Notes on Solow and Romer Model PDF

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Notes on Solow Model and Romer ModelJubo YanThis note summarizes the derivation of the Solow model, Romer model, and other relevant model specifications. The note also helps to organizethe growth theories in a way so you can see their connections. Please note that the math is used to connect the dif...

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Notes on Solow Model and Romer Model Jubo Yan This note summarizes the derivation of the Solow model, Romer model, and other relevant model specifications. The note also helps to organize the growth theories in a way so you can see their connections. Please note that the math is used to connect the different theories and help you to grasp the key ideas. The most important thing is to understand the intuitions of the theories rather than the technical details.

1

Extension of the Harrod-Domar Model

In the Harrod-Domar model, we focus on the aggregate output and its growth rate. The population size is assumed to be fixed and irrelevant for the aggregate output growth. What if we consider the change in population size and the per capita output? From the Harrod-Domar model, we have ∆Y s =g +δ = +δ c Y s Y (t + 1) − Y (t) or −δ = c Y (t)

(1)

Hence, we can rewrite Equation 1 in the following form cY (t + 1) = sY (t) + (1 − δ)cY (t)

(2)

Now, we assume the population size to be L(t) in period t and grows at rate of n (i.e., L(t + 1) = (1 + n)L(t))1 . To discuss the growth pattern at Y (t) per capita level, we use the definition that y(t) ≡ L(t) to denote per capita 1 I am using L(t) to denote the population size to keep this note consistent with your lecture slides and textbook. I used P (t) to denote population size when I discuss this in class.

1

output level. Dividing both sides of Equation 2 by L(t), we get L(t + 1) = (1 − δ)cy (t) + sy (t) L(t) y(t + 1) L(t + 1) s = (1 − δ) + y(t) L(t) c s (1 + g ∗ )(1 + n) = (1 − δ) + c

cy(t + 1) or or

(3)

where g ∗ is the per capita output growth rate. As an approximation, we have g ∗ ≈ cs − n − δ. If we compare this with the one from the original HarrodDomar model g = sc − δ, this result emphasizes the effect of population growth n on the per capita output growth.

2 2.1

Deriving the Solow Model The discrete version

To derive the discrete version of the Solow model, we first write down the dynamic of the capital accumulation. Note that this is from the definition of capital accumulation. K(t + 1) = (1 − δ)K(t) + sY (t)

(4)

If we divide both sides of Equation 4 by L(t) and use the same trick as we derive Equation 3, we have (1 + n)k(t + 1) = (1 − δ)k(t) + sy(t)

(5)

This is the discrete version of the Solow model.

2.2

The continuous version

To derive the continuous version, we need to replace the discrete time periods with continuous time. Consider a small time interval (∆) that approaches 0 infinitely. K (t + ∆) = I(t)∆ + (1 − δ∆)K (t) ˙ ≡ Because K(t)

d K(t) dt

=

K (t+∆)−K (t) , ∆

we have

˙ K(t) = I(t) − δK(t) = sY (t) − δK(t) 2

(6)

(7)

Divide both sides of Equation 7 by the population size L(t), ˙ I(t) K(t) K(t) = −δ = i(t) − δk(t) L(t) L(t) L(t)

(8)

Using chain rule we also have the following K(t) ˙ ˙ d ( L(t) ) K(t)L(t) − L(t)K(t) ˙ = d k(t) = = k(t) 2 dt L(t) dt ˙ ˙ ˙ L(t) K(t) K(t) K(t) − = − nk(t) = L(t) L(t) L(t) L(t)

(9)

Plugging Equation 8 into Equation 9, we have the continuous version of the Solow model ˙ ) = i(t) − (n + δ)k(t) = sy(t) − (n + δ)k(t) k(t

(10)

Note that both the discrete and continuous versions of the model give us the same story. We can rewrite the discrete version (1 + n)[k(t + 1) − k(t)] = sy(t) − (n + δ )k(t)

(11)

Notice that the production function y = f (k) is concave in k but the second component of the equation (n + δ)k is linear in k, we will always reach a point at which k˙ (t) = 0 or k(t +1) − k (t) = 0. In other words, the per capita capital stops increasing and thus the per capital output growth rate will be 0.

3

Endogenous Growth Model (Human Capital)

The fundamental reason for the Solow model not being able to sustain long run growth is the decreasing marginal return of capital (i.e., y = f (k) is a concave function). Is there a way to change the Solow model so long run growth can be sustained? One possibility is to allow for investment in human capital. The intuition can be thought as the following: 1. Unskilled labor uses capital less efficiently than skilled labor. An example is how well workers can operate machines. We can keep on allocating more machines to workers, but the marginal output from an additional machine must decrease as workers are becoming more and more occupied. 3

2. If we allow for human capital investment, it can be thought as we increase the workers’ capacity so they can operate more machines efficiently and simultaneously so the marginal product of capital (machine) does not drop. 3. We certainly shouldn’t expect any worker to increase his/her ability to operate more machines infinitely but we can always imagine it being the case in which workers acquire more skills and they use more advanced machines (e.g., a basic machine costs 100, a more advanced one costs 200, so on and so forth). The production function of human capital is given by y = k αh1−α

(12)

where h is the human capital input. Note that h is an endogenous variable. In other words, the amount to invest in human capital is determined by the economic agents. We then have k (t + 1) − k (t) = sy(t)

Investment in physical capital

(13)

h(t + 1) − h(t) = qy(t)

Investment in human capital

(14)

If we define r as the human to capital ratio in the long run, we have k(t + 1) − k(t) = sr 1−α k(t) h(t + 1) − h(t) = qr−α h(t) It can be proved that sr 1−α = qr −α in the long run (i.e., steady state). The proof is below y(t) k(t + 1) =s + 1 = sr(t)1−α + 1 k(t) k(t) y(t) h(t + 1) =q + 1 = qr(t)−α + 1 h(t) h(t) h(t) then let r(t) ≡ k(t) r(t + 1) qr(t)−α + 1 so = sr(t)1−α + 1 r(t) α

r(t) α−1 q q 1+ q r(t) + r(t) = r(t) r(t + 1) = s 1 + r(t)α−1 s + r(t)α−1 s

4

(15)

Now, assume r(t) > qs , then we have r(t)α q r(t)α−1 s

1+ 1+

>1

(16)

Combine Equation 16 with the first equality in Equation 15, we see that r(t + 1) > qs . Similarly, it is easy to see that when r(t) > qs q r(t)

+ r(t)α−1

s + r(t)α−1

r(t + 1). We have hence proved the following If at any date t, we have r(t) > qs , then it must be that r(t) > r(t+1) > q . s Following the same logic, we can prove If at any date t, we have r(t) < qs , then it must be that r(t) < r(t+1) < q . s Therefore, in the long run steady state, we must have r = sr 1−α = qr −α. This leads us to the following prediction

q . s

In other words,

k (t + 1) − k (t) = sr 1−α = sαq 1−α k(t) h(t + 1) − h(t) = qr−α = sαq 1−α h(t) We have just proved that the human resource endogenous growth model predicts a positive long run growth rate and the per capita physical and human capital grow at the same rate. The human capital model is a simple example of endogenous growth model. In this sense, the Harrod-Domar model is also an endogenous growth model. The Harrod-Domar model, however, only has one input (capital) which features constant return to scale. In the human capital model, we have two inputs–physical capital and human capital–and they behave constant return to scale jointly.

4

Romer Model

The Romer model generates positive growth through a different mechanism– externality or, more precisely, technology spillover. 5

There are two inputs in the production function–physical capital and labor. To derive the prediction of the Romer model, we need to start from the production function of a single firm or industry Y (t) = A(t)K(t)αL(t)1−α

(18)

A(t) stands for some measure of overall productivity, and it is a macroeconomic parameter common to all firms (industries) in the economy. So far, the production is no different from the standard Solow model. Our focus of this model is the determination of A(t). Unlike the Solow model in which A(t) is determined exogenously, A(t) in the Romer model is a positive externality generated by the joint capital accumulation of all firms (industries) in the economy.2 The technology or knowledge accumulation can then be modeled at the economy level A(t) = aK ∗ (t)β

(19)

Note the difference between Equation 19 (i.e., economy level) and Equation 18 (i.e., firm or industry level). The technology or knowledge is taken as given at the firm (industry) level but can be accumulated at the economy level. At the economy level, the total output function is3 Y (t) = aK(t)α+β L(t)1−α

(20)

Hence, the Romer model has constant return to scale at firm (industry) level but it behaves increasing return to scale at the economy level. We now derive the prediction of the growth rate from the Romer model.4 dY ∂Y ∂K ∂Y ∂L Y˙ (t) = + = ∂L ∂t ∂K ∂t dt

(21)

Because ∂Y = a(α + β )K (t)α+β−1 L(t)1−α ∂K ∂Y = aK(t)α+β (1 − α)L(t)−α ∂L 2 There are also endogenous growth models that make the investment in knowledge A(t) a deliberative endogenous choice. Note that is the case of the Romer model. Knowledge is generated by positive externality in the Romer model. 3 ¯ instead of a and K ∗ here. I use different In the textbook, the notations are A and K notations as they better connect the Romer model to the Solow model because we can understand the A(t) here as the same A in the Solow model. 4 This is the same derivation as the one on page 163 of the textbook.

6

Combine the equations, we have ˙ ˙ K(t) L(t) ) Y˙ (t) = (aK α+β L1−α)((α + β) + (1 − α) L(t) K(t) ˙ K(t)

˙ L(t)

(22)

Y˙ (t)

In the long run steady state, K(t) , L(t) , and Y (t) (the growth rates) are all ˙ ) = I(t)−δK (t) = sY (t)− δK (t). Dividing constant. We also know that K(t this equation by K(t), we have ˙ sY (t) K(t) −δ = K(t) K(t) Because

˙ K(t) K(t)

is a constant,

Y (t) K(t)

˙

(t) must also be a constant. Recall that YY (t)

is also a constant, so we must have

˙ K(t) K(t)

=

Y˙ (t) Y (t)

of the total output. Using Equation 22 and let

= g.5 g is the growth rate ˙ L(t) L(t)

≡ n, we have

g = (α + β)g + (1 − α)n This gives us the prediction of the Romer model: g=

5

n(1 − α) 1−α−β

K and Y must grow at the same rate to keep

7

Y (t) K(t)

(23)

a constant....