Deriving Growth in the Romer Variety Model

Why derive the growth rate?

The Romer variety model explains sustained growth through the creation of new productive designs. Its central result is not simply that research adds ideas: it specifies how the existing stock of knowledge and the labor allocated to research determine the rate at which new ideas arrive. Deriving that relationship makes it possible to distinguish changes in the flow of innovation from changes in the growth rate of the idea stock.

We use continuous time. Let \(A(t)\) denote the stock of available designs at time \(t\), and let \(\dot A(t)\) denote the flow of newly created designs per unit of time. The innovation growth rate is \(g_A(t)=\dot A(t)/A(t)\). Thus, \(\dot A\) and \(g_A\) are different quantities: one is a number of designs per unit of time, while the other is a proportional rate.

Definition: The Romer variety model is a research-based growth model in which new designs expand the range of intermediate inputs available for production. In its standard research specification, the flow of new designs is proportional to both the existing stock of designs and the number of researchers: \(\dot A=\delta H_A A\), where \(\delta>0\) is research productivity and \(H_A\) is labor allocated to research.

Research technology and labor allocation

Begin with a fixed total supply of labor \(H\). Suppose a constant share \(s_R\), with \(0\leq s_R\leq1\), works in research. Then

$$ H_A=s_R H. $$

In the standard Romer specification, each researcher is more productive when the stock of existing knowledge is larger. The research equation is

\dot A=\delta H_A A.

Dividing the flow of new designs by the stock of designs gives the proportional growth rate:

g_A=\frac{\dot A}{A}=\delta H_A=\delta s_R H.

This relationship assumes constant research productivity \(\delta\), a fixed total labor supply \(H\), a constant research share \(s_R\), and the standard specification in which research productivity is proportional to \(A\). Under these conditions, the idea stock grows at a constant proportional rate. Solving the differential equation gives \(A(t)=A(0)e^{g_A t}\). The flow \(\dot A\), in contrast, rises over time because it equals \(g_A A\).

Allocate labor.
Find the number of researchers from \(H_A=s_R H\). Check that \(s_R\) is a share, not a percentage entered without conversion.

Apply the research technology.
Use \(\dot A=\delta H_A A\) for the flow of new designs in the standard model.

Convert the flow to a growth rate.
Divide by the current idea stock: \(g_A=\dot A/A=\delta H_A\).

Connect ideas to output.
On the balanced growth path described below, output and the idea stock grow at the same rate when labor allocation is fixed.

Why variety growth carries through to output

A simple production representation shows how an expanding range of designs affects aggregate production. Let \(x(i)\) be the quantity of intermediate input associated with design \(i\), let \(L_Y\) be labor used in final production, and let \(0<\alpha<1\). Suppose final output is

Y=L_Y^{1-\alpha}\int_0^A x(i)^\alpha\,di.

Suppose one unit of each intermediate input requires one unit of capital, so total capital is \(K=\int_0^A x(i)\,di\). On a symmetric allocation, every active variety has \(x(i)=K/A\). Substituting this quantity into the production integral gives

Y=L_Y^{1-\alpha}A\left(\frac{K}{A}\right)^\alpha =L_Y^{1-\alpha}A^{1-\alpha}K^\alpha =(A L_Y)^{1-\alpha}K^\alpha.

Taking proportional growth rates yields \(g_Y=(1-\alpha)(g_A+g_{L_Y})+\alpha g_K\). If production labor is fixed and capital and output grow at the same rate on a balanced growth path, then \(g_{L_Y}=0\) and \(g_K=g_Y\). Rearranging gives \(g_Y=g_A\). This result is conditional: it uses the specified variety production structure, fixed production labor, and balanced growth. It is not a claim that output must grow at the same rate as ideas during every transition or under every production technology.

Derivation: Under the standard research technology, \(g_A=\dot A/A=\delta H_A\). With \(H_A=s_RH\), constant research productivity and fixed labor supply, \(g_A=\delta s_RH\). In the symmetric variety production setup, balanced growth with fixed production labor implies \(g_Y=g_A\). Therefore the model’s balanced-growth output rate is \(g_Y=\delta s_RH\), under these assumptions.

Worked implications

Worked Example: From Research Allocation to Idea and Output Growth

A hypothetical economy has a fixed labor supply of \(H=100\), with \(s_R=0.15\) allocated to research. Research productivity is \(\delta=0.020\) per researcher per year in the standard Romer model. The initial design stock is \(A(0)=40\). Find the researcher count, the annual growth rate of designs, the design stock after two years, and the balanced-growth output rate.

Method. First convert the labor share into researchers, then use the standard research equation to find the proportional growth rate. The model implies \(A(t)=A(0)e^{g_A t}\).

The number of researchers is \(H_A=s_RH=0.15(100)=15\). The design growth rate is

g_A=\delta H_A=0.020(15)=0.300

per year, or \(30.0\%\) per year as a continuous-time growth rate. After two years,

A(2)=40e^{0.300(2)}=40e^{0.600} =40(1.8221188)=72.884752.

Rounded to two decimal places, \(A(2)=72.88\) designs in the continuous model. On the balanced growth path with fixed production labor, the output growth rate is also \(30.0\%\) per year. Check: the design stock rises from 40 to about 72.88, consistent with positive exponential growth. The common mistake this method avoids is reporting \(\dot A\) as the growth rate; \(\dot A\) is the flow of designs, while \(g_A=\dot A/A\) is the proportional rate.

Allowing knowledge spillovers to vary

The standard equation assumes that the research advantage from accumulated knowledge is proportional to \(A\). To examine this assumption, consider the generalized research technology

\dot A=\delta H_A^\lambda A^\phi, \qquad \delta>0,\quad \lambda>0.

Here \(\lambda\) describes how the research flow responds to the number of researchers, and \(\phi\) describes how it responds to the existing knowledge stock. Dividing by \(A\) gives

g_A=\delta H_A^\lambda A^{\phi-1}.

The standard Romer specification is \(\lambda=1\) and \(\phi=1\). When \(\phi=1\), the knowledge-stock term cancels from the proportional growth rate, leaving \(g_A=\delta H_A^\lambda\). With the standard linear researcher effect as well, this becomes \(g_A=\delta H_A\). When \(\phi<1\), the current growth rate falls as \(A\) rises, holding \(H_A\) fixed; when \(\phi>1\), it rises as \(A\) rises. These are statements about the generalized equation’s instantaneous growth rate. A constant long-run growth rate does not follow from it when \(\phi\ne1\) and researcher input is fixed.

The parameter \(\delta\) captures research productivity: a larger \(\delta\) raises the innovation flow and growth rate for given inputs. The researcher effect depends on \(\lambda\). For \(\lambda=1\), a proportional increase in researchers produces the same proportional increase in research output. In the generalized model, the elasticity of the research flow with respect to researchers is \(\lambda\). Knowledge spillovers are represented by \(\phi\): they determine how existing ideas change research productivity. In the standard case, \(\phi=1\) means a larger idea stock exactly offsets the need to spread research effort across more designs in the expression for \(g_A\).

Worked Example: A Weaker Knowledge Spillover

Suppose a hypothetical research sector follows \(\dot A=\delta H_A A^\phi\), with \(\delta=0.030\), \(H_A=16\), \(A=256\), and \(\phi=0.75\). Find the current innovation growth rate and the flow of new designs. Give the growth rate as a percentage to one decimal place.

Method. Divide the research flow by \(A\) to obtain the growth rate. Since \(\phi-1=-0.25\),

g_A=\delta H_A A^{\phi-1} =0.030(16)256^{-0.25}.

Because \(256^{0.25}=4\), it follows that \(256^{-0.25}=1/4=0.25\). Therefore,

$$ g_A=0.030(16)(0.25)=0.48(0.25)=0.120 $$

per year, or \(12.0\%\) per year. The flow of new designs is \(\dot A=g_AA=0.120(256)=30.72\) designs per year. Check: dividing the flow by the stock recovers \(30.72/256=0.120\). The common mistake is to confuse the flow \(30.72\) with the growth rate; another is to use \(A^{-\phi}\) instead of \(A^{\phi-1}\) when converting the generalized research technology into a growth rate.

Worked Example: Comparing Two Research Allocations

A hypothetical economy has fixed total labor \(H=200\), and research productivity is \(\delta=0.015\). Use the standard Romer equation. Compare the design growth rates when the research share rises from \(0.08\) to \(0.11\). State the change in percentage points and the proportional increase in the growth rate.

Method. Compute researchers at each allocation, then apply \(g_A=\delta H_A\). At the first allocation, \(H_A=0.08(200)=16\), so \(g_{A,1}=0.015(16)=0.240\), or \(24.0\%\) per year.

At the second allocation, \(H_A=0.11(200)=22\), so \(g_{A,2}=0.015(22)=0.330\), or \(33.0\%\) per year. The increase is \(0.330-0.240=0.090\), which is \(9.0\) percentage points. Relative to the initial growth rate, the proportional increase is

\frac{0.090}{0.240}=0.375=37.5\%.

Check: researcher supply rises from 16 to 22, an increase of \(6/16=37.5\%\), and with a linear researcher effect the growth rate rises by the same proportion. This comparison holds other parameters fixed; it does not account for any separate effect that reallocating labor might have on current production.

Worked Example: Inferring Research Productivity From a Doubling Time

In a hypothetical economy using the standard Romer research technology, the design stock doubles over 10 years while the number of researchers remains fixed at \(H_A=20\). Estimate \(\delta\), reporting four significant figures. Assume continuous time and a constant growth rate over the period.

Method. Under constant growth, \(A(10)/A(0)=e^{g_A(10)}\). A doubling means this ratio is 2, so \(2=e^{10g_A}\). Taking natural logarithms gives \(g_A=\ln(2)/10\).

g_A=\frac{\ln(2)}{10} =\frac{0.693147}{10} =0.0693147

per year. The standard research equation gives \(g_A=\delta H_A\), so

\delta=\frac{g_A}{H_A} =\frac{0.0693147}{20} =0.003465735.

To four significant figures, \(\delta=0.003466\) per researcher per year under this model’s units. Check: \(20(0.003465735)(10)=0.693147=\ln(2)\), so the implied design stock doubles in 10 years. The common mistake is to calculate \(1/10=10\%\) as the continuous growth rate. Doubling time implies \(g_A=\ln(2)/10\), not \(1/10\).

What the result does—and does not—say

The standard result \(g_A=\delta s_RH\) is a conditional relationship, not a universal law. It requires a research technology linear in both the existing knowledge stock and the number of researchers, a fixed research share, a fixed total labor supply, and constant research productivity. The balanced-growth implication \(g_Y=g_A\) additionally uses the variety production structure and the balanced-growth condition for capital.

These distinctions matter when interpreting comparisons. Raising \(\delta\) improves research productivity directly. Raising \(s_R\) increases the researcher supply in this setup, but it also changes how much labor remains outside research; the growth equation alone does not measure that allocation’s effect on current output or welfare. Changing \(\phi\) alters the knowledge spillover mechanism and can make the current innovation growth rate depend on the existing idea stock. A candidate should therefore identify which parameter or assumption changed before applying the standard growth formula.

Do not confuse levels and rates. \(A\) is the stock of designs, \(\dot A\) is the flow of new designs, and \(g_A=\dot A/A\) is the proportional growth rate.
Convert labor shares correctly. A research share of \(15\%\) means \(s_R=0.15\), not \(15\).
Check the spillover exponent. In the generalized model, the growth-rate exponent is \(\phi-1\), not \(\phi\).
State the balanced-growth conditions. Output growth equals idea growth here only under the specified production structure and balanced path.

Key takeaway: In the standard Romer variety model, \(\dot A=\delta H_AA\), so \(g_A=\delta H_A\). With fixed labor \(H\) and research share \(s_R\), this is \(g_A=\delta s_RH\). Under the stated variety production and balanced-growth assumptions, output grows at that same rate. The generalized spillover equation changes the result to \(g_A=\delta H_A^\lambda A^{\phi-1}\).

Check Your Understanding

For each problem, use the specified model and keep growth rates distinct from innovation flows.

A hypothetical economy has \(H=120\), research share \(s_R=0.10\), and standard-model productivity \(\delta=0.040\). Find the number of researchers and the annual design growth rate. (Answer: \(H_A=12\); \(g_A=0.48\) per year, or \(48\%\).)
A research sector has \(\dot A=0.020H_AA^{0.5}\), with \(H_A=25\) and \(A=625\). Find the current design growth rate. (Answer: \(g_A=0.020(25)625^{-0.5}=0.50/25=0.020\) per year, or \(2.0\%\).)
Under the standard model, \(H=200\), \(\delta=0.015\), and the research share rises from \(0.08\) to \(0.11\). Find the initial and final growth rates, their percentage-point difference, and the proportional increase. (Answer: \(24.0\%\) and \(33.0\%\); increase \(9.0\) percentage points, or \(37.5\%\).)
A standard-model economy targets a design growth rate of \(18\%\) per year. If \(H=150\) and \(\delta=0.020\), what number of researchers and research share are required? (Answer: \(H_A=0.18/0.020=9\); \(s_R=9/150=0.06\), or \(6\%\).)
In the standard model, \(A(0)=50\) and \(g_A=0.040\) per year on a balanced growth path. Find the design stock after three years and the output growth rate. (Answer: \(A(3)=50e^{0.12}\approx56.38\); output growth is \(4.0\%\) per year under the stated balanced-growth assumptions.)

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