THE SOLOW MODEL OF ECONOMIC GROWTH, MASTERS MACROECONOMICS

THE SOLOW MODEL

History:-

This is the simplest macroeconomics model of economic growth given by Robert Solow and Trevor Swan in 1956. An important feature of Solow model is that it is simple and abstract representation of a complex economy.

Assumptions:- 

 It focuses on four variables Output (Y), Capital (K), Labour (L), and Knowledge or effectiveness of labour (A). At any point of time economy has some amount of capital, labour and knowledge and these are combined to produce output.

Production function is given by:-

Y(t) = F(K(t), A(t)L(t)) …………………(1)

Here t denotes time. Also you need to notice that time does not enter production function directly but through K, L and A. That is, output changes over time only if input to production changes.

Amount of output rises over time and there is technological progress only if the amount of knowledge increases. Also A and L enter here in multiplication. Hence AL is known as labor-augmented.

Assumptions in the production function.

It has constant returns to scale in its two arguments, capital(K) and effective labour (L). That is, doubling the quantities of capital and effective labour with A held fixed doubles the amount produced.

F(cK, cAL)= cF(K,AL) for all c 0. ………(2)

So, in this model inputs other than capital, labour and knowledge are relatively unimportant.

Assumption 2:- it satisfies Euler’s theorem and Inada condition.

Inada condition:- lim k→0, f'(k) =∞ and lim k→∞, f'(k) = 0. 

In equation 2 set  c = 1/AL, this gives

F(k/AL, 1)  = 1/AL F(K, AL)  …………………..(3)

Here K/AL is the amout of capital per unit of effective labour, and F(K,AL) / AL is Y/ AL output per unit of effective labour.

 Now, define k = K/AL, (small k is equal to capital K/AL) and  y = Y/AL, and f(k) = F(k, 1). ( here AL/AL gives 1 (one)).

Thus, Y = f(k) …………………………….(4)

Equation 4 is output per unit of effective labour as a function of capital per unit of effective labour.

The intensive form production function, f(k) is assumed to satisfy f(0)=0,

f ’(k) > 0, f ”(k) < 0.

Endowments, Market structure and market clearing:-

The initial level of capital, labour and knowledge are taken as given and are assumed to be strictly positive. Labour and knowledge grow and constant rates. L …………….(5)

A(t) =gA(t) ……………………..(6)

Where n and g are parameters and L(t) and A(t) denotes derivative w.r.t time.   (that is L(t)= dL(t)/dt).

The growth rate of L refers to the quantity L(t) / L(t) and is equal to n, similarly growth rate of A is g.

(A key fact about the growth rate is that the growth rate of a variable equals the rate of change of its natural log that is, L(t) / L(t) equals d ln L(t) / dt)

Applying this in equation 5 and 6 we get.

ln L(t) = [ln L(0)] + nt ……………..(7)

ln A(t) = [ lnA(0)] + gt, …………….(8)

where L(0) and A(0) are value of L and A at time 0.

Solow model in discrete time

Output is divided between consumption and Investment.

Y = C + I ……………….(9)

The fraction of output devoted to investment, s, is exogenous and constant.

The exciting capital (C) depreciates at rate . So,

K(t) = sY(t) - δK(t) ……………(10)

(sum of n, g and δ is assumed to be positive.)

Also, in this model there is only a single good, government is absent, no fluctuation in employment, rate of saving, technological progress, population growth and depreciation are constant.

The dynamics of the model:-

The evolution of labour and knowledge is exogenous.

The dynamics of K

As the economy is growing over time. So we will focus on capital stock per unit of effective labour k.

Since k = K/AL, we use the chain rule to find

K(t) = K(t)/A(t)L(t) - k(t) /[A(t)L(t)]² [A(t)L(t) + L(t)A(t)]…………….(11)

=  K(t)/A(t)L(t) - k(t)/A(t)L(t) (L(t) / L(t)) -k(t)/A(t)L(t) ( A(t) / A(t)) …………..(12)

Here K/ AL is simply equal to k.  from 5 and 6 L / L and A /A are n and g respectively.

K is given by eqn 10. Substituting this value in eqn 12 we get.

K(t) = sY(t) - δk(t)/A(t)L(t) - nk(t) - gk(t)

= sY(t)/A(t)L(t) - δk(t)-nk(t) - gk(t) ……………….(13)

By using the fact that Y/AL is given by f(k), we have

K(t) = sf(k(t)) – (n + g +δ )k(t) ……………….(14)  Equation 14 is the key equation of the solow model. It states that the rate of change of the capital stock per unit of effective labour is the difference between sf(k) which is the actual investment per unit of effective labour and ( n + g + )k, is breakeven investment, the amount of investment which must be done to keep k at existing level.

         Figure-1, Actual and break-even investment

 This figure plots the two terms of the expression for K as function of K. Break-even investment, (n + g + δ)k is proportional to k. Actual investment sf(k), is constant times output per unit of effective labour.

 

Figure 2, The phase diagram for k in Solow model

This figure shows the k as a function of k. when k is less than k*, actual investment exceeds break-even investment, and so k is positive. If k exceeds k*, k is negative. If k equals k*, then k is zero. Thus, regardless of where k starts, it converges to k* and remains there.

The balanced growth path in Solow model

Since k converges to k*. How variables behave when k equals k*.

By assumption, labour and knowledge are growing at rate n and g respectively. The capital stock k equals ALk, since k is constant at K*, K is growing at rate (n + g). (that is k/k equals n + g). with both capital and effective labour growing at rate n + g, the assumption of constant return implies that output, Y, is also growing at that rate.

We can also find speed of convergence which determines that how rapidly k approaches k*.

 References:- 
Acemoglu, Daron (2009). "The Solow Growth Model". Introduction to Modern Economic Growth. Princeton: Princeton University Press.
Romer, David (2006). Advanced Macroeconomics. McGraw-Hill.