Romer states his strategy to demonstrate a) the endogeneity of growth in the very long run, and b) the balanced growth equilibrium in the very long run.

The strategy for characterizing the model that is followed here is to solve for an equilibrium in which the variables A, K, and Y grow at constant exponential rates.  This is generally referred to as a balanced growth equilibrium.  [Romer, 1990, S90]

Verifying that a balanced growth exists therefore reduces to the problem of showing that prices and wages are such that H_Y and H_A remain constant as Y, K, C, and A grow. [Romer, 1990, S90]

Review of certain assumptions in the proof:

Romer’s assumptions are listed in the previous section.  We repeat, so as to highlight, certain key assumptions as to linearity, level-of-use x-bar, and the definition of total K, supplying the connectors within the proof and assuring constant exponential growth.

• • The evolution of the store of economically useful knowledge is given by Å= δH_AA or (same factors but different collocations) H_AδA, with δ being a productivity parameter for the development of new designs. (S83)
  • … the output of new designs is linear in each of H_A and A when the other is held constant. (S84) (assumed for analytical convenience). This linearity and nonconvexity in A makes unbounded growth of A possible.
  • x-bar: All existing producer durables are  used at the same level, symbolized comprehensively by x-bar (S90-91)  verbatim: “All the producer durables that have been designed up to time t are used at the same level, symbolized by x-bar, … Over time x-bar remains constant and A grows at a constant exponential rate.” (N15) all ideas (designs) are equally good and have identical costs of production, forgone η consumables.  

• • η is a parameter representing the primitive unit in all products and the correlative monetary accounting cost: It symbolizes
    • a unit – whether alone or as the primitive component of some final output; whether in a consumable or in a producer durable – . (S80)
    • the number of units of forgone consumption required to produce one unit of durable goods (S82);
    • thus, total capital used is K = ηΣ_i^Ax_l (S82) or K = x(i) = (A)(x-bar) = A(K/(ηA)) (S88-89)(7)
    • the unit of denomination for the cost and for the exchange value of capital goods
    • the unit of denomination for loans

    • “Once it owns the design, the manufacturer of producer durables can convert η units of final output into one durable unit of good i. … The correct interpretation of this formal description is that the forgone consumption is never manufactured. The resources (inputs) that would have been used to produce the foregone output (consumption) are used instead to manufacture capital goods.  It is possible to exchange a constant number of consumption goods for each unit of capital goods if the production function used to manufacture capital goods has exactly the same functional form as the production function used to manufacture consumption goods.” (S81)

  • The wage for human capital, w_H, i.e. “the rental rate per unit of human capital” (S85), is its marginal product; w_H= P_AδA. (S91) and W_H= αH_Y^α-1 … (S91)
  • H_A is constant (by assumption of definition of H as a constant cumulative effect (S79)),

Outline of the Proof:

 Strategy: “The strategy is to solve for an equilibrium in which the variables A, K, and Y grow at a constant exponential rate, … generally referred to as a balanced growth equilibrium.” (S90)

If, by the Cobb-Douglas production function for final output (7) (S89) (S92), L, H_Y, and x-bar are fixed, then the production function Y(H_Y, L, X) is an ordinary differential equation and, by the partial first derivative with respect to time, output Y grows at the same rate as the variable A.

If (7) (S89) (S92) x-bar is fixed by assumption (S90-91) (N15) and because total usage of capital is K = (A)(x-bar)(η), then per the Cobb-Douglas production function (7) (S89) (S92) and the partial first derivative of Y with respect to time, K must grow at the same rate as A

 Conversely, if (S90, line 29) by equation (3) (S83) the productivity δA of human capital H_A in research R&D grows in proportion to linear A, then the wage paid for output of designs in R&D will grow in proportion to A.

 If (S90,line 26) total capital, K, and A cumulate at the same rate, the wage paid for human capital in the final output sector will also grow in proportion to A

If the price P_A of a design in a balanced growth equilibrium equals the present discounted value of the stream of monopoly profit to monopolist sellers (renters out) of producer variables,  then – by equation (5, 6, 6’, 8) (S86-91) and the partial differentials in the Cobb-Douglas form (linear in the first degree) – the price P_A of a design is constant.

If the productivity δA of human capital, H_A, grows at the same rate in both sectors, R&D and final output, and, because the growth of K and A are equal, is compensated at the same rate in both sectors, and if the price for new designs, P_A, is constant, the allocations of H_A and H_Y between R&D and durables manufacturing will remain constant

In other words, if prices and wages are such that H_Y and H_A remain constant as Y, K, C, and A grow, then H_A devoted to research is constant.

If the H_A factor in the evolution of A devoted to research is constant, then by the form of the evolution of A (3) (S83), Å = H_AδA, A will grow at constant rate symbolized by δ.

If (Uzawa, 1965) (S88) the evolution of A, Å = H_AδA, is by assumption linear and determined by the allocation of resources H_A and H_Y  between the research sector and the final goods sector, then A will grow at a constant exponential rate.

 If (Solow, 1956) (S88), as we said above, A grows at a constant exponential rate, then A, K, and Y grow at a constant exponential rate.

 Then, with A, K, and Y growing at an equal constant exponential rate, we have solved for a balanced growth equilibrium.

Thus, IF K/Y is constant, then it is easy (S92) to derive the “equilibrium of growth rates”; i.e.

g = C°/C = Y°/Y = K°/K = A°/A…  (S92)

g = dH_A= dH – Γr… (S92)

and, by using key assumptions as to linearity, level-of-use x-bar, and the definition of total K, we have demonstrated the conditions and the possible achievement of balanced-growth equilibrium at a constant exponential rate in the very long run.

Note in the excerpt below that Romer and Lonergan are in agreement as to the general nature of a balanced growth equilibrium in the very long run; however, Lonergan explains, in addition, the normative pure cycle of expansion in the short and intermediate term, which Romer calls “transient dynamics” (S90) or “transition dynamics” (S97) and which is not within the purpose or scope of Romer’s essay.  But we emphasize that Romer’s omission of the intrinsic cyclicality of the transitional dynamics of minor expansions and major “revolutions” is not grounds for criticism of Romer’s work. Romer defined his purpose and scope and he achieved what he set out to do.

One may expect the increment of a volume to stand to the increment of a surface as the volume does to the surface.  To suppose the contrary leads to absurd conclusions…  If, for instance, dQ”/Q” were on a long-term aggregate much greater or much less than dQ’/Q’, then a series of long-term periods would make this difference multiply in geometrical progression to effect a convergence of Q”and Q’or else a mounting divergence.  Such a convergence or divergence would imply that the more roundabout methods of capitalist progress were increasingly less efficient or increasingly more efficient in expanding the supply of consumer goods. Neither view is plausible.  New ideas and new methods increase existing efficiencies in both the surplus and the basic stages; the ratio between the quantity of surplus and the quantity of basic products per interval is not a matter of efficiency but of the point-to-line correspondence involved in any more roundabout method, in the fact that a single surplus product gives a flow of basic products. [CWL 15, 124]