2.1 The Solow Model
2.1 The Solow Model 2.1.1. The basic model There are two input factors, capital K and labor L, which are combined according to a Cobb-Douglas production technology to produce final output Y : Y = FHK, LL = K a L1-a
with
0 0 generates skilled labor according to H = ey u L with y > 0
(112)
Note that if u = 0, then H = L, i.e. all labor is unskilled. By increasing u, a unit of unskilled labor increases the effective units of skilled labor H according to d logHHL dH ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = y ï ÅÅÅÅÅÅÅÅÅÅÅÅÅ = y H du du
(113)
Suppose that u increases by one unit (an additional year of schooling) and assume y = 0.1. In this case, H rises by 10%. This formulation is motivated by a large literature in labor economics that find that an additional year of schooling increases wages by something like 10%. As in the standard Solow model, physical capital is accumulated by investing a fraction of output ° K = sK Y - d K
(114)
The steady state This model can be solved for the steady state as the basic model. First express the production function in terms of output per worker (divide by L) y = k a HA hL1-a
(115)
with h := ey u . As for the saving rate we simply assume that agents use a constant fraction of their time for education, i.e. u is constant. Hence, h is also constant and thus the above intensive production function is very similar to the one used above. Dividing by A h we get èa yè = k è è with yè := y ê HA hL and k := k ê HA hL. The capital accumulation equation in terms of k can be written as è° è k = sK yè - Hn + g + dL k
(116)
(117)
è è° The steady state values of k and yè are found by k = 0, which yields è sK k ÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ èyÅ = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ n+g+d èa Substituting this into yè = k gives
sK i y yè * = jj ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ zz n + g + d k {
(118)
aêH1-aL
(119)
Rewriting this in terms of output per worker gives aêH1-aL sK i y h AHtL y* HtL = jj ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ zz kn+g+d{
(120)
12
econ_dyn_Heidelberg.nb
Countries are rich and others are poor because they [1] have high investment rates in physical capital, [2] spend a large fraction of time accumulating skills (education), [3] have low population growth rates, and [4] have high levels of technology. In addition, as in the Solow model, in the long run economies grow at constant rates.
The steady state How well does this model perform empirically in terms of explaining why some countries are richer than others? Since incomes are growing over time, we consider relative incomes. If we define per capita income relative to the USA to be y* êêy = ÅÅÅÅÅÅÅÅ ÅÅÅÅÅ y*US
(121)
then from (120), relative (steady state) income is given by sê K aêH1-aL êê êêê êêy* = J ÅÅÅÅ hA êêxÅÅÅÅÅ N
(122)
where x := n + g + d and a bar above a variable is used to denote a variable relative to its US value. In answering the question how well the extended Solow model performs in explaining cross-country differences we assume that g is the same across countries. This seems to be a valid first approximation (why?). The figure below compares the actual levels of (relative) GDP per worker in 1997 to the levels predicted by the model. The following parameter values have been assumed: a = 0.3, y = 0.1, and g + d = 0.075 for all countries. Finally, it is assumed that A is the same in all countries. It can be seen that the fit of the model is fairly good for rich countries (which are close the 45-degree line), but gets weaker for poorer countries (which are above the 45-degree line).
Source : Jones H2002L
Ñ
with
0 0 generates skilled labor according to H = ey u L with y > 0
(112)
Note that if u = 0, then H = L, i.e. all labor is unskilled. By increasing u, a unit of unskilled labor increases the effective units of skilled labor H according to d logHHL dH ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = y ï ÅÅÅÅÅÅÅÅÅÅÅÅÅ = y H du du
(113)
Suppose that u increases by one unit (an additional year of schooling) and assume y = 0.1. In this case, H rises by 10%. This formulation is motivated by a large literature in labor economics that find that an additional year of schooling increases wages by something like 10%. As in the standard Solow model, physical capital is accumulated by investing a fraction of output ° K = sK Y - d K
(114)
The steady state This model can be solved for the steady state as the basic model. First express the production function in terms of output per worker (divide by L) y = k a HA hL1-a
(115)
with h := ey u . As for the saving rate we simply assume that agents use a constant fraction of their time for education, i.e. u is constant. Hence, h is also constant and thus the above intensive production function is very similar to the one used above. Dividing by A h we get èa yè = k è è with yè := y ê HA hL and k := k ê HA hL. The capital accumulation equation in terms of k can be written as è° è k = sK yè - Hn + g + dL k
(116)
(117)
è è° The steady state values of k and yè are found by k = 0, which yields è sK k ÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ èyÅ = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ n+g+d èa Substituting this into yè = k gives
sK i y yè * = jj ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ zz n + g + d k {
(118)
aêH1-aL
(119)
Rewriting this in terms of output per worker gives aêH1-aL sK i y h AHtL y* HtL = jj ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ zz kn+g+d{
(120)
12
econ_dyn_Heidelberg.nb
Countries are rich and others are poor because they [1] have high investment rates in physical capital, [2] spend a large fraction of time accumulating skills (education), [3] have low population growth rates, and [4] have high levels of technology. In addition, as in the Solow model, in the long run economies grow at constant rates.
The steady state How well does this model perform empirically in terms of explaining why some countries are richer than others? Since incomes are growing over time, we consider relative incomes. If we define per capita income relative to the USA to be y* êêy = ÅÅÅÅÅÅÅÅ ÅÅÅÅÅ y*US
(121)
then from (120), relative (steady state) income is given by sê K aêH1-aL êê êêê êêy* = J ÅÅÅÅ hA êêxÅÅÅÅÅ N
(122)
where x := n + g + d and a bar above a variable is used to denote a variable relative to its US value. In answering the question how well the extended Solow model performs in explaining cross-country differences we assume that g is the same across countries. This seems to be a valid first approximation (why?). The figure below compares the actual levels of (relative) GDP per worker in 1997 to the levels predicted by the model. The following parameter values have been assumed: a = 0.3, y = 0.1, and g + d = 0.075 for all countries. Finally, it is assumed that A is the same in all countries. It can be seen that the fit of the model is fairly good for rich countries (which are close the 45-degree line), but gets weaker for poorer countries (which are above the 45-degree line).
Source : Jones H2002L
Ñ
