MACROECONOMICS FORMULAS EDITION A.A. 2018 - 2019 Written and edited by Amina Costanzo and Valentina Tuveri 1 2 This material has been written by students without any possible intention of substituting the official teaching materials provided by the University. Therefore, it should be merely seen as an useful tool for the study of the subject. It does not provide such complete exam preparation as the teaching material Implemented by the University. FINANCIAL MARKETS AND EXPECTATIONS REAL INTEREST RATE ) π" ≈ π" − π"'( EXPECTED PRESENT DISCOUNTED VALUE (PDV) (Or present discounted value of the sequence of payments) 1 1 €π" = €π§" + €π§ ) + €π§ ) + β― = π(€π§" , €π§ ) , π" , π ) ) ) 1 + π" "1( (1 + π" )(1 + π(,"1( ) "15 [It is an increasing function of both current and expected future payments and it is a decreasing function of both current and expected future interest rates] With ) ) ) (€π§" , €π§"1( , €π§"15 , … , €π§"19 , )= sequence of current and expected future payments ) ) (π" , π"1( , π"15 , … )= current and expected future interest rates By which we get €π" = €π" × π" BOND PRICES 1. One-year bonds (short-term): €π(," = 2. Two-years bonds (long-term) €π5," = 100 1 + π(," 100 100 = ) (1 + π5," )5 =1 + π(," >(1 + π(,"1( ) 3. N-years bonds (long-term): €π9," = 100 (1 + π5," )9 ARBITRAGE CONDITION 1. With no risk premium (x=0) 1 + π(," = By which we get: €π5," = 2. With risk premium (x>0) 3 ) €π(,"1( €π5," C €?@,AB@ (1D@,A 1 + π(," + x = By which we get: ) €π(,"1( €π5," €?C €π5," = (1D@,AB@ 1F @,A EXPECTED PRICE OF A SHORT-TERM BOND ) €π(,"1( = By which we get: €π5," = 100 ) 1 + π(,"1( 100 ) (1 + π(," )(1 + π(,"1( ) BOND YIELDS TO MATURITY 1. Relation between long-term, short-term and next year's expected one-year interest rates 1 ) π5," = =π(," + π(,"1( > 2 2. General case (n-years) 9'( π9," = 1 1 ) ) ) ) =π(," + π(,"1( + π(,"15 + β― + +π(,"19'( > = I π(,"1J π π JKL STOCK MARKET AND MOVEMENTS IN STOCK PRICES 1. In nominal terms ) ) ) €π·"1( €π·"15 €π·"19 €π" = + + β― + ) ) 1 + π(," + π₯ =1 + π(," + π₯>=1 + π(,"1( + π₯> =1 + π(," + π₯> … =1 + π(,"19'( + π₯> With €π" = nominal price of the stock in the current period ) ) €π·"1( , €π·"15 ,…= expected future dividends 2. In real terms ) ) ) π·"1( π·"15 π·"19 π" = + + β― + ) ) 1 + π(," + π₯ =1 + π(," + π₯>=1 + π(,"1( + π₯> =1 + π(," + π₯> … =1 + π(,"19'( + π₯> [the stock price depends positively on expected future dividends and negatively on expected future interest rates and risk premiumi] ARBITRAGE CONDITION BETWEEN STOCKS AND CONDIZIONE DI ARBIRTRAGGIO TRA AZIONI E TITOLI ANNUALI 1 + π(," + x = ) ) €π·(,"1( + €π(,"1( €π" Stock dividends = bond yields 4 THE ROLE OF EXPECTATIONS IN THE REAL ECONOMY: CONSUMPTION AND INVESTMENT WHEALTH 1. Non-human wealth (WFI) π WFI = πΎπ + πΎππ with π πΎπ = financial wealth πΎππ = housing wealth 2. Human wealth (V(π¦ ) )) π(π¦") ) = π¦" + ) ) π¦"1( π¦"15 + ) )+β― 1 + π" (1 + π" )(1 + π"1( 3. Total wealth (πΎ"X" π ) π πΎ"X" = πΎπ + πΎππ + π(π¦") ) π CONSUMPTION BUDGET CONSTRAINT 1. Present consumption πΆ" + π = (π" − π" ) 2. Future consumption ) ) πΆ" + π = π"1( − π"1( + (1 + π)π With S= private saving 3. Intertemporal consumption a. If the individual has only human wealth πΆ" + By which we get ) ) πΆ"1( π"1( − π"1( = π" − π" + 1+π 1+π ) ) πΆ"1( = −(1 + π)πΆ" + (1 + π)(π" − π" ) + (π"1( − π"1( ) b. If the individual has also non-human wealth πΆ" + ) ) πΆ"1( π"1( − π"1( = π" − π" + + ππΉπΌ 1+π 1+π DEMAND FUNCTION FOR INVESTMENT π(π") ), π" πΌ" = πΌ[ ] + 5 With π(π") ) = present value of the exected profits (following the investment) 6 π(π") ) = ) ) ) (1 − πΏ)π"15 (1 − πΏ)5 π"1c π"1( + + +β― ) ) ) )(1 ) (1 + π" ) + (1 + π"1( 1 + π" (1 + π" ) + (1 + π"1( + π"15 ) Where d= depretiation rate The firm invests in the current period if and only if: π(π") ) ≥ πf With πf = real price of the investment So we get πΌ" = πΌ g π(π") ) πf π) , h = πΌ[ − + + π π ) πf − − − d ] − OPEN ECONOMY AND GOODS MARKET REAL EXCHANGE RATE Price of domestic goods in terms of foreign goods π= Where πΈπ π∗ P= domestic price level (expressed as domestic currency) E= nominal exchange rate P*= general foreign price level (expressed as foreign currency) UNCOVERED INTEREST PARITY (U.I.P.) 1 + π" = The U.I.P can be rewritten as follows: 1 + π" = By which we get 1 + π" = Where C 'l lAB@ A lA (1 + π"∗ ) ) πΈ"1( + πΈ" − πΈ" πΈ" = nominal expected tasso di depreciation/appreciation rate π" = π"∗ 7 (1 + π"∗ ) − πΈ" +1 πΈ" ) πΈ"1( A good approximation is Where (1 + π"∗ )πΈ" ) πΈ"1( ) πΈ"1( − πΈ" πΈ" π" = domestic interest rate π"∗ =foreign interest rate πΈ" =current nominal exchange rate ) πΈ"1( = expected future exchange rate [we do not consider transaction costs and insolvency risk] EQUILIBRIUM CONDITION OF THE GOODS MARKET IN AN OPEN ECONOMY Where π−π π π π∗ π =πΆm n+πΌm n+πΊ +πm + + − + πΌπ π π π n− ( ) − π + + π∗ π πm n= demand for exports + − st π π ( )= demand for exports u + + st π π π∗ π ππ = π m n− u ( )= net exports + + + − SAVING, INVESTMENT AND THE CURRENT ACCOUNT BALANCE π = πΆ + πΌ + πΊ + ππ By subtracting to both sides C+T Knowing that S=Y-C-T We get π − πΆ − π = πΆ + πΌ + πΊ + ππ − πΆ − π π = πΌ + πΊ − π + ππ ππ = S + πxyz − πΌ Ø NX>0= trade surplus (an excess of savings over investments) Ø NZ<0= trade deficit (an excess of investments over savings) 8 THE IS-LM IN AN OPEN ECONOMY UNCOVERED INTEREST PARITY πΈ" ) πΈ"1( Domestic interest rates= foreign interest rates 1+π ) πΈ= πΈ{ 1 + π∗ (1 + π" ) = (1 + π"∗ ) Where πΈ{ ) = expected exchange rate MUNDELL- FLEMING MODEL 1. In a flexible exchange rates regime IS LM π−π π π π∗ π =πΆm n+πΌm n + πΊ + ππ m + + − + π = π€Μ UIP πΈ= 1+π ) πΈ{ 1 + π∗ By which we get the equilibrium condition π−π π π π∗ π =πΆm n+πΌm n + πΊ + ππ ~ + + − + 2. In a fixed exchange rates regime IS LM π 1 + π πΈ{ ) π∗ • − 1 +− π−π π π π∗ π =πΆm n+πΌm n + πΊ + ππ m + + − + π π n − − π = π × πΏ(π, π) By which we get the equilibrium condition π−π π π π∗ π =πΆm n+πΌm n + πΊ + ππ ~ + + − + 9 π πΈ n − − π 1 + π πΈ{ ) π∗ • − 1 +− EXCHANGE RATE REGIMES EXHANGE RATE MOVEMENTS UNDER FLEXIBLE EXCHANGE RATES πΈ" = 1 + π" ) πΈ 1 + π ∗ " "1( Continuing to solve forward in time in the same way we get: πΈ" = πΈ ) "1( (1 + π" )(1 + π" ) ) … . (1 + π"19 ) ) (1 + π ∗ " )(1 + π" ∗) ) … . (1 + π"19 ∗) ) 10 PUBLIC DEBT PUBLIC DEFICIT π·" = ππ΅"'( + πΊ" − π" Where π·" = government deficit at time t π΅"'( = government debt at the end of the year t-1 π" =taxes minus transfers during year t πΊ" = government spending on good and services during year t πΊ" − π" = primary deficit (if πΊ" > π" government spending is larger that net taxes incomeà deficit) (if πΊ" < π" net taxes income is larger than goverment spendingà surplus) ππ΅"'( = interest payments GOVERNEMENT BUDGET CONSTRAINT (in real terms) In order to fit the budget constraint the government must run a new deficit On the basis of the intertemporal budget constraint: π΅" − π΅"'( = π·" By using the definition of deficit π΅" = (1 + π)π΅"'( + πΊ" − π" FULL REPAYMENT OF THE DEBT IN YEAR t π" − πΊ" = (1 + π)"'( (−βπL ) With π" − πΊ" = primary deficit DEBT STABILISATION IN YEAR t π" − πΊ" = π(−βπL ) THE EVOLUTION OF THE DEBT-TO-GDP RATIO π΅" π΅"'( πΊ" − π" = (1 + π) + π" π"'( π" With g= real growth rate of output π= π" − π"'( π"'( π" =1+π π"'( π΅" π΅"'( πΊ" − π" = (1 + π − π) + π" π"'( π" Where ‡A 'ˆA ‰A 11 = primary deficit and GDP ratio=d 1 + π=interest rate ŠA = the evolution of the debt-to-gdp ratio=bt ‰ A So we get π" = (1 + π − π)π"'( + π THE EVOLUTION OF THE DEBT-TO-GDP RATIO π΅" π΅ˆ'( π΅ˆ'( πΊ" − π" − = (π − π) + π" π"'( π"'( π" If ŠA ‰A − Š•Ε½@ ‰AΕ½@ =0à ŠA ‰A = π{ (there are no changes of the debt-to-GDP ratio) πΊ" − π" π" 1 πΊ" − π" π{ = π − π π" π{ = (1 + π − π)π{ + Where ‡A 'ˆA ‰A =d π{= steady debt-to-GDP ratio 12