Consider a pair of countries, "host" and "source", in a world of free capi- tal mobility which …xes the world rate of interest, denoted by r. We will now describe the host country, whose economic variables will be subscripted by "H". The description of the source country is similar with a subscript "S". Variables with neither an H nor S subscript are identical for the two countries. There is a representative industry whose product serves both for consumption and investment. As before, …rms last for two periods. In the …rst period there exists a continuum ofNH …rms which di¤er from each other by a productivity factor ". The number NH of …rms (or entrepreneurs) is
…xed. We refer to a …rm which has a productivity factor of " as an " …rm. The cumulative distribution function of " is denoted byG(:);with a density functiong(:):
As before, we assume for simplicity that the initial net capital stock of each …rm is the same and denote it by (1 )K0
H, where is the rate of physical depreciation. If an" …rm invests I in the …rst period, it augments its capital stock to K = (1 )K0
H +I, and its gross output in the second period will be AHF(K; L)(1 +"), where L is the labor input (in e¤ective units) andAH is a country (H) - speci…c productivity parameter. Note that
4.2. WAGE DETERMINATION 79
As before, we assume that there exists a …xed setup cost of investment,
CH; which is the same for all …rms (that is, independent of "). We assume that, due to some (suppressed) …xed factor, F is strictly concave, exhibiting diminishing returns to scale in K and L. Note that the average cost curve of the …rm is U-shaped, so that perfect competition, which we assume, can prevail.1 Consider an "-…rm which does invest in the …rst period an amount
I =K (1 )K0
H in order to augment its stock of capital toK. Its present value becomes V+(A H; KH0; "; wH) CH; where V+(AH; KH0; "; wH) = max (K;L) AHF(K; L)(1 +") wL+K 1 +r (K K 0 H) ; (4.1) and where we assume for notational simplicity that = 0.
The demands of such a …rm for K and L are denoted by K+(A
H; "; wH) and L+(A
H; "; wH). They are given by the marginal productivity conditions:
AHFK(K; L)(1 +") =r; (4.2)
and
AHFL(K; L)(1 +") =wH; (4.3)
where FK and FL denote the partial derivatives of F with respect to K and
L, respectively. As before, we assume that " is bounded from below by -1, so that output is always nonnegative; and bounded from above by one.
Note, however, that an "-…rm may chose not to invest at all [that is, to stick to its existing stock of capital (KO
H)] and avoid the lumpy setup cost
CH. Naturally, a …rm with a low " may not …nd it worthwhile to incur the setup cost CH. In this case, its present value is:
V (AH; KH0; "; wH) = max
L
AHF(KH0; L)(1 +") wHL+KH0
1 +r : (4.4)
The labor demand of such a …rm, denoted byL (AH; KHO; "; wH); is de…ned by:
AHFL(KHO; L)(1 +") = wH: (4.5)
It is straightforward to show that (@V+=@") (@V =@")>0(see Appen-
dix 4A.1). Therefore, there exists a cuto¤ level of", denoted by"0;such that
an"-…rm will make a new investment, if" > "0. This cuto¤ level of"depends
onAH; CH; KH0;and wH. We write the cuto¤" as"0(AH; CH; KH0; wH): It is de…ned implicitly by:
V+(AH; KH0; "0; wH) CH =V (AH; KH0; "0; wH): (4.6)
That is, the cuto¤ productivity level is the level at which the …rm is just indi¤erent between making a new investment and incuring the setup cost or sticking to its existing capital stock.
4.2. WAGE DETERMINATION 81
The wage ratewH is determined in equilibrium by a clearance in the labor market. We assume that labor is con…ned within national borders. Denoting the country’s endowment of labor in e¤ective units by L~0
H , we have the following labor market-clearing equation:
NH "0(AH;CZH;KH0;wH) 1 L (AH; KH0; "; wH)g(")d"+ NH 1 Z "0(AH;CH;KH0;wH) L+(A H; "; wH)g(")d"= ~L0H: (4.7)
Dividing the latter equation through by NH, yields:
"0(AH;CZH;KH0;wH) 1 L (AH; KH0; "; wH)g(")d" + 1 Z "0(AH;CH;KH0;wH) L+(A H; "; wH)g(")d" =L0H; (4.8) where L0
H L~0H=NH is the e¤ective labor per …rm.
Note that no similar market-clearing equation is speci…ed for capital, because we assume that capital is freely mobile internationally, and its rate of return is equalized internationally. The same description with the subscript "S" replacing "H" holds for the source country.
Note that di¤erences in labor abundance between the two countries are manifested in the wage di¤erences. To see this, suppose that the two countries are identical, except that e¤ective labor per …rm is more abundant in the
host country than in the source country, that is: L0
H > L0S. Note also that
the number of …rms in the economy is also a measure of the abundance of entrepreneurship. Thus, the abundance (respectively, scarcity) of labor is also relative to the scarcity (respectively, abundance) of entrepreneurship. If wages were equal in the two countries, then e¤ective labor demand per …rm were equal and the market-clearing condition [equation (4.7)] could not hold for both countries. Because of the diminishing marginal product of labor, it follows that the wage in the relatively labor-abundant country is lower than in the relatively labor-scarce country, that is: wH < wS2. Thus, equal returns to capital (through capital mobility coexist with unequal wages3. This provide a complementary reconciliation of the Lucas (1990) paradox of why capital does not ‡ow from rich to poor countries.4