Transcript Investment
Investment Mid-term Exam • Tuesday, November 17th 9AM • ? • Semi-open Book (Bring 1 A4 size paper with handwritten notes)) • Coverage. Lecture notes through November 15th. this one. Interest Rates • Most observed interest rates are money interest rates, i.e. how much money you can get in the future in exchange for some money today. 1+it • We are more interested in goods interest rates i.e. how many goods you can get in the future in exchange for some goods today. 1+rt Real Interest Rates and Nominal Interest Rates • The goods interest rate is derived from the nominal interest rate. If the price of goods is Pt then giving $1 to the bank is equivalent to saving 1 P goods. t • When you get the money back with interest after 1 year you can buy 1 it goods. Pt 1 Real Interest Rate • Goods gotten per goods given up is 1 it Pt 1 1 Pt 1 it Pt 1 Pt 1 rt • Future price level is not known when the savings decision is made. Must be forecast by savers (and borrowers). Only ex post real interest rate observed by economists. Interest Rates: The Data • Main source of international interest rate data is IMF’s International Financial Statistics which is only available with a subscription fee. • Interest rate data is typically available from Central Banks website (List of Websites) • US Data is most conveniently obtained from the FRED database at the St. Louis Federal reserve. Real GDP: Yt • GDP aka Nominal GDP aka Current Dollar GDP is the weighted sum of the number of goods produced using their current prices as the weight. • Real GDP aka Constant Dollar GDP aka GDP adjusted for inflation is the weighted sum of the number of goods produces using the Base Year prices as yardsticks. Price Indices: Pt • Two most commonly used price indices are GDP Deflator and Consumer Price Index (CPI) • The CPI is the price of a representative market basket of goods relative to the price of that same basket during a benchmark/base year (multiplied by 100). • The GDP deflator is the ratio of nominal GDP to Real GDP (multiplied by 100). Nominal GDP GDP P GDP Deflator Real GDP Y The Data • The U.N. maintains annual statistical databases of GDP and its components (investment, consumption, etc.) for constant and current dollar measures for a wide variety of countries UN Main Aggregates Data Base • Data on aggregates at higher frequencies are typically available from national statistical agencies Examples – USA – Hong Kong – Japan Investment is a small share of GDP 50.00% 45.00% 40.00% 35.00% 30.00% 25.00% 20.00% 15.00% 10.00% 5.00% 0.00% China France Germany Hong Kong SAR of China Japan Republic of Korea United States Investment Shares Japan Current Dollar or Constant Dollar 0.37 0.35 0.33 0.31 0.29 0.27 0.25 19 70 19 72 19 74 19 76 19 78 19 80 19 82 19 84 19 86 19 88 19 90 19 92 19 94 19 96 19 98 20 00 20 02 20 04 0.23 Current Constant 19 70 19 72 19 74 19 76 19 78 19 80 19 82 19 84 19 86 19 88 19 90 19 92 19 94 19 96 19 98 20 00 20 02 20 04 Relative Price of Investment goods. Japan 1.15 1.1 1.05 1 0.95 0.9 Gross Domestic Investment (and Components) Table 5.2.5. Gross and Net Domestic Investment by Major Type [Billions of dollars] Bureau of Economic Analysis Downloaded on 10/23/2006 At 8:46:47 AM Last Revised August 02, 2006 Line Gross Depreciation Net % of GDI 1 Gross domestic investment 2454.5 1604.8 849.7 100.00% 4 Gross private domestic investment 2057.4 1352.6 704.8 83.82% 22 Change in private inventories 7 Fixed investment 2036.2 1352.6 683.6 82.96% 10 Nonresidential 1265.7 1045.6 220.1 51.57% 13 Structures 338.6 260.6 78 13.80% 16 Equipment and software 927.1 785 142.1 37.77% 19 Residential 770.4 307 463.4 31.39% 23 Gross government investment\1\ 397.1 252.2 144.8 16.18% 30 Structures 248.9 127.4 121.5 10.14% 37 Equipment and software 148.1 124.8 23.3 6.03% BEA NIPA Table 5.2.5 Volatility: Investment and GDP Annual Growth Rates Hong Kong 30.00% 25.00% 15.00% 10.00% 5.00% 0.00% 19 65 19 68 19 71 19 74 19 77 19 80 19 83 19 86 19 89 19 92 19 95 19 98 20 01 % Growth Rates 20.00% -5.00% -10.00% -15.00% -20.00% GDP I Demand for Capital • A Cobb-Douglas firm that hires labor and capital in rental markets maximizes profits. 1 max PK t t ( At Lt ) Wt Lt Rt Kt Yt Rt K t Pt Optimal K R/P R/P MPK K K* Cost of Capital Rises R/P R/P MPK K K** K* Productivity Rises R/P R/P MPK’ MPK K* K*** K What is the cost of capital? • Examine an optimizing odel of investment. – Firm starts in period 0 with K0. – Firms hire workers at wage rate Wt in every period. – Firms sell goods at price Pt I P – Firms buy new investment goods at price t – Firms use new investment goods to accumulate capital K (1 ) K I t 1 t t Firm’s Intertemporal Problem • Rather than optimizing profits in each period, the firm managers will construct a plan to maximizes the real net worth of the firm. Nominal Present Value • Assume that the firm will last until period T. The firm will generate a certain cash flow. The present value (in $ terms is) 1 1 I V0 0 PK ( A L ) W L P I t t t t t t t t 1 i t 0 t • We can think of this as the financial value of the firm. Real Present Value • The real present value is V0 0 1 1 I K ( A L ) w L p I t t t t t t t P0 1 r t 0 • Where the relative prices Wt wt Pt I Pt p Pt I t Optimization Problem • Write the optimization problem as choosing a stream of investment, capital, and labor to maximize the net worth of the firm. 1 Kt ( At Lt ) wt Lt p I max t K ,I ,L (1 r ) t 0 I t t • Subject to restrictions on the path of capital Kt 1 It (1 ) Kt t Lagrangian Multiplier • We can solve this by assuming that there is some large cost q to the firm of setting tomorrows capital stock greater than the sum of today’s capital stock. max K ,I ,L t 0 Kt ( At Lt )1 wt Lt ptI It qt Kt 1 It (1 ) Kt (1 r )t • How large should this cost be? The minimum necessary to insure the firm never violates the constraint. • The first order conditions are I wt pt qt (1 ) Yt t t t (1 r ) Lt (1 r ) (1 r ) (1 r )t (1 r )t 1 Rt 1 Pt 1 Yt 1 qt 1 qt (1 ) t 1 K t 1 (1 r ) (1 r )t I 1 t 1 I I (1 r ) pt pt 1 (1 ) 1 r (1 ) ptI 1 t 1 Cost of Capital • The rental cost of capital is the cost of owning capital for one period. • This cost includes: – The foregone interest that you could have received had you lent goods. – The depreciation of the capital – Any “capital loss” that occurs Cost of Capital • The user cost of capital is sometimes approximated as Rt 1 (1 r ) ptI ptI1 (1 ) Pt 1 I 1 tI1 I I 1 r (1 ) p p r t t t 1 t 1 1 t 1 Dynamics of Investment Capital Adjustment Costs • Assume that when investment is large, some of the investment goods are wasted. Number of goods purchased per investment goods installed • Assume pI =1 for all t. c 2 Costt I t ( I t ) 2 Lagrangian Optimization • Choose Investment and capital to maximize value of the Firm max K ,I , t 0 * 1 t t Kt ( A L ) c 2 I t I t qt K t 1 (1 ) K t I t 2 (1 r )t FOC • Optimize 1 + c I t qt = t (1 + r) (1 + r) t FOC Yt 1 K t 1 qt 1 qt (1 ) t 1 t 1 (1 r ) (1 r ) (1 r )t Investment Equation • We can write investment as a function of the marginal q (i.e. the discounted benefit of adding a unit of capital) 1 I t = q t 1 c qt Yt 1 (1 ) qt 1 Kt 1 (1 r ) Recursive Substitution q0 q0 Y1 K (1 ) q1 q1 1 (1 r ) Y1 K (1 ) Y2 K (1 ) q2 2 (1 r ) Y1 K (1 ) q2 1 (1 r ) 1 (1 r ) q0 q2 Y1 K 1 (1 r ) (1 ) Y2 (1 r )2 Y3 K (1 ) q3 3 (1 r ) K2 (1 )2 q3 (1 r )3 Marginal Q • Marginal q is the discounted sum of future marginal product of capital discounted by the real interest rate. • The benefit of having capital inside the firm relative to the opportunity cost of not having funds outside the firm invested in the bank. (1 ) q0 MPK j j j 1 (1 r ) j 1 Intuition of Marginal q • The variable xt is the benefit of having an extra unit of capital at the end of time t which includes marginal product of capital. • But any investment today will last for many periods. The marginal benefit of investing is the discounted sum of having capital in future periods. Note that capital depreciates at a rate of (1-δ) leaving (1-δ)t after t periods. Implications of Marginal q • Long-lasting changes in capital productivity or interest rates have more effect on investment. • Investment is more sensitive to long-run interest rates. Steady State • Dynamic Equation 1 1 I t = q t 1 c • Dynamic Equation 2 q MPK (1 ) q MPK (r ) q MPK q (1 r ) r Investment Curve • Whenever q > 1, then invest and capital stock increases. • Whenever q < 1, then disinvest and capital stock decreases. • Whenever q = 1, leave capital stock alone. q ΔK > 0 1 ΔK < 0 K Lagrangian • Optimize 2 max K ,I ,L K t ( At Lt )1 t 0 I c It pt K t wt Lt ptI I t qt K t 1 I t (1 ) K t 2 Kt (1 r )t FOC wt (1 - α)Yt = (1 + r)L t (1 t + r) 2 It ptI + ptI c t (1 + r) K t -δ (1 t+ r) = qt t I I Yt 1 cI ptI1 t ptI1c t t K t 1 2 Kt Kt K t qt 1 (1 ) qt (1 r )t 1 (1 r )t 1 (1 r )t Investment Equation • We can write investment as a function of the marginal q (i.e. the discounted benefit of adding a unit of capital) It p + cp - δ K t = qt It - δ 1 ( qt 1) I 1+ ( r) t 1+ ( r) K t p c t t I t I t It 1 qt ( I 1) pt Kt c Marginal q • Recursively substitute marginal benefit of owning capital I I 1 Y cI 2 t 1 ptI1 t 1 cptI1 t 1 t 1 MBK t 1 1 r Kt 1 2 Kt 1 Kt 1 Kt 1 (1 ) (1 ) qt MBKt 1 qt 1 , qt 1 MBKt 2 qt 2 , (1 r ) (1 r ) (1 ) qt 2 MBKt 3 qt 3 (1 r ) 2 3 (1 ) (1 ) (1 ) qt MBKt 1 MBKt 2 MBKt 2 MBKt 3 ..... (1 r ) (1 r ) (1 r ) Intuition of Marginal q • The variable xt is the benefit of having an extra unit of capital at the end of time t which includes marginal product of capital less capital adjustment costs plus the degree to which having more capital reduces future capital adjustment costs. • But any investment today will last for many periods. The marginal benefit of investing is the discounted sum of having capital in future periods. Note that capital depreciates at a rate of (1-δ) leaving (1-δ)t after t periods. Implications of Marginal q • Long-lasting changes in capital productivity or interest rates have more effect on investment. • Investment is more sensitive to long-run interest rates. Tobin’s q • Difficult to measure q for an individual firm. • Researchers typically use Tobin’s q as a proxy for marginal q: tq Market Value of Equity+Book Value of Debt Replacement Cost of Capital Equivalency • Multiply the first order condition by qt K t 1 MBK t 1 K t 1 (1 ) qt 1 K t 1 (1 r ) 2 1 c I I t 1 I I t 1 MBK t 1 K t 1 K t 1 cpt 1 I t 1 Yt 1 pt 1 1 r 2 K K t 1 t 1 Dt 1 2 q I Pt 1 t 1 t 1 1 c I I t 1 I wt Lt pt 1 qt 1 I t 1 Yt 1 pt 1 1 r 2 K 1 r t 1 q [(1 ) K t 1 I t 1 ] (1 ) 1 Dt 1 qt K t 1 MBK t 1 K t 1 qt 1 K t 1 t 1 Pt 1 (1 r ) 1 r 1 r qt K t 1 q K 1 Dt 1 t 1 t 2 Pt 1 1 r 1 r – Recursive substitution qt 1 K t 2 qt 2 K t 3 1 1 qt K t 1 t 1 , qt 1 K t 2 t 2 1 r 1 r 1 r 1 r 2 3 1 1 1 qt K t 1 t 1 t 2 t 3 .... 1 r 1 r 1 r Vt qt Vt qt K t 1 Vt qt I I K t 1 pt pt K t 1 Tobin’s q • A firm’s q is measured as the financial value of a firm divided by the replacement cost of capital. • When a firm is valued by financial markets at levels greater than the sale value of its capital, then capital is worth more inside the firm than out and the firm should add capital. • When financial value of a firm exceeds value of its capital, capital is worth more inside the firm than outside so the firm should move more capital inside the company. Data • Book Value of Debt can be obtained from a firms balance sheet • Market Value of Stock is the product of the stock price and the number of shares • Market value of physical capital is defined recursively. Define pKt+1 as the end of period replacement cost of capital. Pt I pKt 1 (1 ) I pKt It Pt 1 Empirical Results • Tobin’s q has only weak effects on investment. – Reason: Fluctuations in the stock market are not taken by firms as a measure of future value of capital. • After controlling for q, cash flow still affects the level of investment. Tobin’s q • Difficult to measure q for an individual firm. • Researchers typically use Tobin’s q as a proxy for marginal q: tq Market Value of Equity+Book Value of Debt Replacement Cost of Capital Tobin’s q • A firm’s q is measured as the financial value of a firm divided by the replacement cost of capital. • When a firm is valued by financial markets at levels greater than the sale value of its capital, then capital is worth more inside the firm than out and the firm should add capital. • When financial value of a firm exceeds value of its capital, capital is worth more inside the firm than outside so the firm should move more capital inside the company. Data • Book Value of Debt can be obtained from a firms balance sheet • Market Value of Stock is the product of the stock price and the number of shares • Market value of physical capital is defined recursively. Define pKt+1 as the end of period replacement cost of capital. Pt I pKt 1 (1 ) I pKt It Pt 1 Empirical Results • Tobin’s q has only weak effects on investment. – Reason: Fluctuations in the stock market are not taken by firms as a measure of future value of capital. • After controlling for q, cash flow still affects the level of investment. Assymmetric Information • There are a set of entrepreneurs with risky investment project. – Each project requires 1 unit of good to implement. – Each entrepreneur has 0 < W < 1 units of goods. – Each entrepreneur must borrows (1-W) from a risk neutral financial system which also accepts deposits at a rate 1+r. Project Risk • Investment projects will have a random payoff with a uniform distribution over the range [0, 2γ] • Expected value of the outcome is γ. • Each entrepreneur has its own level of γ - Riskier projects have higher expected returns. 1 2 γ 2γ Symmetric Information • To make them willing to provide loans, banks demand an expected return of 1+r. • If all information is public and free, banks can lend (1-W) and collect an equity payment of (1 r ) W Payoff • This will pay an expected value of (1+r)(1-W) for banks. • The expected payoff for entrepreneurs is γ -(1+r)(1-W). • Entrepreneurs will invest if γ -(1+r)(1-W) > (1+r)W Asymmetric Information • Assume that entrepreneurs naturally observe their own level of output. B • Banks must pay a cost of c in order to monitor output. This makes equity contract expensive to implement. • Risky debt is the efficient contract – Banks receive a predetermined payment D. – If entrepreneurs do not pay D, the banks pay monitoring cost and collect entire output level. Expected return of banks • If D < 2 γ , the probability that an entrepreneur will be able to fully repay the loan is the probability of default is D 2 1 2 γ D 2γ 2 D 2 and Payoff if there is a default • What is the average payoff of payoffs of project which are low enough that the entrepreneur must default? D 2 • The entrepreneur will only default if their payoff is less than D. • Given a uniform distribution, half of projects with payoffs less than D, will have payoffs less than D 2 Expected return of banks • If D > 2 γ , the entrepreneur will always default. The payoff to the banker would be γ-c. • If D < 2 γ, the payoff would be 2 D D D 2 c 1 2 D ( c) D D 2 2 2 2 4 • The payoff is a quadratic function of D with a maximum at MAX D 2 c If the lender asks for debt repayments beyond that point, they will be paying monitoring costs so frequently that they will lose money Returns as a function of Debt RMAX 2γ-c 2γ D Expected payoff to banker • Bankers demand an expected payoff at least equal to (1+r)(1-W). • They will never lend if RMAX < (1+r)(1-W). • If RMAX > (1+r)(1-W) then in a competive market, the debt levels will be 2 c 1 2 D D (1 r )(1 W ) 2 4 D 2 c 2 c 2 4 (1 r )(1 W ) Payoff to the entrepreneur • The entrepreneurs will have to pay returns to bankers to compensate them for their expected monitoring costs. The expected value of monitoring costs is 2 c D A c 2 2 c 2 4 (1 r )(1 W ) 2 c • The expected return to the entrepreneur is γ-(1+r)(1-W) – A. They will invest if γ-(1+r)(1-W) – A > (1+r)W. Investment Decision • We can write this function as (1 r )(1 W ) A (1 r )W • Hurdle for investment is higher when W < 1. • The greater is W, the less will be the average monitoring costs. Midterm Exam 2 • Tuesday, November 16 9-10:50 – Lecture Theater E • Coverage: Frictional, Unemployment, Consumption, Investment, • Bring Calculator, Writing Instruments, 1 A4 sized paper with written notes.