_{1}

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A Samuelsonian serendipity theorem for an endogenous growth model is derived. The formula for optimal population growth rate deviates from those of the model with exogenous population growth rates in a third best endogenous growth model of the Lucas type with imperfect international capital movements and human capital externalities. Calibration shows that the effect of variation of the exogenous population growth rates on other variables and the deviation of population growth rates from its optimal value are small. The reason is that labour supply, interest rates and technical change are endogenous. There is not much of an incentive for population growth policy unless Frisch parameters change with ageing.

Samuelson [^{1} We extend the theorem to endogenous growth with endogenous labour supply, imperfect international capital movements and human capital externalities in a third-best extension of Lucas’s [

The maximization program for the consumers is

max c t , e t , L t , B t + 1 , K t + 1 , h t + 1 ∑ t = 0 ∞ β t ( N t ( c t 1 − σ ( 1 − σ ) − ξ ( L t N t ) 1 + ϑ 1 + ϑ ) − μ t [ N t c t + K t + 1 − ( 1 − δ k ) K t − ω t ( 1 − e t ) h t L t − r k t K t − B t + 1 + ( 1 + r ( B t Y t ) ) B t ] − μ h t [ h t + 1 − F e t γ h t − ( 1 − δ h ) h t ] )

The expression above shows the utility function of the entire population, N_{t}, where 0 < β < 1 is the subjective discount factor, σ > 0 is the intertemporal elasticity of substitution for per-capita consumption, c_{t}, ϑ is the Frisch parameter for the active part of the population or labour supply, L_{t}, expressed as a share of the population, ξ > 0 is a parameter which measures the disutility of participation in the active population relative to the consumption part of utility. The households decide between spending their time in production ( 1 − e t ) for immediate output generation and education, e t , to increase their productivity for later production. Income from labor with human capital h_{t} is ω t ( 1 − e t ) h t L t , the income from capital rent is r k t K t , and the debt from outside the economy’s

borders minus the interest and re-payment is, B t + 1 − ( 1 + r ( B t Y t ) ) B t . r ( B t Y t ) is the debt dependent interest rate. Spending on consumption is N t c t , and gross capital investment is K t + 1 − ( 1 − δ k ) K t .

Consumption is not differentiated in regard to age or (not) working. This is implicit in the assumption of equal consumption of all for a given point in time. By implication, we do not model pay-as-you-go pension systems with defined benefits or contributions [

The economy is assumed to consist of output-producing firms and labor- and capital-supplying consumers. Output is formed by a Cobb-Douglas production function and is determined by physical capital, K t , and efficient labour, ( 1 − e t ) h t L t . A human capital externality is added as, h ¯ t ϵ , modelled after Lucas [

This forms the production function

Y t = A ( K t ) 1 − α ( ( 1 − e t ) h t L t ) α h ¯ t ϵ (1)

The demand for physical and human capital is determined in a firm which maximizes profits:

max ( 1 − e t ) , K t π = A ( K t ) 1 − α ( ( 1 − e t ) h t L t ) α h ¯ t ϵ − ω t ( 1 − e t ) h t L t − r k t K t

ω t = α Y t ( 1 − e t ) h t L t (2)

r k t = ( 1 − α ) Y t K t (3)

Equations ((2) and (3)) represent first-order conditions, equating marginal productivity of labor and capital to wages for efficient labour and rental rates.

In the utility maximization, given the initial value N_{t}, only the level of the population, N_{t}_{(+i)}, appears.^{2} It has three effects. Higher N_{t}_{+1} leads to higher temporary utility N_{t}_{+1}u_{t}_{+1}; any given labour time L_{t}_{+1} is shared among more people; more people have to be fed with additional consumption N_{t}_{+1}c_{t}_{+1}.^{3} The first-order condition for N_{t}_{+1}, given N_{t}, and using (6) is

u t + 1 ≡ c t + 1 1 − σ 1 − σ − ξ ( L t + 1 N t + 1 ) 1 + ϑ 1 + ϑ ≥ ( ≤ ) − ξ L t + 1 1 + ϑ N t + 1 − 1 − ϑ + c t + 1 1 − σ (4)

This condition is similar to that of [

c 1 − σ ( 1 1 − σ − 1 ) ( > ) = ( < ) ( L N ) 1 + ϑ ( ξ 1 + ϑ − ξ ) (4’)

(4’) is a condition for the optimal population level. If σ >(<) 1, ξ 1 + ϑ − ξ = ξ − ϑ 1 + ϑ <(>) 0 is required and fulfilled for any positive Frisch parameter.

As the Hamiltonian of the households dynamic problems defined above is already maximized for given N_{t}_{+i}, when we derive with respect to N_{t}_{+1} ? analogous to indirect utility in [

Equation (4’) will only be fulfilled for specific values of N_{t}_{+1} as c and L are already optimally chosen by the household; it ensures that the two parts of utility are well balanced and consumption or labour supply are never too high or too low. However, N can adjust only slowly without migration and therefore temporarily this may perhaps hold only with inequality.

First-order conditions from household’s utility maximization for consumption c and labour L can be derived to find, together with (2),

c t = ξ − 1 / σ ( L t N t ) − ϑ / σ ( α Y t L t ) 1 / σ (5)

This can be compared to our serendipity condition (4’) in order to find the difference with the market equilibrium after making exponents comparable to (4’):

c t 1 − σ = ξ ( σ − 1 ) / σ ( L t N t ) − ϑ ( 1 − σ ) / σ ( α N L Y N ) ( 1 − σ ) / σ = ξ ( σ − 1 ) / σ ( L t N t ) ( − ϑ − 1 ) ( 1 − σ ) / σ ( α Y N ) ( 1 − σ ) / σ (5’)

The equality form of the serendipity condition (4’) is

c 1 − σ = ( L N ) 1 + ϑ ξ ( − ϑ 1 + ϑ ) ( 1 − σ σ ) (4’’)

Equating right-hand sides of (4’’) and (5’) we get

( L N ) ϑ + 1 σ ξ ( − ϑ 1 + ϑ ) ( 1 − σ σ ) = ξ ( σ − 1 ) / σ ( α Y N ) ( 1 − σ ) / σ (6)

Other first-order conditions for the dynamic problem of the household are shown in [

( 1 + g Y ) ( 1 + g N ) = ( 1 + g h ) 1 + ϵ α ( 1 + g L ) ( 1 + g N ) (7)

where g_{h} is the endogenous growth rate of productivity h and it is constant in the steady state, which has no transition as the model can jump to its solution for e, b, r(b,) X, and all other variables. Whereas the model solution (7) links the growth rate ratios of Y/N and L/N linearly, in (6) comparable growth rates have exponents.

The growth rate of N can contribute to welfare by ensuring that the level and growth rates of c and L/N do not get too far apart. Equality of growth rates of both sides of (4’) requires

( 1 + g c ) 1 − σ = ( 1 + g L 1 + g N ) 1 + ϑ or 1 + g N = 1 + g L ( 1 + g c ) 1 − σ / 1 + ϑ (4’’)

The comparable equation from the model solution is again linear in growth rates of c and L/N.

( 1 + g c ) = ( 1 + g L 1 + g N ) ( 1 + g h ) 1 + ϵ α (8)

Comparison of (6) and (7) as well as (4’’) and (8) show that the serendipity condition is not redundant and could determine the population growth rate.

The relevance of the serendipity theorem stems from the current problem of ageing based on the fall of population growth in the second half of the 1960s. The question then is how far population levels and growth rates are away from the optimum. We extend the calibration in Gaessler [

The last column of

gN_{ } | gL | g_{1+D} | e | b | X | 1 + r(1 + η) | 1 + gc | 1 + gh | (1 + g_{Y})/(1 + g_{N}) | 1 + gN opt | |
---|---|---|---|---|---|---|---|---|---|---|---|

0.003 | 0.00256 | 0.00044 | 0.3953 | 0.0473 | 0.688 | 0.0507 | 1.0298 | 1.0129 | 1.0306 | 1.003002 | |

0.002 | 0.00157 | 0.00043 | 0.3801 | 0.0383 | 0.694 | 0.0496 | 1.0287 | 1.0124 | 1.0295 | 1.001995 | |

0.001 | 0.00059 | 0.00041 | 0.3658 | 0.0297 | 0.700 | 0.0484 | 1.0277 | 1.0120 | 1.0285 | 1.001000 | |

0 | −0.0004 | 0.00040 | 0.3523 | 0.0216 | 0.706 | 0.0473 | 1.0267 | 1.0116 | 1.0275 | 0.999995 | |

−0.001 | −0.00138 | 0.00038 | 0.3396 | 0.0137 | 0.713 | 0.0463 | 1.0257 | 1.0112 | 1.0265 | 0.999000 | |

−0.002 | −0.00237 | 0.00037 | 0.3276 | 0.0062 | 0.720 | 0.0453 | 1.0248 | 1.0108 | 1.0256 | 0.997997 | |

−0.003 | −0.00327 | 0.00027 | 0.3213 | −2E−05 | 0.724 | 0.0444 | 1.0240 | 1.0106 | 1.0252 | 0.997085 | |

If the population gets older, this can perhaps be captured by a higher Frisch parameter if labour supply reacts more sluggishly to wage increases.

The comparison of the theoretical model with the additional serendipity result (4), (4’), (4’’) and columns “1 + gN opt” of Tables 1-3 show that conditions (4) - (4’’) are not redundant and the choice of the population growth rates might lead to a better choice of growth paths. The result is interesting because all parts of the traditional golden rule are endogenous and optimized: the interest rate, the

vartheta | gN | e | b | r | η | 1 + r(1 + η) | 1 + gh | 1 + gL |
---|---|---|---|---|---|---|---|---|

1 | 0.002 | 0.3838 | 0.0423 | 0.0473 | 0.0586 | 1.0501 | 1.012551 | 1.001122 |

2 | 0.002 | 0.3836 | 0.0447 | 0.0474 | 0.0615 | 1.0504 | 1.012545 | 1.001410 |

3 | 0.002 | 0.3837 | 0.0459 | 0.0475 | 0.0630 | 1.0505 | 1.012547 | 1.001559 |

4 | 0.002 | 0.3834 | 0.0466 | 0.0476 | 0.0639 | 1.0506 | 1.012539 | 1.001644 |

5 | 0.002 | 0.3834 | 0.0471 | 0.0476 | 0.0644 | 1.0507 | 1.012538 | 1.001703 |

6 | 0.002 | 0.3834 | 0.0475 | 0.0476 | 0.0649 | 1.0507 | 1.012537 | 1.001745 |

7 | 0.002 | 0.3833 | 0.0477 | 0.0476 | 0.0652 | 1.0507 | 1.012536 | 1.001777 |

8 | 0.002 | 0.3833 | 0.0479 | 0.0477 | 0.0654 | 1.0508 | 1.012536 | 1.001801 |

vartheta | gN | 1 + gY | X | 1 + gc | 1 + gw | 1 + g(1 + D) | 1 + gY/1 + gN | 1 + gN opt |

1 | 0.002 | 1.03141 | 0.69250 | 1.02937 | 1.017488 | 1.00088 | 1.02936 | 1.0015567 |

2 | 0.002 | 1.03170 | 0.69221 | 1.02965 | 1.017480 | 1.00059 | 1.02964 | 1.0018493 |

3 | 0.002 | 1.03185 | 0.69201 | 1.02980 | 1.017483 | 1.00044 | 1.02979 | 1.002000 |

4 | 0.002 | 1.03192 | 0.69198 | 1.02988 | 1.017472 | 1.00036 | 1.02986 | 1.0020862 |

5 | 0.002 | 1.03198 | 0.69193 | 1.02993 | 1.017470 | 1.00030 | 1.02992 | 1.0021459 |

6 | 0.002 | 1.03202 | 0.69189 | 1.02997 | 1.017469 | 1.00025 | 1.02996 | 1.0021887 |

7 | 0.002 | 1.03205 | 0.69185 | 1.03000 | 1.017468 | 1.00022 | 1.02999 | 1.0022209 |

8 | 0.002 | 1.03208 | 0.69183 | 1.03003 | 1.017467 | 1.00020 | 1.03002 | 1.002246 |

depr | gN | e | b | r | η | 1 + r(1 + η) | 1 + gh | 1 + gL |
---|---|---|---|---|---|---|---|---|

0.01 | 0.002 | 0.341 | 0.540 | 0.074 | 0.347 | 1.09953 | 1.031 | 1.00091 |

0.02 | 0.002 | 0.361 | 0.271 | 0.061 | 0.235 | 1.07482 | 1.022 | 1.00123 |

0.03 | 0.002 | 0.384 | 0.046 | 0.048 | 0.063 | 1.05052 | 1.013 | 1.00156 |

0.033 | 0.002 | 0.391 | −0.008 | 0.044 | −0.013 | 1.04331 | 1.010 | 1.00166 |

0.034 | 0.002 | 0.393 | −0.025 | 0.043 | −0.042 | 1.04091 | 1.009 | 1.00169 |

0.035 | 0.002 | 0.396 | −0.041 | 0.042 | −0.072 | 1.03852 | 1.008 | 1.00172 |

0.036 | 0.002 | 0.398 | −0.057 | 0.040 | −0.105 | 1.03613 | 1.007 | 1.00175 |

0.037 | 0.002 | 0.401 | −0.072 | 0.039 | −0.140 | 1.03375 | 1.006 | 1.00179 |

0.038 | 0.002 | 0.403 | −0.086 | 0.038 | −0.177 | 1.03137 | 1.005 | 1.00182 |

0.04 | 0.002 | 0.408 | −0.114 | 0.036 | −0.259 | 1.02662 | 1.003 | 1.00189 |

0.05 | 0.002 | 0.435 | −0.228 | 0.026 | −0.878 | 1.00312 | 0.994 | 1.00221 |

0.06 | 0.002 | 0.464 | −0.314 | 0.016 | −2.216 | 0.98003 | 0.985 | 1.00254 |

0.07 | 0.002 | 0.495 | −0.383 | 0.008 | −6.371 | 0.95735 | 0.976 | 1.00288 |

depr | gN | 1 + gY | X | 1 + gc | 1 + gw | 1 + g(1 + D) | (1 + gY)/(1 + gN) | 1 + gN opt |

0.01 | 0.002 | 1.077 | 0.683 | 1.075 | 1.044 | 1.00109 | 1.075 | 1.0019999 |

0.02 | 0.002 | 1.054 | 0.685 | 1.052 | 1.031 | 1.00076 | 1.052 | 1.0020002 |

0.03 | 0.002 | 1.032 | 0.692 | 1.030 | 1.017 | 1.00044 | 1.030 | 1.0019999 |

0.033 | 0.002 | 1.025 | 0.695 | 1.023 | 1.014 | 1.00034 | 1.023 | 1.0019999 |

0.034 | 0.002 | 1.023 | 0.696 | 1.021 | 1.012 | 1.00031 | 1.021 | 1.0020001 |

0.035 | 0.002 | 1.021 | 0.698 | 1.019 | 1.011 | 1.00028 | 1.019 | 1.0020000 |

0.036 | 0.002 | 1.019 | 0.699 | 1.016 | 1.010 | 1.00025 | 1.016 | 1.0019999 |

0.037 | 0.002 | 1.016 | 0.700 | 1.014 | 1.008 | 1.00021 | 1.014 | 1.0020001 |

0.038 | 0.002 | 1.014 | 0.702 | 1.012 | 1.007 | 1.00018 | 1.012 | 1.0019999 |

0.04 | 0.002 | 1.010 | 0.705 | 1.008 | 1.005 | 1.00011 | 1.008 | 1.0020000 |

0.05 | 0.002 | 0.988 | 0.723 | 0.986 | 0.992 | 0.99979 | 0.986 | 1.0020001 |

0.06 | 0.002 | 0.966 | 0.752 | 0.964 | 0.979 | 0.99946 | 0.964 | 1.0020002 |

0.07 | 0.002 | 0.945 | 0.800 | 0.943 | 0.966 | 0.99913 | 0.943 | 1.0019997 |

rate of technical change and labour supply growth. However, the deviation of the assumed from the optimal population growth rates are substantial only for variations of the Frisch elasticity, that is if ageing changes labour supply elasticities.

Equation (4) and its variations therefore broaden the serendipity results discussed in the introduction to the area of endogenous growth.

The serendipity theorem, suggesting that some rates of population growth may be better than others, turns out to be relevant in the third-best endogenous growth model with imperfect international capital movements as foreign debt, interest rates, labour supply growth and consumption paths are chosen optimally in the third best sense. We have considered the Benthamite case where the utility function is multiplied by the population size N. Other variants of utility functions could be analyzed in the same way.

Ziesemer, T. (2018) The Serendipity Theorem for an Endogenous Open Economy Growth Model. Theoretical Economics Letters, 8, 720-727. https://doi.org/10.4236/tel.2018.84049