The Romer 1990 growth model - growth at steady state

Romer 1990 增长模型 - 稳态增长

     In Romer 1990, growth is at “steady state” when the marginal product of human capital is equal for research and for final output production,  PA𝞭A = 𝜶Y(HY, L, xi)/HY.  At steady state, human capital allocated to research (“HA”) and to final output production (“HY”) are in stable proportions for the following reasons:

• The price of technology (“PA”) is constant, while the stock of technology (“A”) and final output (“Y”) both grow at the same rate:

  • PA is the present value of the stream of monopoly profits, PA = 𝝅(max)/r = p̄ix̄i(1 - 𝜸)/r where the profit-maximising price and quantity (p̄i, x̄i) are constants.  If “r” and “𝜸” are constant, “PA” is also constant;

  • The stock of technology “A” grows at a constant rate, Ȧ = 𝞭HAA if “HA” is constant; and

  • Final output production, Y(L, HY, SUM(𝜂xi)) = Y(L, HY, 𝜂(Ax̄i)) grows at the same rate as “A” if “L” and “HY” are fixed.  [Note 1]

• Since “A” and “Y” grow at the same rate, and “PA” is constant, “HY” is also constant; hence, human capital allocated to research and to final output production are stable.  

      Romer 1990 does not address the transient dynamics of an economy that is not yet at steady state.  Presumably some transversality condition is necessary for an economy to reach steady state, but whether the transversally condition (whatever it is) is also sufficient to achieve steady state is not known.   

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Note 1:  Since the intermediate durables, “xi” are symmetrical, SUM(𝜂xi)) = 𝜂SUM(xi)) = 𝜂(Ax̄i) where “A” is the number of technologies accumulated, and “x̄i” is the profit-maximising amount of each intermediate durable, and therefore  the total capital stock, K = 𝜂(Ax̄i).  

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     在 Romer 1990 中，当研究和最终产出的人力资本的边际产出相等时，增长处于稳定：PA𝞭A = 𝜶Y(HY, L, xi)/HY。在稳定状态下，分配给研究（“HA”）和最终产出（“HY”） 的分别人力资本比例是稳定的，原因如下：

• 技术价格（“PA”）不变，而技术存量（“A”）和最终产出（“Y”）均以相同的速率增长：

  • “PA”是垄断利润流的现值：PA = 𝝅(max)/r = p̄ix̄i(1 - 𝜸)/r，其中利润最大化价格和数量（p̄i, x̄i）均不变。如果“r”和“𝜸”恒定，则“PA”也恒定；

  • 如果“HA”恒定，技术存量“A”以恒定速率增长： Ȧ = 𝞭HAA；

  • 如果“L”和“HY”固定，最终产出：Y(L, HY, SUM(𝜂xi)) = Y(L, HY, 𝜂(Ax̄i)) 的增长速率与“A”相同。[注1]

• 由于“A”和“Y”以同样的速率增长，且“PA”恒定，“HY”也恒定；因此，分配给研究和最终产出的人力资本是稳定的。

     Romer 1990 没有讨论尚未达到稳定状态的经济的瞬态动态（transient dynamics）。据推测，某些横向性条件（transversality condition）对于经济达到稳态是必要的，但横向性条件（无论它是什么）是否也足以实现稳态尚不清楚。

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注1: 由于中间耐用品（“xi”）都是对称性的，SUM(𝜂xi)) = 𝜂SUM(xi)) = 𝜂(Ax̄i)，其中“A”是积累的技术数量，“x̄i”是每一个中间耐用品的利润最大化数量。因此，总资本存量：K = 𝜂(Ax̄i)。