Case 3: Country is Maintaining Fixed Exchange Rate and Running Fiscal Deficit Every Period36Stages of BOP Crisis38Lecture 639Model of Production39Production Function39Returns to Scale40Intensive Form of Production Function40Allocation of Resources40Solving the Model41General Equilibrium41Capital and Labour Share42Analysing the Production Model43Model where TFP is Same Across Countries43Improving the Model44Implied Total Factor Productivty44Understanding Differences in Output Across Countries45Implied TFP Relative to the US45TFP vs Capital in Explaining Differences45TFP Differences46Evaluating the Production Model47Lecture 748Solow Growth Model48Capital Accumulation48Solow Model Set-up49Differences between Solow Model and Production Model49Solving the Solow Model50Transition Dynamics of Output51Solving for Steady State Capital51Solving for Steady State Output52Analysis of One-Time Change in Productivity52Lecture 853Explaining Capital in Solow Model53Differences in Output per Capita53Steady State and Problems with the Solow Model54Capital Accumulation54Population Growth in the Solow Model54Solving the Per-Capita Model55Steady State Growth55Analysis of Changes in Variables56Change in Savings/Investment Rate56Change in Depreciation Rate57The Principle of Transition Dynamics57Strengths and Weakness of the Solow Model58Saving Rate and Consumption58Dynamics of Steady State Consumption and Saving Rate58Varying the Savings Rate59Problems with Changing the Savings Rate59Lecture 960Romer Model60Returns to scale60Example of Fixed Costs and Returns to Scale61Returns to Scale in Romer Model61Pareto optimal allocation62Ways to Promote Innovation62Romer Model Set-up63Production Function63Ideas and Labour63Flow of Ideas63Solving the Romer Model64Balanced Growth64Growth in the Romer Model65Case Study: A Model of World Knowledge65Analysis of Changes in Variables66Change in Population66Change in Research Share 66Growth Effects vs Level Effects67Model with Diminishing Returns to Idea67Combining Romer and Solow Models67Growth Accounting68Summary68Combined Romer and Solow Models Set-up69Solving the Combined Model71Transition Dynamics71Combined Model Output Function (Ratio Scale)72Analysis of Changes in Variables72Change in Research Share 72Change in Savings/Investment Rate73Change in Depreciation Rate74Miscellaneous Items75Growth Rate75Growth Rate75Properties of Growth Rates75Ways to Calculate75Summary of the Different Models76Set -up of Model76Solving the Model77Growth Rates78Why Combined Solow-Romer Model?78

Lecture 1

Utility

Let . Then, utility is increasing at a decreasing rate. That is,

· Strictly increasing in each good individually

· Diminishing marginal utility in each good individually

Indifference Curve

The set of all consumption bundles that deliver a particular level of utility. That is,

Marginal Rate of Substitution

is the maximum quantity of one good a consumer is willing to give up in order to obtain one extra unit of the other good. Graphically, it is the (negative of the) slope of an indifference curve.

Notations

· consumption in period

· nominal price of consumption in period

· nominal income in period

· real income in period ,

· nominal wealth at the beginning of period /end of period

· nominal interest rate between periods

· real interest rate between periods

· net inflation rate between periods 1 and 2

· lump sum nominal tax in period

· lump sum real tax in period

· real government spending in period

· government asset position at the beginning of period /end of period

Consumer Budget Constraint

Consider a 2 period model.

The consumer’s lifetime budget constraint is given by its present discount value of lifetime expenditure and disposable (after-tax) income.

Assuming ,

Government Budget Constraint

Assume the government exists for 2 periods, and has spending in each period it can finance via taxes and/or issuing government debt/assets.

The government’s lifetime budget constraint is given by its present discount value of lifetime expenditure and income.

Assuming ,

Government Savings

A government’s savings during a given period is the change in its wealth during the period. That is, in period , its savings is . A government runs a

· Fiscal surplus if its savings is positive, Borrowing < Spending

· Fiscal deficit if its savings is negative, Borrowing > Spending

Economy-Wide Resource Frontier

From the Consumer lifetime budget constraint and the Government lifetime budget constraint

We have the economy-wise resorce frontier

Assuming ,

Consumer Optimisation

Method 1: = Price Ratio

Method 2: Lagrangian

Given , we maximise . We thus set up the lagrangian,

Solving for the first order conditions, at maximum, we have,

We thus have the same result as method 1,

Price Ratio

Note that is the price ratio because

· is the cost of consuming 1 unit in period 1 from an opportunity cost point of view in period 2, that is, in terms of period 2 goods

· is the cost of consuming 1 unit in period 2

National Savings

·

·

·

Note that is not affected by . The national savings determine the market equilibrim real interest rate , which appears in the consumer’s budget constraint.

Policy Experiment: Effect of change in tax on national savings.

To determine if a change in tax affects national savings, since and remain unchanged, we check if changes with . Suppose and are unchanged, and decreases. Thus, must rise.

Thus, from the government budget constraint, we have that

Now, to determine if a change in will affect , we must check if the consumer’s utility and budget constraints change. Utility captures a person’s preferences and a change in tax will not affect it. As for the budget constraint, we have

With a change in tax, similar to the above, since we have shown ,

Thus, assuming remains unchanged, both utility and budget constraint are unchanged, and so do the optimal consumption choices. Thus, national savings are unchanged.

Ricardian Equivalence

For a given PDV of government spending, that is , the timing of lump-sum tax changes do not affect consumption or national savings.

This is because rational consumers understand that a tax cut today means a tax increase in the future (since government spending is unchaged). Consumers thus save the entire tax cut in order to pay higher taxes in the future.

Effects of Tax Policy – In depth analysis

Lump Sum Tax

A tax whose total incidence (total amount paid) does not depend in any way on any decisions/choices an individual makes. This is seen earlier.

Proportional/Distortionary Tax

A tax whose total incidence depends on the decisions/choices an individual makes. Suppose there is a consumption tax rate of . Then, assuming PDV is unchanged,

and the price ratio/slope of budget constraint is

If the timing of tax rate changes, for instance, a tax rate cut today means a tax rate increase in the future, since the budget line slope changes, consumption does change, and so do national savings.

Ricardian equivalence does not apply here because the consumer budget constraint has changed.

Lecture 2

Inifinite Period Framework

Suppose there is only one real asset, stock, and a infinite number of periods.

Notations

· consumption in period

· nominal price of consumption in period

· nominal income in period

· real income in period ,

· number of stocks held at the beginning of period /end of period (wealth brought to period )

· nominal price of a unit of stock in period

· nominal dividend paid in period by each unit of stock held at the start of period

· inflation rate between period and period

· Let denote the consumer’s subjective discount factor. This captures how much the consumer cares about future utility relative to the current period. The lower it is, the less the consumer values the future, and is more impatient.

We have the consumer’s life-time utility to be either one of

and the budget constraint in period to be

Note that in period , the consumer has 2 sources of income.

and the interest rate is

Consumer Optimisation using Lagrangian

Case 1: case

We set up the Lagrangian as follows,

Case 2: case

We set up the Lagrangian as follows,

Note that we can find the general solution by treating the lagrangian as a 2-period problem. Hence, we compute the necessay first order conditions. At maximum, we have,

Case 1: case,

Case 2: case

Thus from the 2nd equation,

Note that we derive the above using Case 2. We can get the same result using Case 1.

Solving further, from the 1st and 3rd equation, since

we have

Thus, since

Thus, note that consumption and inflation affects stock prices. In fact, any factor that affects consumption and inflation also affects stock prices, such as monetary and fiscal policy, globalisation, and GDP.

Consumer Optimisation using MRS

Note that, by treating the optimisation as a 2-period problem (as we did earlier), we can also solve for the optimality conditions using and price ratio. From the above,

Since the between period and period consumption (ratio of marginal utilities) is

and the price ratio is

Fisher’s Equation

Price Ratio as determined earlier

We therefore have

Steady State

In our model, it is when all real variables settle down to constant values. Nominal variables, however, may still move over time. Assume and . Then,

can be seen as the price of consumption in a given period, in terms of consumption in the next period. Thus, in the long run, can be seen as the degree of impatience in an economy.

We can use to explain why positive interest rates exist in the long run. However, in the short run, we can have negative interest rates.

Lecture 3

Roles of Money

· Medium of exchange: Eases double coincidence of wants

· Unit of account: Common language for prices to be quoted in

· Store of value: Money will not perish in a long time

Money in Utility

Suppose now that money directly yields utility. Then, we have money in utility function (MIU),

Note that it is real money, , not nominal money,, in the function.

Bonds

Suppose that now there are bonds in the economy. Further suppose that these bonds are all single-period zero-coupon government bonds.

Assume that, bonds are bought in period at price , at face value of . Thus,

Notations

· nominal price of a one-period bond

· nominal interest rate between period and period

· face value of bond repaid in period , normalised to 1

and are inversely related. If price of bond increases, then the return on it (interest rate) falls.

Money and Bond Market

Short term bond markets and money markets are tightly linked to each other, and are linked by

Interest rate can be thought of in 2 ways:

1. Interest payoff of a bond

2. Opportunity cost/Price of holding money, since each unit of wealth held as a dollar is a unit of wealth not held as a bond, which entails the loss of the chance to earn interest

Open Market Operations

The government sets a certain number of bonds to be sold. Some are bought by the central bank, and the rest are left in the open market for the private sector.

The central bank can adjust money supply by changing the amount of bonds in the open market through open market operations.

Expansionary and Contractionary Monetary Policy

· Central bank buys bonds from the financial sector by printing new money

· Open market supply of bonds falls, and money supply in the money market rises

Thus, falls. This is Expansionary Monetary Policy.

Similarly, if the central bank sells bonds to the open market in exchange for money, it reduces money supply. increases, and this is a contractionary monetary policy.

Money in Utility (MIU) Function

In the money market equilibrium,

Notations

General

· consumption in period

· nominal price of consumption in period

· nominal income in period

· real income in period ,

· inflation rate between period and period

Stock

· real stocks held at the beginning of period /end of period

· nominal price of a unit of stock in period

· nominal dividend paid in period by each unit of stock held at the start of period

Money

· nominal money held at the beginning of period /end of period

Bond

· nominal bonds held at the beginning of period /end of period

· nominal price of bond in period

· nominal interest rate on bond purchased in period and pays off in period

Consumer Optimisation

Budget Constraint

The budget constraint in period is

Denote

Lagrangian

We want to find

We set up the Lagrangian as follows,

Solving for the first order conditions, at maximum, we have,

For the last one, note that

Therefore, we have

Thus from the 2nd equation,

From the 3rd equation,

From the 1st and 4th equation, and since ,

Note that the price of the short-term bond is the pricing kernel. Most, if not all, asset prices are connected to bond prices. In finance, the pricing kernel reflects the price/return of the least risky asset in the economy, in this case, bonds.

The Consumption-Money Optimality Condition

can be used to determine the real money demand in period , . Note that money demand is increasing in and decreasing in .

Analysing Monetary Policy

Money (and monetary policy) is neutral if changes in the money supply (changes in monetary policy) have no effect on the real economy.

· New Keynesian view

Money is non-neutral (because prices are rigid/sticky, often for long periods of time)

· RBC view

Money is neutral (because prices are not rigid/sticky in any important way)

Money Shock

Consumers set and “planned” . The central bank sets “actual” . If

then a money shock has occurred.

· : positive money shock

· : negative money shock

In this scenario, assume does not adjust. However, according to different views, and (note that prices are nominal, while inflation is real) adjust differently. Since ,

· New Keynesian view

· cannot adjust (in period ) because prices are sticky

· A positive (negative) money shock leads to a rise (fall) in (real variable)

· Money (and monetary policy) is hence not neutral

· RBC view

· can adjust (in period ) because prices are not sticky

· A positive (negative) money shock leads to a rise (fall) in (nominal variable)

· There is no change in , and no reason to change since they reflect optimal choices

· Money (and monetary policy) is hence neutral

Money and Inflation in the Long-Run

We want to find out what determines inflation in the long run (steady-state). Assume that

Then, from the Money Demand Functions in periods and

From the defintion of inflation,

we similarly define money growth as

Thus,

Imposing a steady state, , we have

Thus, in the long run, rate of money growth = rate of inflation . In the steady state, inflation is determined solely by how quickly the central bank changes nominal money supply.

Therefore, in the short run, changes in monetary policy might affect consumption and GDP (depending on whether we use Keynesian or RBC point of view). However, in the long run, changes in money growth only affects inflation.

Lecture 4

Monetary and Fiscal Policy Interactions

Monetary policy and fiscal policy interact with each other, and place restrictions on what the other can achieve.

Fiscal Authority

· Controls government spending

· Collects taxes

· Issues (sells) new bonds for financing needs

· Receives profits from central bank (because it legally charters central bank)

The fiscal authority budget constraint in period is given by

· price of bonds sold in period

· total amount of one period bonds government sells in period

· total amount of one period bonds government must repay in period at face value 1

· receipt (profits) turned over from the central bank to the fiscal authority in period

Monetary Authority

· Controls money supply of economy by engaging in open market operations

· Turns over any profits it earns to fiscal authority

The monetary authority budget constraint in period is given by

· amount of one period bonds central bank buys on the open market in period

· pay-offs of one period bonds central bank receives in period with face value 1

· change in money supply engineered by the central bank in period

Consolidated Government Budget

From

And by combining and eliminating , we get

Let , the total quantity of fiscally-issued bonds held by the private sector (quantity available on the open market). Then, the consolidated government budget constraint becomes

Active vs Passive Policies

Fiscal Policy

Has 3 instruments

· Government spending

· Taxes

· Bonds

Monetary Policy

Has 1 instrument

· Money supply

Definition

A policy authority is active (passive) if every (not every) instrument at its disposal can be completely freely chosen, without concern for the consolidated government budget constraint.

An active (passive) authority does not (must) engage in policy in such a way as to make sure the consolidated government budget constraint balances.

Present Value Analysis

There are limitations or restrictions that each policy-setting authority places on the actions of the other. The period choices of one policy authority may restrict the choices of the other policy authority in periods

Seignorage Revenue

Dividing the consolidated government budget constraint by ,

is the seignorage revenue, the real quantity of resources the government raises for itself through the act of money creation. This is important in developing countries beause of poorly-developed tax collection systems and corruption.

Lifetime Consolidated Government Budget Constraint

After much hardwork and derivation, we obtain

That is, the real value of government debt that must be repaid at the start of period , , is equal to sum of present discounted seignorage revenues and present discounted fiscal surpluses.

This debt must be repaid by either period and/or later

· seignorage revenues (money creative policies – nominal repayment)

· fiscal surpluses (fiscal adjustment – real repayment)

or both. “or later” implies rolling over maturing debt, which is borrowing anew to repay debt due.

However, money creative typically sparks inflation (Friedman effect), as the expansion of money supply reduces the value of each unit of money (prices of goods rise).

Ricardian Policy

In considering dyanmic interatctions between fiscal and monetary policy, most relevant case is usally when the fiscal authority is active (the leader).

Definition

A Ricardian fiscal policy is in place if the fiscal authority sets its planned sequence of tax and spending policy to ensure that the present-value consolidated GBC is balanced

A non-Ricardian fiscal policy is in place if the fiscal authority sets its planned sequence of tax and spending policy without regard for whether the present-value consolidated GBC is balanced.

Cases of Polices

Suppose that both the fiscal and monetary authority have planned the sequence for and and respectively. If the fiscal authority changes the precise timing of collection in a Ricardian way, then the monetary authority does not need to react.

However, if the fiscal authority does so in a non-Ricardian fiscal policy, then the central bank must balance the consolidated GBC. Assuming falls,

Case 1: Fiscal Theory of Inflation

· Monetary authority alters its plan for (increase)

· Reacted passively to ensure present value consolidated GBC holds

· Money creation leads to inflation, but the central bank can print money at anytime, so inflation can occur at anytime or over a long period (long and sustained)

Case 2: Fiscal Theory of Price Level

· Monetary authority does not alter its plan for

· Since does not change, the entire adjustment must come via an increase in the period price level (not sustained change)

· One time rise in price causes inflation to occur in period

Case 3: Mix of FTI and FTPL

If the monetary authority can only help partially, that is the increase in money creation does not generate enough seignorage revenues to balance the consolidated GBC, then fiscal pressures are relieved through both a change in current price level and future inflation

Inflationary Finance

In reality, the divison of fiscal pressure on nominal prices into current pressure versus future pressure is hard to disentable as different economics experience different combinations.

Fiscal Theory of Inflation

Effects of inflationary finance felt as a long and sustained (thought not necessarily very sharp) rise in inflation, in period and/or future periods.

Fiscal Theory of Price Level

Effects of inflationary finance felt as a short-lived but very sharp rise in inflation, in period , and no further inflation in future periods

Lecture 5

Definitions

Exchange Rate

A nominal exchange rate is the price of one currency in terms of another currency.

Appreciation and Depreciation

A currency appreciates (depreciates) against another currency when it becomes stronger (weaker) compared to the other currency. Alternatively, if a currency becomes “more expensive” in terms of another currency, it has appreciated against said currency.

Foreign Reserves

Foreign reserves are a central bank’s holding of foreign currencies for the purpose of government international transactions (most commonly, exchange rate interventions). It is stock variable.

Foreign Reserves Changes

A country’s foreign reserves change during a given time period measures by how much its foreign reserves have changed during that time period. It is a flow variable.

Types of Exchange Rate

Floating

A nominal exchange rate is floating if it is determined solely by the forces of market demand and supply. It is the equilibrium price of the demand and supply of currencies.

Fixed

A nominal exchange rate is fixed (pegged) if it is determined through government intervention in exchange markets in order to fix the rate at some value.

This is not the equilbirum price. For instance, in 1991, the Argentine government adopted a fixed exchange rate at 1 Peso/USD to control hyperinflation, while the equilibrium price was at 2 Peso/USD.

The Argentine Currency Peg

At 1 Peso/USD, private-market demand for dollars > private-market supply of dollars.

Hence, the Argentine central bank has to sell some of its foreign dollar reserves to fill the supply shortfall in the Forex Market for USD. Resultingly, from 1991 to 2001, the Argentine CB ran a BOP deficit.

In 2001, the Argentine CB was very low on foreign reserves, and the peg is expected to not last much longer, so depreciation (devaluation) to the floating rate was imminent. This resulted in a currency run against the Argentine Peso as holders try to switch to the more stable currency, the USD, before the Peso depreciates.

Hence, the demand for USD increases, further increasing the floating exchange rate from 2 Peso/USD to 3 Peso/USD.

Fixed exchange rate systems are inherently dynamic phenomenas, and require certain combination of monetary policy and fiscal policy to be sustainable over the long run, which wasn’t the case in Argentina.

Singapore Foreign Exchange Policy

In 1980, Singapore adopted an exchange rate-centered monetary policy.

Singapore exchange rate is trade-weighted, that is, the Singapore dollar is managed against a basket of currencies of its major trading partners and competitors. The weights capture different degrees of importance of the various currencies which is determined by the extent of Singapore’s trade dependence on the respective countries.

Building Blocks of the Fiscal Theory of Exchange Rates

1. Money Demand Function

Recall that real money demand depends positively on and negatively on (the opportunity cost of holding money). Expressed in the genral form, for closed-economies, as

Note that consumption is constant in every period (steady-state). That is, in every period.

Nominal and Real Exchange Rates

A nominal exchange rate is the price of one currency in terms of another currency, denoted .

A real exchange rate is the price of one country’s consumption basket in terms of another country’s. In particular, it is how many units of domestic consumption one unit of foreign consumption can buy, denoted .

· domestic currency/foreign currency

· foreign currency/unit of foreign consumption (foreign price level)

· domestic currency/unit of domestic consumption (domestic price level)

· domestic consumption/foreign consumption

2. Purchasing Power Parity

Averaged over long periods of time (steady-state), . That is, in the long run, one basket of goods in one country can buy one basket of goods in another. Assume PPP always holds, .

Further suppose that in every period to remove the effect of foreign inflation. Thus,

This illustrates the primary motivation behind why countries like Argentine adopt fixed exchange rates. It eliminates domestic inflation, since never changes if never changes.

3. Interest Rate Parity

The interst rate parity condition states that

· Current nominal exchange rate

· Future nominal exchange rate

· Domestic nominal interest rate

· Foreign nominal interest rate

Assuming that domestic and foreign financial markets are functioning well, that is, there is no arbitrage present, interest rates are equalised after adjusting into common currencies.

Suppose that that foreign real interest rate never changes, that is, each period. Coupled with zero foreign inflation rate, from the Fisher equation,

Hence, in every period, replacing future exchange rate with expected future exchange rate,

Summary of Auxillary Assumptions

i. Consumption constant in every period,

ii. No foreign inflation. , and thus,

iii. Foreign real interest rate never changes,

Government Budget Constraint

Recall that the period- consolidated (fiscal-monetary) budget constraint is

where the governments’ revenue to fund spending is from tax, , printing money, , and net borrowing from the public, .

Assumptions of government

In developing countries, most government assets are foerign reserves, typically held as foreign-dominated bonds, rather than foreign currency.

Assume that the government does not borrow from the public. Denote the government’s foreign reserves at period to be (not ). Then, in period , the government relies on only four sources for spending.

· Printing money:

· Tax:

· Interest earned from foreign reserves (bonds):

· Drawing from foreign reserves to spend:

4. Government Budget Constraint

Notations

· Nominal exchange rate (domestic currency/foreign currency)

· Domestic price level (domestic currency/domestic goods)

· Domestic holdings of foreign reserves at the beginning of period /end of period

(units are in foreign currency)

· Real domestic government purchases

· Nominal tax revenue (assumed lump-sum)

· Nominal domestic money in circulation at the beginning of period /end of period

Thus, the period- consolidated (fiscal-monetary) budget constraint is

Since , dividing throughout by and rearranging,

, government using reserves, , government accumulating reserves.

is known as the fiscal deficit, where

· is the expenditure

· is the real tax revenue

· is the real income/revenue earned from investing the foreign reserves in foreign countries.

, government running surplus, , government running deficit.

Four Building Blocks of the Fiscal Theory of Exchange Rates

i. Money Demand Function

ii. Purchasing Power Parity

iii. Interest rate parity

iv. Consolidated Flow Budget Constraint

Definition of Balance of Payments

Balance of Payments is defined as the change in foreign reserves in period t, .

Changes in Exchange Rate

Three cases that track the dynamics over time of various macroeconomic measures by looking at all the four building blocks in each period.

Case 1: Fixed Exchange Rate in Place and Individuals Expect it to Remain in Place

Case 2: Unanticipated, one-time devaluation

Case 3: Country is Maintaining Fixed Exchange Rate and Running Fiscal Deficit Every Period

Case 1: Fixed Exchange Rate in Place and Individuals Expect it to Remain in Place

Let . Then, for any period , we have

i.

ii.

.

iii. (for )

iv.

Hence, as long as the peg is in place and is expected to remain in place,

i. Domestic nominal interest rate is equal to foreign real interest rate

ii.

iii. Seignorage revenue is zero

Central bank is not permitted to print money when peg is in place. A fixed exchange rate “ties the hands” of the central bank, imposing discipline on money-creation, which also brings down inflation.

iv. If the fiscal authority is running a deficit, , foreign reserve level falls since .

Case 2: Unanticipated, one-time devaluation

Suppose in period , the government unexpectedly weakens the domestic currency to a new fixed rate and promises (credibly) to never again change the exchange rate – new rate is .

Then, in , since , and ,

i.

ii.

.

iii.

iv.

Since this is a one time change, from period onwards, everything goes back to Case 1.

i. Domestic nominal interest rate does not change when an unexpected, one-time change in exchange rate occurs, since it is linked expected changes in the exchange rate thorugh IRP.

ii. A devaluation of the domestic currency causes positive seignorage revenue for the government, only in the period of devaluation.

iii. Even if the fiscal authority is running a deficit, , if , can be positive.

Balance of Payment Crisis

Definition

A situation in which a government is unable or unwilling to meet its international financial obligations, often brought about by an unsustainable mix of fiscal and monetary policies.

· Recall that denotes the domestic holdings of foreign reserves at the beginning of period /end of period

Case 3: Country is Maintaining Fixed Exchange Rate and Running Fiscal Deficit Every Period

Suppose that the peg actually collapses at the start of period , and individuals/markets expect/understand the collpase will occur then.

Period

i.

ii.

iii.

iv.

v. unknown

vi.

(Assume government has reserves)

Period

i.

ii.

iii.

iv.

v.

vi.

Since

However, this changes expectations of the exchange rate in period , affecting the IRP, as people expect the currency to depreciate at a rate .

Note that the collapse of the peg will occur when the country runs out of foreign reserves, that is, , since it is no longer able to maintain the fixed exchange rate by selling foreign reserves. Thus, we can further conclude that , since foreign reserves must be 0 at the start of .

Since the government is running a fiscal deficit every period, for all .

Period , since ,

i.

ii.

iii.

iv.

However, since , and money demand is decreasing in ,

v.

vi.

Since

Notice that there are now two sources of drain on foreign reseres. Both and results in foreign reserves falling even faster than before.

i. Domestic nominal interest rate rises when a devaluation is imminent

The domestic nominal interest rate rises when the devaluation is imminent because in order to be induced to hold an asset denominated in a currency that is about to weaken, investors have to be compensated with a higher return.

ii. Domestic nominal money supply falls when collapse is imminent

Currency run occurs. Holders of domestic currency try to switch to the more stable foreign currency before domestic currency depreciates. This causes domestic nominal money supply to fall as indivduals trade in their domestic currency for foreign currency at the fixed rate.

Residents get foreign currency, Central bank gets domestic currency, falls.

iii. Seignorage revenue is negative when collapse is imminent

Stages of BOP Crisis

The three stages of a BOP Crisis are

i. Pre-collapse phase

· Persistent fiscal deficits, no political will to balance budgets

· Persistent decline in foreign reserves

· Low inflation – direct consequence of fixed exchange rate

ii. BOP crisis

· Decline in foreign reserves woresened due to currency run

· Fixed exchange rate is abandoned

iii. Post-collapse phase: Whatever happens here depends on monetary conducted post-collapse. In the case of Argentina

· Mixed record on inflation

· Strong GDP growth folllwing severe recession in 2002

· High unemployment

· Political independece of central bank still in question

Lecture 6

Model of Production

Production Function

Consider the model of a single, closed economy with no outside trade, and has one consumption good/representive good, a basket of goods. The inputs in the production process are capital and labour, denoted by and respectively. Then, the production function is

where is the output of the economy given a fixed productivty and inputs and .

This function, in particular, is the Cobb-Douglas production function, but there can be other types of production functions. By convention, is assumed to be , and is assumed to be .

Let . Since ,

Note that is increasing in both and . That is, the marginal product of captial and labour are positive. Futher note that since ,

we can see that is increasing at a decreasing rate in and . That is, there are diminishing marginal returns of both capital and labour to production. Summarising, for ,

Function

Notice that is increasing in and is increasing in .

Returns to Scale

For a Cobb-Douglas Production function, there are 3 types of returns to scale it can exhibit. Returns to scale refer to the increase in production (output) as a result of a change in the inputs, both by the same factor. Suppose that both and are multiplied by a constant . Denote,

i. Constant returns to scale:

ii. Increasing returns to scale:

iii. Increasing returns to scale:

When doing comparison, make sure that you compare the two outputs using only , the factor that has been scaled by.

Intensive Form of Production Function

is the per capita production function, for . This is also known as the intensive form. The intensive form is only possible for production functions with constant returns to scale.

Allocation of Resources

The objective function to maximise is , where is the rental wage of capital, and is the wage rate. That is, we want to find

The rental rate and wage rate are taken as given under perfect competition. At maximum, when the firm maximises profits, we have the following first-order conditions,

For simplicty, assume that the price of output is normalised to 1.

Solving the Model

The model has 5 endogenous variables and 5 equations

Variable

Equation

1

Output

2

Captial

3

Labour

4

Wage

Supply = Demand for Labour

5

Rental price of capital

Supply = Demand for Capital

Supply of Labour and Capital are fixed, and hence, constant at and .

General Equilibrium

A general equilbirium is a solution to the model when more than a single market clears. In this context, there are two markets, captial and labour.

Since firms hire capital and labour based on and , we have that the demand curves for capital and labour are based on these hiring rules, and hence, trace out exactly the marginal product schedules for and . Note that for , we have

Moreover, since supply of capital and labour are fixed, we have the following markets,

Hence, at equilbrium, we have

Variable

Equation

1

Output

2

Captial

3

Labour

4

Wage

5

Rental price of capital

where the superscript denotes the equilibrium values.

This solution implies that

· Firms employ all the supplied capital and labour in the economy

· The production function is evaluated with the given supply of inputs

· and evaluated at the equilibrium values of and

· Equilbrium rental rate and wage are proportional to the output per captial and output per labour.

where is output per capital and is output per worker

Capital and Labour Share

Capital share and labour share are given by

the amounts paid to capital and labour. All income is paid to capital or labour, resulting in zero profit in the economy, verifying the assumption of perfect competition. This also verfies that production equals spending equals income. This can be seen where

That is, income = production.

Analysing the Production Model

We can use the model to explain differences in income across countries. Note that from empirical data obtained, usually . Thus, .

If the productivty parameter, is 1, then the model over-predicts the GDP per capita. Diminishing returns to capital implies that countries with low will have a high , and those with a lot of will have a low , and cannot raise GDP per capita by much through capital accumulation. Assuming for all countries,

Since we have that poor countries have a higher and at equilibrium, we have that poor countries have a higher rental rate, which means higher interest rates as well. Hence, we should observe flow of funds from rich to poor countries to seek for higher returns. However, this is not usually the case in reality, which implies that might not be the same for all countries.

Model where TFP is Same Across Countries

Assuming , we have that the predicted GDP per capita for all countries is .

The graph shows the model’s predicted values for GDP/capita against the actual GDP/capita relative to the US. The blue line is when predicated GDP = actual GDP.

Poorer countries are further from this line, that is, predicated GDP/capita actual GDP/capita.

Hence, the model predicts that most countries should be substantially richer than they are.

Improving the Model

A simple production model with no difference in productivty across countries is misguided. In reality, it is usually the case that richer countries have a higher productivity than poorer countries. Considering this, we can see that

Due to the higher productivty, , the slope of the against is now flatter for the poor countries, which means that it is now possible for both the poor and rich countries with different level of capital to have the same , and hence, there is no fund transfer.

Notice that for the same level of capital as the poor country, , the rich country is able to command a higher . This is because higher productivty allows capital to be used more effectively or efficiently, increasing its marginal output, .

Implied Total Factor Productivty

Data on Total Factor Productivty, TFP (a.k.a residual), is not collected, but can be calculated from data present on actual output and capital per person. Since we have , the implied TFP is

Using the above method, we can see in the graph that the implied TFP relative to the US of poor countries are much lower than that of rich countries. This explains the difference observed in actual GDP per capita and predicted GDP per capita.

Understanding Differences in Output Across Countries

Implied TFP Relative to the US

In the above example, we found the “implied TFP relative to the US”

· Since we are calculating (TFP) using given values of and , what is actually found is the implied TFP, and not the actual TFP.

· This value is relative to the US if the output, , and capital, , of a country used to calculate the implied TFP are relative to the US, and not the actual output and capital of that nation.

TFP vs Capital in Explaining Differences

Let the subscripted variables denote the respective values relative to those of the rich nation, which are all, hence, normalised to 1. Since , we have

Thus, due to the normalisation, , and hence, denote

The value of denotes how much more important TFP differences are than capital per person differences in accounting for the difference in GDP (output) per capita across nations.

i. TFP, , differences explain of the difference in .

ii. Captial per capita, , differences explain the remaining of the difference in .

Rich countries are therefore rich because they

· Have more capital per person

· Use labour and capital more efficiently due to higher TFP

TFP Differences

Some of the reasons why countries may have TFP differences are because of

i. Human Capital

· Stock of skills (Education and training) individuals accumulate to increase productivity

· Examples are people attending college, workers learning to operate machinery, doctors mastering a new technique.

In the US, each year, education seems to increase future wages by 7%. In developing countries, returns are usually higher, from 10-13% per year.

This may be because typically, students learning basic skills may have higher returns due to diminishing marginal returns to education, as compared to students in higher-level education.

ii. Technology

Richer countries may use more modern and efficient technologies than poor countries, increasing their productivty parameter

Goods such as

· state-of-the-art computer chips and software

· new pharmaceuticals

· military arsenal

· skyscrapers

and production techniques such as

· just-in-time inventory methods

· information technology

· tightly integrated transport networks

are much more prevelant in rich countries than in poor.

iii. Institutions

· Institutions are in place to foster human capital and technological growth

· Examples are

·

· Property rights

· Rule of Law

· Government Systems

· Contract Enforcement

·

· Some challenges in countries with uncertain institutions are

· No well-defined set of laws to follow to establish business

· Rules are not the same for everyone

· Licensing fees and taxes may vary over time and without warning

· Corruption and Bribes

· Imports may be challenging to receive

· Profits may be taxed or stolen due to insufficient property rights

· A coup or war could change the environment overnight

iv. Misallocation

· Resources may not be put to their best use in poor countries

· Examples are inefficiency of state-run-resources and politcal interference

Evaluating the Production Model

· Per capita GDP is higher if capital per person is higher and if factors are used more efficiently

· Constant returns to scale imply that output per person can be written as a function of capital per person (intensive form)

· Capital per person is subject to strong diminishing returns because the exponent is much lower than one.

A weakeness of the model is that, in the absence of TFP, the production model incorrectly predicts differences in income, and does not provide an answer as to why countries have different TFP levels.

Lecture 7

Solow Growth Model

Augments the production model with capital accumulation. That is, captial stock is no longer exogenous. It is now endogenised, and to be solved for in the model.

The accumulation of capital is a possible engine of long-run economic growth. That is, perhaps, some countries are richer than others because they invest more in accumulating capital.

Let the production model, time subscripted, be

where we assume it is Cobb-Douglas, with constant returns to scale.

Capital Accumulation

Capital can grow from investment, which comes from household savings. That is, in a closed economy, ( is open economies). Output can be used for consumption or investment,

This is called the resource constraint, and assumes there are no imports or exports.

Goods invested for the future determine the accumulation of capital, given by the capital accumulation equation

· Next period’s capital

· This period’s capital

· This period’s investment

· Depreciation rate

, the amount of capital level depreciated, is also known as the break even investment. Net investment is , the change in capital stock.

Change in capital stock is defined as

That is, future capital depends on investment now. For simplicity, assume that labour supply is not included, and is given exogenously at a constant level .

Solow Model Set-up

The economy consumes a fraction of output and invests the rest

where is the investment and is the fraction of total output invested (also exogenous). Since consumption is the share of output not invested, we have

The model has 5 endogenous variables and 5 equations

Variable

Equation

1

Output

2

Captial

3

Labour

4

Consumption

5

Investment

The parameters are , which are exogenous (they are given).

Differences between Solow Model and Production Model

In the Solow Model, there are

· Added dynamics of capital accumulation

· No capital and labour market interaction and their prices

Solving the Solow Model

Combine the investment allocation and capital accumulation equations, and we have the change in capital equals to net investment,

Substitute the fixed amount of labour into the production function, and we have

Then, we end up with 3 graphs,

i. Output:

ii. Savings = Investments:

iii. Depreciation:

At , the amount of investment > depreciation. That is, . There is positive net investment.

Hence, capital stock will increase until , . At this point, change in captial stock is , and thus, capital stock will stay at this value of capital forever, the steady-state. Similarly, if depreciation is greater than investment, negative net investment, the economy converges as capital decreases to the same steady state, since .

That is, capital stock changes until amount of investment undertaken = amount of capital lost through depreciation. This is called transition dynamics.

Transition Dynamics of Output

When not in the steady state, the economy exhibits a change in capital toward the steady state. As moves to its steady state, , output also moves to its steady state. At the rest point of the economy, all the endogenous variables, are steady. Consumption is seen as the difference between output and investement at any capital level.

Note that in the graph, since . Transition dynamics take the economy from its initial level of capital to the steady state.

Solving for Steady State Capital

At steady state, investment equals depreciation, . Substituting this into the production function, , at the steady state, we have , and thus,

Thus, we have that the steady state of capital as a function of the exogenous variables, given to be

It is positively related to the investment or savings rate, , size of the workforce, , and productivity of the economy, . It is negatively related to the depreciation rate .

Solving for Steady State Output

At the steady state output, we have . Then,

Thus, we have that the steady state of ouput as a function of the exogenous variables, given to be

A higher steady state production is caused by higher productivity and investment rate, while a lower steady state production is cause by faster depreciation.

The output per capita is also thus given as,

Analysis of One-Time Change in Productivity

The exponent on productivity is different here than in the production model, and the higher productivity has additional effects in the Solow model by leading the economy to accumulate more capital. Supppose there is a one-time increase in productivity, . Then, there is a proportional increase in both the production function and savings curve.

There are two effects on output due to the change in productivity, from to , that explain the fast growth.

i. Direct effect (increase) from increase in

ii. Indirect effect (increase) from an increase in , causes capital accumulation since

In advanced countries, is slow, which explains why they have lower growth rates.

Lecture 8

Explaining Capital in Solow Model

In the steady state in the Solow Model, . Hence,

That is, capital-output ratio should be linear in . While investment rates vary across countries, depreciation rate is assumed to be relatively constant.

A key determinant of a country’s capital-output ratio is its investment/savings rate. As predicted by the Solow model, these variables are positively related, and this predication holds up remarkably well in data.

Differences in Output per Capita

Since , taking , we have that

Notice that since , . The Solow model thus shows that TFP is more important in explaining differences in per capita output, as it has both a direct and indirect effect on output, while still plays an, albeit smaller, role.

Steady State and Problems with the Solow Model

In the Solow model, the economy reaches a steady state because investment has diminishing returns, and the rate at which production and investment rise is smaller as capital stock increases. Moreover, a constant fraction of capital stock depreciates every period, which is not diminishing. Eventually, net investment is zero and the economy rests at a steady state.

Thus, no long-run growth in the Solow model since growth stops at the steady state, and output, capital, and consumption are constant.

Empirically, however, economies appear to continue to grow over time, which highlights a drawback of this model.

Capital Accumulation

Hence, according to the Solow model, capital accumulation is not the engine of long-run economic growth. While saving and investment are beneficial in the short-run, they do not sustain long-run growth, particularly due to diminishing returns.

Population Growth in the Solow Model

Growth in the labour force can lead to aggregate economic growth, but not growth in output per person. Eventually, diminishing returns lead and to approach the steady state.

To solve the Solow model graphically, we convert all 3 graphs – ouput, investment, and depreciation curve to the per-capita form. This stops the graphs from shifting up every period. In the steady state, will be a constant since

and in the long run, we can show that . That is, and are both constant, and this pins down the steady state for analysis.

Solving the Per-Capita Model

We have the following per-capita equations, where .

Note that both and are concave functions in . Note that to account for the change in population, the per-capita break-even/depreciation curve is given by .

Suppose the economy is at point . At this time, the amount of

· output per capita produced is

· saving per capita is

· required investment to maintain the same level of capital per capita is

Thus, the net investment per capita is .

Steady State Growth

At the steady state, we have and are constant. That is, . Thus,

Thus, at the steady state, .

Analysis of Changes in Variables

Change in Savings/Investment Rate

Suppose the investment rate, , increases permanently for exogenous reasons at . Then, the investment curve rotates upwards, while the depreciation curve is unchanged.

The capital stock increases by transition dynamics to reach the new steady state because investment exceeds depreciation, leading to capital accumulation over time. The higher capital also causes output to rise to a new steady state. The new steady state is located to the right.

When increases, the initial growth rate of income is immediately higher as at this point, capital accumulation is the largest, and has the lowest diminishing returns.

Note that as opposed to the change in output when increases, there is no new output curve. Thus, increase only comes from capital accumulation, that is, there is no direct increase.

Change in Depreciation Rate

Suppose the depreciation rate, , increases permanently for exogenous reasons at . Then, the depreciation curve rotates upwards, while the investment curve is unchanged.

The capital stock decreases by transition dynamics to reach the new steady state because depreciation exceeds investment, leading to capital deccumulation over time. The lower capital also causes output to fall to a new steady state. The new steady state is located to the left.

The Principle of Transition Dynamics

If an economy is below the steady state, it will grow. If it is above the steady state, the growth rate will be negative. As shown above in the ratio scale graph, output changes more rapidly when an economy is further from the steady state. As it approaches the steady state, growth shrinks to 0.

This allows us to understand why economies grow at different rates. The farther below an economy is from its steady state, the faster it grows. This was true for OECD countries.

However, for the world as a whole, on average, rich and poor countries grow at the same rate. This implies that

· most countries, both rich and poor, have already reached their steady states

· countries are poor not because of a bad shock, but because they have parameters that yield a lower steady state, such as savings/investement rates and productivity

Strengths and Weakness of the Solow Model

Strengths

· Provides a theory that determines how rich a country is in the long-run or steady state

· Explains principle of transition dynamics that allows for an understanding of differences in growth rate across countries

Weaknesses

· Focuses on investment and capital, but the much more important factor of TFP is still unexplained

· Does not explain why different countries have different investment and productivty rates, and a more complicated model could endogenise the investment rate

· Does not provide a theory of sustained long-run economic growth

Saving Rate and Consumption

Dynamics of Steady State Consumption and Saving Rate

Notice that when or , there is no consumption at the steady-states. In the case of , there is no production in the economy, and if , all savings go towards capital investment.

Consumption per capita, the difference between and , thus increases from 0 to some maximum value as increases, and then decreases back down to 0 as reaches 1.

The level of capital per capita associated with the value of saving rate that yields the highest level of consumption per capita in steady state is known as the golden-rule level of capital.

Varying the Savings Rate

The government can affect the savings rate in various ways

· Vary public saving: if public saving is positive, overall, saving would be higher and vice-versa

· Vary taxes: tax breaks to provide more incentive to save

Governments should care about the consumption per worker, and not output per worker, because consumption reflects the welfare of the people

Problems with Changing the Savings Rate

Suppose that the capital per capita is below the golden rule level, that is, current savings is some . To maximise the welfare of future generations, savings rate should increase to .

Then, the capital accumulation proccess takes place, and the economy shifts to the new steady state with a higher level of consumption, the maximum, in time to come.

However, for the current generation, they face a drop in consumption since some income is now used for capital accumulation instead.

This raises the question on whether then should governments care more about the future or current generation. Politically, it is unlikely that governments will ask the current generations to make sacrifices for the sake of future generations, and hence, the economy remains at a steady state below that of the golden-rule level of capital

Lecture 9

Romer Model

The Romer model divides the world into (finite) objects, which are capital and labour from the Solow model, and (infinite) ideas, which are items used in making objects.

This distinction forms the basis for modern theories of economic growth, and sustained economic growth occurs because of new ideas.

Ideas are, in fact, club goods, because they are

· Non-rival: One person’s use of ideas does not reduce the usefulness to someone else

· Excludable: Legal restrictions on a use of a idea may exclude people from using it

Returns to scale

Increasing returns to scale

· Average production per dollar spent is rising as scale of production increases

· Double inputs more than doubles outputs

· High initial fixed development costs

Constant returns to scale

· Average production per dollar spent is constant

· Double inputs exactly doubles outputs

· The standard replication argument implies constant returns to scale

Example of Fixed Costs and Returns to Scale

Suppose a new antibiotic is created, and the cost of developing new drugs is large. The initial start-up costs for research and development produce no actual good. To create a single dose of the antibiotic, it costs

where is a fixed cost used for R&D.

Every dose thereafter costs only to produce. The production function with and without the initial research (fixed costs), where is the amount spent and is the number of antibotics are,

Notice that as increases, decreases. Since is constant, average production per dollar spent, , increases when there are initial fixed costs present. Thus, the scale of production increases, that is, there are increasing returns to scales.

Returns to Scale in Romer Model

In the Romer model, has been endogenised into the production function. Let

Multiply all variables by . Then,

That is, in objects ( alone, there are constant returns to scale. But in objects and ideas , there are increasing returns to scale.

Pareto optimal allocation

A pareto optimal allocation is one that cannot be changed to make someone better off without making someone else worse off. Perfect competition results in pareto optimality since .

Under increasing returns to scale, a firm faces intial fixed costs and marginal costs. But, if , no firm will research to invent ideas as fixed research costs cannot be recovered. To create incentives for firms to innovate and produce, there must be a wedge between price and marginal cost, implying we cannot have perfect competition and innovation.

Ways to Promote Innovation

Patents are able to grant monopoly power over a good for a period of time. This generates positive profits, and provide incentive for innovation. However, results in welfare loss.

Other methods are such as government funding or prizes to reduce fixed costs.

Romer Model Set-up

Production Function

In the Romer model, the production function is given to be

Ideas and Labour

Unregulated markets underprovide ideas. New ideas depend on

· Existence of ideas in the previous period

· The number of workers producing ideas

· Worker productivity

The population is made up of workers producing

· Ideas (researchers, innovators):

· Output (production/line worker):

Thus, we have the resource constraint

Expressing the endogenous variables in terms of the parameters, let

where and are fixed parameters,

· amount of labour

· fraction of labour producing ideas.

Flow of Ideas

Researchers use their own labour and old ideas to generate new flow of ideas. That is,

Solving the Romer Model

The model has 4 endogenous variables and 4 equations

Variable

Equation

1

Output

2

Idea Production Function

3

Resource Constraint

4

Allocation of Labour

The parameters are , which are exogenous (they are given)

We have that output per person depends on the stock of knowledge, ,

as opposed to the Solow model, where output depends on capital per person.

Since growth rate of knowledge is constant a

and stock of knowledge depends on its initial value and its growth rate

Combining, since

Plotting against , the graph is linear with -intercept and gradient . Hence, is constant, and we have

Balanced Growth

The Romer model does not exhibit transition dynamics, as does the Solow model. Instead, it has a balanced growth path, where all endogenous variables grow at a constant rate. .

Growth in the Romer Model

The Romer model produces long-run growth because it does not have diminishing returns to ideas due to its non-rivalry. In , has exponent 1.

In fact, labour and ideas together have increasing returns, and returns to ideas are unrestricted. This is as opposed to the Solow model, where capital has diminishing returns, which causes capital and income to stop growing.

Case Study: A Model of World Knowledge

The US has several hundred more times more researches than Luxembourg. According to the Romer model, the growth rate of the US, must hence grow at a rate several hundred times that of Luxembourg. However, Luxembourg has had a higher growth rate.

When ideas are perfectly shared across the globe, the original Romer model needs adjustment.

where the two countries have access to the same knowledge. Thus,

The ratio can be very high for Luxembourg. This means that although Luxembourg has a smaller number of researches , it can have a higher growth rate of ideas, .

However, it is not accurate to think of ideas as remaining in a country. Workers in Luxembourg can benefit from ideas invented in the US. International trade, MNCs, licensing agreeents, international patents, migration of students and workers, and the open flow of information ensure that an idea created in one place can impact economies worldwide.

The Romer model is better applied to the world’s stock of ideas, as opposed to a country-by-country basis. Through the spread of idea, growth in the world’s store of knowledge drives long-run growth in every country.

Analysis of Changes in Variables

Recall that . Using the log scale and letting , we have

Change in Population

A change in the population changes the growth rate of knowledge as an increase in population can immediately and permenently increase the growth rate of per capital output.

Since increases, increases, and when labour increases, the ratio scale per capita production function has a steeper gradient.

Change in Research Share

Suppose that increases. That is, the share of the labour force in the research sector increases. Then, there are two effects.

· -intercept decreases

· Gradient increases

The growth rate is higher as more researchers produce more ideas which leads to faster growth (growth effects). However, there is an immediate initial drop in output per capita when increases since there are less people in the production sector (level effects).

Growth effects explain changes to the rate of growth of per capita output, and level effects explain changes in the level of per capita GDP.

Growth Effects vs Level Effects

Growth effects are changes to the rate of growth of per capita output. Level effects are changes to the level of GDP per capita.

The degree of increasing returns matters for growth effects. If the exponent on ideas is less than 1, there will still be sustained growth, but growth effects are eliminated due to diminishing returns.

Model with Diminishing Returns to Idea

Suppose instead that we have

Variable

Equation

1

Output

2

Idea Production Function

3

Resource Constraint

4

Allocation of Labour

The parameters are , which are exogenous (they are given) and .

Combining Romer and Solow Models

Combined model has properties of both models.

· Non-rivalry of ideas in long-run growth along a balanced growth path.

· Exhibits transition dynamics if economy is not on balanced growth path.

For short periods of time, countries can grow at different rates. But, in the long-run, countries all grow at the same rate.

Growth Accounting

Growth accounting determines the sources of growth in an economy and how they may change over time. Consider the production function , where total factor productivity is the stock of ideas. Then, we have

Adjusting growth rates by labour hours. That is, , we have,

where TFP growth is often called the residual.

Summary

Institutions (property rights, laws) play an important role in economic growth. The Solow and Romer models provide a basis for analysing differences in growth across countries, but do not explain why investment rates and TFP differences across countries.

· Solow: Explains short-run differences in growth across countries

· Romer: Explains long-run growth, but has no transition dynamics

Ideas per capita don’t matter, but rather the total stock, due to its non-rivalrous nature.

Each economy in terms of production is exactly the same with what Solow recommended. But in terms of TFP or ideas, each country enjoys the overall trend in world knowledge that is generated by a Romer model. Transition dynamics associated with the Solow model help us to be able to explain or understand differences in growth rates across countries.

Solow and Romer models have made many additional valuable contributions, such as the modern theory of monopolistic competition and new understanding of exogenous technological progress.

Combined Romer and Solow Models Set-up

The combined model is set-up by adding capital into the Romer model production function. It has 5 endogenous variables and 5 equations

Variable

Equation

1

Output

2

Captial

3

Knowledge

4

Workers

5

Researchers

The parameters are , which are exogenous (they are given).

We can hence deduce the following

i. The production function has constant returns to scale in objects and increasing returns in ideas and objects together

ii. The change in the capital stock is investment minus depreciation

iii. Researchers are used to produce new ideas

The combined model will result in

· A balanced growth path, since increases continually over time

· Transition dynamics

· Long-run growth: To be on a balanced growth path, output, capital, and stock of ideas all must grow at constant rates.

Starting at the production function, we have

Along a balanced growth path, we have that all endogenous variables grow at a constant rate. That is, and . Further notice that from

we have that, since , , and are constant, is constant as well. That is, , and hence,

Since and , given that the labour force and fraction of workers is constant,

Thus, since , we have

Compare this to just the Romer model, where . Output here is higher as

· Ideas have a direct and indirect effect

· Increasing productivty raises output because productivity has increased, and higher productivity leads to higher capital stock through capital accumulation

While capital itself canot serve as an engine for economic growth, it helps to amplify the underlying growth of knowledge. Thus, for the long-run combined model, this equation pins down the growth rate of output and growth rate of output per person.

Solving the Combined Model

The capital-output ratio is proportional to the investment rate along a balanced growth path

Substituting this back into the production function, we have

Comapre this to just the Solow model, where . There is no and .

· Growth in

Leads to sustained growth in output per person along a balanced growth path

· Output

Depends on the square root of the investment rate

· A higher investment rate

Raises the level of output per person along the balanced growth path

Transition Dynamics

The Solow model and the combined model both have diminishing returns to capital. Transition dynamics applies to both models. In the combined model

· The further below its balanced growth path an economy is, the faster the economy grows

· The further above its balanced growth path an economy is, the slower the economy grows.

Changes in any parameter results in transition dyanmics. Long-run growth is achieved through ideas, and explains differences in growth rates across countries.

Combined Model Output Function (Ratio Scale)

Since we have,

Since ,

Letting , and noting that

Analysis of Changes in Variables

Change in Research Share

This is similar to the pure Romer case, except now, changes as well. There are both growth effects due to the change in growth rate, and level effects due to the fall in output.

Change in Savings/Investment Rate

Suppose the investment rate, , increases permanently for exogenous reasons. Then, the level of increases. However, growth rate is unaffected. Hence, there is a parallel shift up of the balanced growth path.

Similar to the pure Solow model, there will be an accumulation of capital by transition dynamics as saving rate rises, leading to a higher output level. Output increases due to both capital accumulation and growth rate in ideas, , (non-diminishing returns). The balanced growth part of income is higher, and the economy is now below the new balanced growth path.

When increases, the initial growth rate of income per capita is immediately higher as the economy enjoys growth from both capital accumulation and . At this point, capital accumulation is the largest, and has the lowest diminishing returns.

In the long run, when there is no more capital accumulation, the economy only enjoys the growth rate from , and thus, the economy is on the new balanced growth path. This new balanced growth path is on a higher level as compared to the old one but has the same growth rate, , as the old one.

Change in Depreciation Rate

Suppose the depreciation rate, , increases permanently for exogenous reasons. Then, the level of decreases. However, growth rate is unaffected. Hence, there is a parallel shift down of the balanced growth path.

Similar to the pure Solow model, there will be a deccumulation of capital by transition dynamics as depreciation rate rises, leading to a lower output level. The balanced growth part of income is lower, and the economy is now above the new balanced growth path

In the long run, when there is no more capital deccumulation, the economy is on the new balanced growth path. This new balanced growth path is on a lower level as compared to the old one but has the same growth rate, , as the old one.

Miscellaneous Items

Growth Rate

Growth Rate

The growth rate of a variable is its proportional rate of change.

Properties of Growth Rates

i.

ii.

iii.

Ways to Calculate

Let denote the time derivative of . Then,

i.

ii.

iii.

Summary of the Different Models

Set -up of Model

Production Function

Solow

Romer

Solow-Romer

1

Output

Output

Output

Output

2

Captial

Captial

Idea Production Function

Captial

3

Labour

Labour

Resource Constraint

Knowledge

4

Wage

Supply = Demand for Labour

Consumption

Allocation of Labour

Workers

5

Rental price of capital

Supply = Demand for Capital

Investment

Researchers

Exogenous Parameters

Solving the Model

Production Function

Equilibrium

Solow

Steady-State

Romer

Equation

Solow-Romer

Balanced Growth Path

1

Output

Output

Output

The version is above

Output

The version is above

2

Captial

Captial

Idea Production Function

Captial

3

Labour

Labour

Resource Constraint

Knowledge

4

Wage

Consumption

Allocation of Labour

Workers

5

Rental price of capital

Investment

Researchers

Capital-Output Ratio

Note that in the Solow Model, while output per person depends on the capital per person, in the Romer Model, output per person depends on the stock of knowledge.

Growth Rates

· Solow

i.

ii.

iii.

Steady state growth rate of variables in the Solow Model are all 0.

· Solow-Romer

i.

ii.

iii.

· Solow-Romer (Balanced Growth Path)

i.

ii.

Why Combined Solow-Romer Model?

· (Romer) Non-rivalrly of ideas results in long-run growth along a balanced growth path

Explains long-run growth

· (Solow) Exhibits transition dynamics if economy is not on balanced growth path

Explains short-run differences in growth across countries, since not all countries have reached their balanced growth paths (or policy differences)

In the short-run, countries can grow at different rates, but they will grow at the same rate in the long-run.

The combined model will result in

· A balanced growth path, since increases continually over time

· Transition dynamics

· Long-run growth: To be on a balanced growth path, output, capital, and stock of ideas all must grow at constant rates.