THE OPEN ECONOMY SOLOW MODEL: CAPITAL MOBILITY Carl-Johan Dalgaard Department of Economics University of Copenhagen
OUTLINE Part I: Assessing international capital mobility empirically. — The Feldstein-Horioka Puzzle (S&W-J, Ch. 4.1.) — The Lucas Paradox (Lucas, 1990) — Resolving the Lucas paradox? (Caselli and Feyrer, 2005) Part II: Open economy Solow model - Capital mobility — The basic model — Empirical issues 2
PART I - THE FELDSTEIN-HORIOKA PUZZLE 3
BACKGROUND In a closed economy setting we know the following must hold Y = C + I ⇔ I = S. Hence, total investments (or the investment share of GDP, I/Y ) must vary 1:1 with total savings (or the savings rate S/Y ). Thus, a simple regression µ I ¶ µ S ¶ = α + β + � i Y Y i i should return β OLS = 1 (and α OLS = 0 in the absence of national accounts mistakes) . 4
BACKGROUND In an open economy, however, things should work di ff erently. In par- ticular, the following must be true: S − I = ∆ F If savings exceed domestic investments, the country is building up net foreign assets. That is, on net the country is investing abroad . As a result S − ∆ F = I. (*) which says domestic investments equal savings minus what we (on net) invest abroad. 5
BACKGROUND Reconsider the regression model from before µ I ¶ µ S ¶ = α + β + � i Y Y i i In light of equation (*) � i ≡ − ∆ F/Y − α. Estimating the above by OLS we get ³ ´ S COV Y , ∆ F/Y ³ ´ β OLS = β − S var Y ³³ ´ ´ ³ ´ S S as COV i , � i = − COV Y , ∆ F/Y . Thus β OLS is expected Y to be (much) smaller than 1. 6
THE PUZZLE The startling fi nding was, however, this Figure 1: Source: Feldstein and Horioka, 1980. Suggests limited capital mobility. In striking contrast to e.g. evidence on very similar interest rates on similar assets dispite being located in di ff erent countries (thus “a puzzle”). 7
THE PUZZLE This fi nding remains something of a puzzle, and is robut to more recent periods (albeit the size of the coe ffi cient shrinks) Figure 2: Source: Obstfeld and Rogo ff (2000). To date: No complete resolution. 1 1 Perhaps in part because no-one seem to know how small β is supposed to be in order to be consistent with capital mobility. 8
THE LUCAS PARADOX 9
SET-UP Another contribution striking a similar cord is Lucas (1990). Lucas’ focus is on rich and poor countries; not just “within the group of rich” Basic point of departure is a one good economy, featuring competitive market. Firms use a Cobb-Douglas production function. They maximize pro fi ts K,L K α L 1 − α max − wL − ( r + δ ) K. | {z } | {z } = Y user cost of capital Focusing on FOC wrt K r + δ = αK α − 1 L 1 − α = αk α − 1 . Suppose this condition holds in any country. 10
THE PARADOX In particular, suppose we consider India and the US. Then (ignoring δ ) r INDIA = αk α − 1 INDIA and r US = αk α − 1 US . Implying µ k INDIA ¶ α − 1 r INDIA = r US k US Capital is hard to measure. But note: y = k α (cf production function). SO µ y INDIA ¶ α − 1 r INDIA α = r US y US Since y US /y IND ≈ 15 and α = . 4 , this implies µ 1 ¶ . 4 − 1 r INDIA . 4 = ≈ 58 r US 15 Why doesn’t capital fl ow to poor countries??? 11
A SOLUTION TO THE PARADOX? Maybe we are getting it wrong because we are missing something. Con- sider the modi fi ed production function Y = XK α L 1 − α where X could be human capital (Lucas’ favorit), or something else (technology). Observe that we now get y = k α X ⇔ k = ( y/X ) 1 /α Hence the fi rst order condition from pro fi t maximization is (still ignoring δ ) α − 1 1 r = MP K = αk α − 1 X = αy α X α 12
A SOLUTION TO THE PARADOX? Now, if r IND ≈ 1 r US then we need µ y IND ¶ α − 1 α µ X IND ¶ 1 r IND α = ≈ 1 r US y US X US or µ y us ¶ 1 − α X us = 15 0 . 6 ≈ 5 . = X ind y ind Lucas manages to motivate “ X ” almost entirely by human capital; h = X. 13
WHY IT MAY NOT BE A RESOLUTION 1. Evidence for external e ff ects of human capital is not strong. 2. Lucas’ calculation is, under reasonable assumptions, not entirely internally consistent. To see this, suppose we rewrite the production function slightly µ K ¶ α 1 − α 1 y = k α X ⇔ y = X 1 − α Y ³ ´ 1 1 1 − α = 5 If X us X us 1 − . 4 ≈ 15! If X is human capital, this X ind = 5 , then X ind implies that we can account for the entire observed di ff erence in labor productivity by this variable alone (growth “multiplier e ff ect”). Not plausible. Of course, things like “A” (TFP) could be included in X . But that violates the calibration. Another look is warranted. 14
A RESOLUTION TO THE LUCAS PARADOX? 15
SET-UP We begin with a set of basic assumptions A1 Y = F ( K, XL ) , X = “e ffi ciency" (human capital, productivity). CRTS: F K · K + F L L = F = Y A2 Competitive markets, and multi-good economy ( p Y 6 = p I ) Implication 1 R · K + w · L = p Y Y , where p Y is the GDP de fl ator, and R is the rental rate of capital (sometimes called: “usercost of capital", Hall and Jorgenson, 1963) R = p I · ( r + δ ) , p I = price of investment good. Implication 2 p Y · F K = R and p Y · F L = w. 16
SET-UP Under these assumptions, we can now obtain an estimate for F K . Let α K ≡ RK/p Y Y (capital’s share). Then K = p I Y F K = α K ( r + δ ) . p Y Using data on capital’s share in national accounts, we can calculate F K for a number of countries. The question is whether marginal products are equalized ... 17
RESULT 1: MARGINAL PRODUCTS ARE NOT EQUAL- IZED They are not ... Figure 3: 18
REFLECTING ON THE RESULT Investors probably do not care about the marginal product per se. They care about the return to their investment, r Perfect capital mobility would require the equalization of the r ’s Fairly easy to calculate the implied r , given the above “view of the world”: K = MP K = p I Y ( r + δ ) ⇔ r + δ = p Y α K MP K , p Y p I it is assumed that δ is about the same in all countries ... 19
RESULT 2: RENTAL RATES ARE INDEED NOT THAT DIFFERENT Figure 4: 20
SUMMING UP Lucas: There are no di ff erences in real rates. Human capital is solely re- sponsible. Shortcoming: Overestimates productivity di ff erences. Ignores relative price di ff erences C&F: marginal products are not equalized. But rental rates are not that di ff erent. — Rich places have a lot of capital (even in a fully integrated world) because: “ X ” is large (not only human capital though), and because the relative price of investment is low ( p I /p y ). — Hence if international capital markets do allocate capital reasonably e ffi ciently, then we better think about how it a ff ects our understanding of the growth process 21
Part II: OPEN ECONOMY SOLOW MODEL - CAPITAL MOBILITY 22
THE BASIC MODEL: SET UP We are considering an open economy, where capital is fully mobile. Labor, however, is not. All markets are competitive. Two new basic relationsships: 1. Savings 6 = Domestic total investment 2. Production and Income are not longer identical The national accounts identity is Y = C + I + NX ⇐ ⇒ Y + rF = C + I + NX + rF where NX represents net exports, and F is holdnings of foreign capital; rF =income in fl ow from foreign capital holdnings. 23
THE BASIC MODEL: SET UP Gross National Income (GNI) is therefore Y + rF , cf 2 . Y is production, or, Gross Domestic Product. Note that if NX> 0 the economy must be building up assets abroad (exports > imports). Hence, in general ( r is assumed constant): NX t + rF = F t +1 − F t Finally, by de fi nition S t = Y t + rF t − C t Combining S t = I t + F t +1 − F t Hence, savings can be used to accumulate domestic capital ( I ), or, foreign assets ( F t +1 − F t ) , cf. 1. 24
THE BASIC MODEL: SET UP As “usual” we have that K t +1 = I t + K t (i.e., here we assume δ = 0 , for simplicity) But, observe that we now have K t +1 = S t − ( F t +1 − F t ) + K t ⇔ K t +1 + F t +1 = S t + K t + F t If we de fi ne total wealth (domestically owned local ( K ) and Foreign ( F ) capital) V t = K t + F t Leaving us with V t +1 = S t + V t . (1) 25
THE BASIC MODEL: SET UP The fundamental assumption about savings behavior is the same: Peo- ple save a constant fraction of total income . In the open economy S t = s · ( Y t + rF t ) , 0 < s < 1 . (2) We will also maintain our basic assumption about production Y t = F ( K t , L t ; A ) = AK α t L 1 − a t From (A) competitive market, and (B) constant returns to scale follows Y t = w t L t + rK t (3) Since w t = ∂F [ · ] , r = ∂F [ · ] ∂L t ∂K t 26
THE BASIC MODEL: SET UP The fact that capital is fully mobile has an important implication. De- note by r w the world real rate of interest. Then at all points in time r w = r = ∂F [ · ] ∂K t Substituting for ∂F [ · ] ∂K t we fi nd µ αA ¶ 1 1 − α r w = αAk α − 1 ⇔ ¯ k = . r w Hence the capital-labor ratio is constant, absent changes in A . Suppose A rises ... Note also, that this implies a constant wage rate w = ∂F [ · ] = (1 − α ) A ¯ k α w t = ¯ (4) ∂L t 27
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