14.452 Economic Growth: Lectures 1 (second half), 2
and 3
The Solow Growth Model
Daron Acemoglu
MIT
Oct. 31, Nov. 5 and 7, 2013.
Daron Ace moglu (MIT ) Econom ic Growth Lecture s 1-3 Oct. 31, Nov. 5 and 7, 20 13. 1 / 88
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Solow Growth Model: Aggregate Production Function and Equilibrium, Study notes of Mechanics
An overview of the Solow Growth Model, focusing on the aggregate production function, labor and capital market clearing conditions, and the equilibrium of the model. It also discusses the continuous time representation of the model and Uzawa's Theorem.
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14.452 Economic Growth: Lectures 1 (second half), 2
and 3
The Solow Growth Model
Daron Acemoglu
MIT
Oct. 31, Nov. 5 and 7, 2013.
Solow Growth Model Solow Growth Model
Solow Growth Model
Develop a simple framework for the proximate causes and the mechanics of economic growth and cross-country income di§erences. Solow-Swan model named after Robert (Bob) Solow and Trevor Swan, or simply the Solow model Before Solow growth model, the most common approach to economic growth built on the Harrod-Domar model. Harrod-Domar mdel emphasized potential dysfunctional aspects of growth: e.g, how growth could go hand-in-hand with increasing unemployment. Solow model demonstrated why the Harrod-Domar model was not an attractive place to start. At the center of the Solow growth model is the neoclassical aggregate production function.
Solow Growth Model Households and Production
Households and Production II
Assume households save a constant exogenous fraction s of their disposable income Same assumption used in basic Keynesian models and in the Harrod-Domar model; at odds with reality. Assume all Örms have access to the same production function: economy admits a representative Örm, with a representative (or aggregate) production function. Aggregate production function for the unique Önal good is Y (t) = F [K (t) , L (t) , A (t)] (1) Assume capital is the same as the Önal good of the economy, but used in the production process of more goods. A (t) is a shifter of the production function (1). Broad notion of technology. Major assumption: technology is free; it is publicly available as a non-excludable, non-rival good.
Solow Growth Model Households and Production
Key Assumption
Assumption 1 (Continuity, Di§erentiability, Positive and Diminishing Marginal Products, and Constant Returns to Scale) The production function F : R^3 +! R + is twice continuously di§erentiable in K and L, and satisÖes
FK (K , L, A)
∂ F ()
∂ K
> 0 , FL (K , L, A)
∂ F ()
∂ L
FKK (K , L, A)
∂^2 F ()
∂ K 2
< 0 , FLL (K , L, A)
∂^2 F ()
∂ L^2
Moreover, F exhibits constant returns to scale in K and L.
Assume F exhibits constant returns to scale in K and L. I.e., it is linearly homogeneous (homogeneous of degree 1) in these two variables.
Solow Growth Model Market Structure, Endowments and Market Clearing
Market Structure, Endowments and Market Clearing I
We will assume that markets are competitive, so ours will be a prototypical competitive general equilibrium model. Households own all of the labor, which they supply inelastically. Endowment of labor in the economy, L¯ (t), and all of this will be supplied regardless of the price. The labor market clearing condition can then be expressed as: L (^) (t) = L¯ (^) (t)
for all t, where L (t) denotes the demand for labor (and also the level of employment). More generally, should be written in complementary slackness form. In particular, let the wage rate at time t be w (t), then the labor market clearing condition takes the form L (t) L¯ (t) , w (t) 0 and (L (t) L¯ (t)) w (t) = 0
Solow Growth Model Market Structure, Endowments and Market Clearing
Market Structure, Endowments and Market Clearing II
But Assumption 1 and competitive labor markets make sure that wages have to be strictly positive. Households also own the capital stock of the economy and rent it to Örms. Denote the rental price of capital at time t be R (t). Capital market clearing condition:
K s^ (t) = K d^ (t)
Take householdsíinitial holdings of capital, K ( 0 ), as given P (t) is the price of the Önal good at time t, normalize the price of the Önal good to 1 in all periods. Build on an insight by Kenneth Arrow (Arrow, 1964) that it is su¢ cient to price securities (assets) that transfer one unit of consumption from one date (or state of the world) to another.
Solow Growth Model Firm Optimization
Firm Optimization I
Only need to consider the problem of a representative Örm:
max L(t ) 0 ,K (t ) 0
F [K (t), L(t), A(t)] w (t) L (t) R (t) K (t).
Since there are no irreversible investments or costs of adjustments, the production side can be represented as a static maximization problem. Equivalently, cost minimization problem. Features worth noting: (^1) Problem is set up in terms of aggregate variables. (^2) Nothing multiplying the F term, price of the Önal good has normalized to 1. (^3) Already imposes competitive factor markets: Örm is taking as given w (t) and R (t). (^4) Concave problem, since F is concave.
Solow Growth Model Firm Optimization
Firm Optimization II
Since F is di§erentiable, Örst-order necessary conditions imply:
w (^) (t) = FL [K (t), L(t), A(t)], (2)
and R (t) = FK [K (t), L(t), A(t)]. (3) Note also that in (2) and (3), we used K (t) and L (t), the amount of capital and labor used by Örms. In fact, solving for K (t) and L (t), we can derive the capital and labor demands of Örms in this economy at rental prices R (t) and w (t). Thus we could have used K d^ (t) instead of K (t), but this additional notation is not necessary.
Solow Growth Model Firm Optimization
Second Key Assumption
Assumption 2 (Inada conditions) F satisÖes the Inada conditions
lim K! 0 FK () = ∞ and lim K !∞ FK () = 0 for all L > 0 all A lim L! 0 FL () = ∞ and lim L!∞ FL () = 0 for all K > 0 all A.
Important in ensuring the existence of interior equilibria. It can be relaxed quite a bit, though useful to get us started.
Solow Growth Model Firm Optimization
Production Functions
F(K, L, A)
0 K 0 K Panel A Panel B
F(K, L, A)
Figure: Production functions and the marginal product of capital. The example in Panel A satisÖes the Inada conditions in Assumption 2, while the example in Panel B does not.
The Solow Model in Discrete Time Fundamental Law of Motion of the Solow Model
Fundamental Law of Motion of the Solow Model II
Note not derived from the maximization of utility function: welfare comparisons have to be taken with a grain of salt. Since the economy is closed (and there is no government spending),
S (t) = I (t) = Y (t) C (t).
Individuals are assumed to save a constant fraction s of their income,
S (t) = sY (t) , (6)
C (t) = ( 1 s) Y (t) (7) Implies that the supply of capital resulting from householdsíbehavior can be expressed as
K s^ (t) = ( 1 δ )K (t) + S (t) = ( 1 δ )K (t) + sY (t).
The Solow Model in Discrete Time Fundamental Law of Motion of the Solow Model
Fundamental Law of Motion of the Solow Model III
Setting supply and demand equal to each other, this implies K s^ (t) = K (t). We also have L (t) = L¯ (t). Combining these market clearing conditions with (1) and (4), we obtain the fundamental law of motion the Solow growth model:
K (t + 1 ) = sF [K (t) , L (t) , A (t)] + ( 1 δ ) K (t). (8)
Nonlinear di§erence equation. Equilibrium of the Solow growth model is described by this equation together with laws of motion for L (t) (or L¯ (t)) and A (t).
The Solow Model in Discrete Time Equilibrium
Equilibrium Without Population Growth and Technological
Progress I
Make some further assumptions, which will be relaxed later: (^1) There is no population growth; total population is constant at some level L > 0. Since individuals supply labor inelastically, L (^) (t) = L. (^2) No technological progress, so that A (t) = A. DeÖne the capital-labor ratio of the economy as
k (t) K (t) L
Using the constant returns to scale assumption, we can express output (income) per capita, y (t) Y (t) /L, as
y (t) = F
K (t) L
, 1 , A
f (k (t)). (10)
The Solow Model in Discrete Time Equilibrium
Equilibrium Without Population Growth and Technological
Progress II
Note that f (k) here depends on A, so I could have written f (k, A); but A is constant and can be normalized to A = 1. From Euler Theorem,
R (t) = f 0 (k (t)) > 0 and w (t) = f (k (t)) k (t) f 0 (k (t)) > 0. (11)
Both are positive from Assumption 1.