To understand the essence of these models and their differences, I have checked the Lucas' book on economic growth (this is actually a volume of collected papers), and found insightful exposition.
In the Introduction of the book, the author first provides a nice summary of Paul Romer's pioneering works.
Paul Romer (1986a, 1986b) worked out an explicit model of a growing economy that reconciled the opposing forces of increasing an diminishing returns, and did so in a way that generated sustained production growth and was at the same time consistent with market equilibrium of many, competing producers. The economics of Romer's model are closely related to the ideas of Allyn Young (1928), but his development of these ideas is entirely new. In the theory, goods are produced with a single kind of capital - Romer called it "knowledge capital" - and each producer's output depends both on his own stock of this capital and on the stock held by other firms. Aggregating over producers, production in the economy as a whole is subject to increasing returns: Every 10 percent increase in the total stock of knowledge capital leads to an output increase of more than 10 percent. But an individual producer, who has no control over the economy's total stock of capital, faces diminishing returns to increases in his own capital. Thus the fact of increasing inequality among the economies of the world is reconciled with the absence of a tendency to monopolization within each economy.
Then, the author relates Romer's idea with his own (Lucas, 1988).
Section 4 of my "On the Mechanics of Economic Development" constructs a model designed to deal with the problem posed by diminishing returns along the lines proposed by Romer. In doing this, I found it more convenient to make use of a model of Uzawa (1965) in which there is both physical and human capital but returns, private and social, depend only on the ratio of these two stocks. The theory replaces the increasing returns assumed by Romer with a kind of constant returns, yielding a system which is easier to analyze than Romer's but which circumvents the problems of diminishing returns in a similar way.
The human capital model I used involves an external effect of human capital, patterned on the external effect of knowledge capital that Romer introduced. But in my analysis, this external effect is not needed to ensure the existence of a competitive equilibrium the way it is in Romer's model. If this effect is removed, the model continues to be internally consistent and is in fact even easier to analyze. [footnote]
[footnote]: Rebelo (1991) stripped the model down to its simplest one-capital-good "Ak" form. Caballe and Santos (1993) provide an elegant analysis of the off-balanced-path dynamics of an Uzawa model without a production externality.
References
Caballe, Jordi, and Manuel S. Santos (1993) "On Endogenous Growth with Physical and Human Capital." Journal of Political Economy, 101: 1042-1067. [313]
Lucas, Robert E., Jr (1988) "On the mechanics Economic Development." Journal of Monetary Economics, 22: 3-42. [13791]
Rebelo, Sergio (1991) "Long Run Policy Analysis and Long Run Growth." Journal of Political Economy, 99: 500-521. [2748]
Romer, Paul M. (1986a) "Increasing Returns and Long-Run Growth." Journal of Political Economy, 94: 1002-1037. [12099]
Romer, Paul M. (1986b) "Cake Eating, Chattering, and Jumps: Existence Results for Variational Problems." Econometrica, 54: 897-908. [58]
Uzawa, Hirofumi (1965) "Optimum Technical Change in an Aggregative Model of Economic Growth." International Economic Review, 6: 18-31. [1142]
Young, Allyn A. (1928) "Increasing Returns and Economic Progress." Economic Journal, 38: 527-542. [2014]
(Number in [ ] shows the citation in Google Scholar.)
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