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Econ 2 Macro Cheat Sheet (DRAFT) by

Econ general sheet with formulas Macroeconomics Nils Gottfries

This is a draft cheat sheet. It is a work in progress and is not finished yet.

Key Equations

 
"_ _" denotes subscript (eg. P_t+1_­/P_t_)
Production and Prices
Production Function
production depends on the inputs of capital, labour and technology factor

Y = F (K, E N)

Y = production
K = capital
N = labour
E = technology factor
Cobb-D­ouglas Production Function
Y = Kα (E N)1-α
Marginal Product of Labour
take derivatice of Y = Kα (E N)1-α with respect to N

MPL = (1 - α) E1-α (K/N)α
Monopo­loistc Compet­ition Price
P = (1+ μ ) MC

μ = mark-up
Marginal Cost
MC = W/MPL
Marginal Cost in term of Cobb-D­ouglas
MPL = Kα (1 - α) E1-α N-α = (1 - α) Y/N

and thus

WN/PY = 1-α/1+μ
Real Interest Rate, Invest­ment, and Consum­ption
Inflation
rate of growth of price level

π_t_ = ∆P_t_/­P_t-1_

One plus real interest rate is the price of goods today divided by the discounted price of goods next year

1 + r_t+1_ = P-t/(P­_t+­1_/­(1+­i_t_)) = 1+i_t_­/(P­_t+­1_/­P_t_) = 1+i_t_­/1+­π_t+1_
or
r_t+1_ ≈ i_t_ - π_t+1_
Firm Investment
to increase capital stock and replace deprec­iated capital

I_t_ = Kd_t+1_ - K_t_ + δK_t_

Kd_t+1_ = desired capital stock next year
δ = depric­itation
Profit Maximising Investment Level
the real marginal revenue product minus deprec­iation is equal to the real interest rate

MPK/I+μ - δ = r
Investment Function
investment depends on real interest rate, expected future income and the existing capital stock at the beginning of the period

I = I (r, Ye, K)

r = real interest rate
Ye = expected future income
K = existing capital stock at beginning of period
Utilit­y-m­axi­mising Consum­pti­on/­Savings Decision
ratio of marginal utility of consuming today divided by discounted marginal utility next year is equal to one plus the real interest rate

u'(C_t­_)/­u'(­C_t­+1_­)/(1+ρ) = 1 + r_t+1_

ρ = subjective discount rate
Real Disposable Income
production minus tax payments plus the real interest rate on net claims on government and foreign households and firms

Yd = Y - T +r(D + F)
Consum­ption Function
consum­ption depends on income today, future expected income, the real interest rate and level of assests

C = C(Yd, Ye - Te, r, A)
Long-run Growth
Constant returns to scale
production per effective worker depends on the capital stock per effective worker

Y/EN = F (K/EN', 1) = f(k)
where k = K/EN'
Steady State Growth Path
capital stock per effective worker is determined by

f'(k*)/1+μ - δ = r ̅
Constant Capital per Effective Worker on Steady State Growth Path
capital stock and production grow at same rate as the effective number of workers

K = k*EN, Y = f(k*)EN
∆K/K = ∆Y/Y = g+n
Long Run Level of Real Interest Rate (closed econ)
is equal to the subjective discount rate plus the techno­logical growth rate

r ̅ ≈ ρ + g
The Labour Market and Phillips Curve
Unempl­oyment Rate
fraction of labour force not employed

u = U/L = L-N/L
Wage-s­etting Equation
if unempl­oyment is above natural level, firms want to raise wages less than the average wage increase, and conversely

∆Wd_t_/W_t-1_ = ∆W_t_/­W_t-1_ - b(u_t_ - un_t)
Unempl­oyment on Natural Level
in the long run desired wages must be equal to actual wage increases, so unempl­oyment must be on a natural level

Nn = (1 - un)L
Phillips Curve
assuming that a share 1- λ of wages is set in advance

∆W/W = ∆We/W - b ̂ (u - un)
; b ̂ = λb/1-λ
Rate of Wage Increase (short run)
depends on the expected wage increase and unempl­oyment
short-run analysis disregard capital, so inflation is the rate of wage increase minus produc­tivity growth

π = ∆W/W - ∆E/E
Phillips Curve (infla­ation)
relates inflation to expected inflation, the output gap and a cost-push shock
{{nl} π = πe + βY ̂+ z

πe = expected inflation
Y ̂ = output gap - has a circumflex
z = cost-push shock
Government Debt
Change in Real Government Debt
equal to the primary deficit plus the real interest rate

General (intro)

Macroe­con­omics
- production
- employment
- price increase
- interest rates
Macroe­conomic models
Three Markets
- labour
- goods - credit (money)
Three Decisi­on-­makers
- typical firm
- typical household
- policy­makers
Monetary Policy
central banks
- set the interest rate
Fiscal Policy
government
- decides taxes and government expend­iture
Basic Model Factors
typical firm
- price-­setting
- wage-s­etting
- investment
typical consumer
- consum­ption
Open Economy
trades with the rest of the world
Keynes Theory
nominal wages are 'rigid­'/'­sticky'
Classical Theory (real business cycle theory)
wages and prices are adjustable to equate supply and demand marets
Neocla­ssical Synthesis
even if wages and prices are sticky in the short run,
we expect them to respond to changes in economic conditions over the long run
National Accounts
- flows of produc­tion, incomes, savings and invest­ments in a period of time (year/­quater)
Balance of Payements Statistics
- flow of payments connected to exports, imports, intern­ational transfers, capital flows
Value of Production (output)
sales of all firms and value of production in public sector added

output not a good measure as large share of output is used as input in other firms
Interm­ediate Goods
goods that are used as inputs in other firms
Value Added
subtract value of interm­ediate inputs from value of output

The Open Economy Long Run (Ch13)

Real Exchange Rate
price level in an open economy relative to the price level abroad, where price levels are converted to the same currency

determ­inant of aggregate demand in the open country
Real Exchange Rate
price level of domestic goods relative to foreign goods

ε=eP/P*
P = price of good production at home in domestic currency
P*= price of good produced abroad in foreign currency
e = nominal exchange rate - price of domestic currency in terms of foreign currency
ε = real exchange rate - price of domest­ically produced goods in terms of goods produced abroad
Current Account
difference between savings and real invest­ments in the country

deficit = borrowing from abroad
government/private sector­/both borrowing to finance consum­ption and real invest­ments in excess of income
(open) Interest Parity Condition
links interest rate differ­entials between countries to expected changes in exchange rate
Interest Parity Condition
for foreign lenders, the expected returns on loans in the currency of the small open economy must be the same as the expected return or loans in the foreign currency

i + ∆ee/e = i*
left= interest rate in small open economy plus expected apprec­iation of currency
right = return on loans in foreign currency
Long-run Analysis
- analyse the effects of changes in exogenous variables
- assume that prices and wages have time to adjust, employment and production at natural levels
- assume intern­ational financial markets are completely integr­ated, free flow of financial capital and interest parity condition holds
Real Interest Rate
determines real cost of borrowing and required return on investment
- open economy it is tied to real in the world financial market
- indepe­ndent of savings and investment in the small open economy
- adjusts in long run to bring equality between aggregate demand and natural level of production
Nominal Exchange Rate
e=ε P*/P
Constant Real Exchange Rate
relative change in the nominal exchange rate is equal to foreign inflation minus domestic inflation

∆e/e = π*-π

r=r*
Long Run Trends in Nominal Exchange rates
shown by ∆e/e = π*-π
- inflation differ­entials between countries
high inflation = deprec­iating nominal exchange rate
high inflation for number of years = constant nominal exchange rate
- exporters will find hard so must depreciate at some point
Long Run Equili­brium
expected change in exchange rate is equal to the actual change

i-π = i* -π*
left= real interest rate in the small open economy
right = world real interest rate

can also be written

i-i*= π-π*
Nominal Interest Rate
a country with high inflation and deprec­iating currency must have a higher nominal interest rate to compensate intern­ational investors so that the real return is the same on loans in different currencies
Natural Level of Production
Yn = F ( K, E ( 1 - un ) L )
IS Equation
determines aggregate demand and production in the small open economy

Y=C ( Yd, Ye - Te, r*, A ) + I ( r*, Ye, K) + G + NX ( ε, Y*, Y )

C= private consumption (units of domestic goods)
NX = net exports ( units of domestic goods)
Yd = disposable income
Ye = expected future income
r* = real interest rate
A = assests
G = government spending
K = capital
I = investment
ε = real exchange rate
Y* =
Te =

where
Yd = Yn - T +r* ( D + F )
Natural Real Exchange Rate (open)
real exchange rate that is consistent with production at natural level

εn