Tutorial 2: the Solow model

We are going to dig into into the seminal model of growth theory in macroeconomics: the Solow model.

Quick model summary

  1. Structure of the economy: firms and consumers (no state). Firms use (and pay) capital and labor to produce an output. Output (=income) is either consumed or saved by households:

    \[ Y_t = C_t + I_t \]

  2. Production Function: a representative firm produces output using a Cobb-Douglas production function: \[ Y_t = F(K_t, A_tL_t) = K_t^\alpha (A_tL_t)^{1-\alpha}\] where \(Y\) is output, \(K\) is capital, \(L\) is labor and \(A\) is technology. where \(0 < \alpha < 1\). Using capital comes at rental price \(r\) and labor comes at price \(w\) (the wage).

  3. Capital accumulation: capital accumulates through savings and depreciates geometrically

    \[ K_{t+1} = I_t + (1-\delta) K_t\]

  1. Saving rate: investment is a constant fraction of output

    \[ I_{t} = sF(K_t,A_tL_t)\]

Definitions

  1. Steady state: trajectory where all variables are constant
  2. Balanced growth path: trajectory such that per capita variables grow at a constant rate (not necessarily the same) and the capital to output ratio is constant

Exercise

1. Warm-up: Returns to scale

Let \(\lambda > 1\), if \(F(\lambda X, \lambda Y) < \lambda F(X,Y)\), the function has decreasing returns to scale (and increasing for \(>\)). If \(F(\lambda X, \lambda Y) = \lambda F(X,Y)\), the function has constant returns to scale. For the following functions, state if they exhibit positive, constant, or negative returns to scale.

  1. \(y_1 = 10x^2y^2\)

\[ y_1(\lambda x, \lambda y) = \lambda^4 y_1(x,y) > \lambda y_1(x,y) \text{ , increasing returns to scale} \]

  1. \(y_2 = \frac{1}{2}x^{1/3}y^{1/2}\)

\[ y_2(\lambda x, \lambda y) = \lambda^{5/6} y_2(x,y) < \lambda y_2(x,y) \text{ , decreasing returns to scale} \]

  1. \(y_3 = x + 2y\)

\[ y_3(\lambda x, \lambda y) = \lambda y_3(x,y) = \lambda y_2(x,y) \text{ , constant returns to scale} \]

  1. \(y_4 = \ln(xy), x>1,y>1\)

\[ y_4(\lambda x, \lambda y) = 2\ln(\lambda) + y_4(x,y) \]

Depends on initial level. Fix \(\lambda = e\) so that \(2\ln(\lambda) =2\))

For \(xy = e\), \(y_4(\lambda x, \lambda y) = 3 > \lambda y_4(x,y)\)

For \(xy = e^3\), \(y_4(\lambda x, \lambda y) = 5 < \lambda y_4(x,y)\)

2. Solve the firm’s maximization problem

Consider the function above (\(Y_t = K_t^{\alpha} (A_tL_t)^{1-\alpha}\)).

  1. Show that if \(K=0\) and/or \(L=0\) production does not occur

Immediate

  1. Show that the production function has constant returns to scale

\[ F(\lambda K_t, \lambda L_t) = \lambda K_t^{\alpha} (A_tL_t)^{1-\alpha} = F(K,L) \]

  1. Show that marginal productivity is positive for capital and labour

\[ F_K(K_t,L_t) = \alpha K_t^{\alpha-1} (A_tL_t)^{1-\alpha} >0 \]

\[ F_L(K_t,L_t) = (1-\alpha) K_t^{\alpha} (A_t)^{1-\alpha} L_t^{-\alpha} >0 \]

  1. Show that marginal productivity is decreasing for capital and labour

\[ F_{KK}(K_t,L_t) = \alpha (\alpha-1) K_t^{\alpha-2} (A_tL_t)^{1-\alpha} <0 \]

\[ F_{LL}(K_t,L_t) = (1-\alpha) (-\alpha) K_t^{\alpha} (A_t)^{1-\alpha} L_t^{-\alpha-1} <0 \]

  1. Write the down the profit maximization program (maximizing profit with respect to K and L) and derive its first order conditions

\[ \max_{K,L} Y_t - rK_t - wL_t \]

FOC with respect to K implies: \(F_K(K_t,L_t) = \frac{dF}{dK} = r\)

FOC with respect to L implies: \(F_L(K_t,L_t)=\frac{dF}{dL} = w\)

  1. Show that all output is needed to pay capital and labor

Differentiating both sides of the constant return to scale equation with respect to \(\lambda\) yields:

\[ F_{K}(\lambda K_t,\lambda L_t)K_t +F_{L}(\lambda K_t,\lambda L_t)L_t = F(K_t,L_t) \]

Evaluating it at \(\lambda = 1\) (constant returns to scale is true whatever \(\lambda\) and in particular for \(\lambda = 1\)) and and plugging the results of the two FOC of the profit maximization problem yields the result.

3. Constant population and technology

For now, we consider that technology and labor (=population) are constant (\(A_t = A\), \(L_t = L\)): no population growth, no technological progress.

  1. Using the capital accumulation equation, show that the ratio \(\frac{K_{t+1}}{K_t}\) goes to \(\infty\) as \(K_t\) goes to 0 and is lower than 1 when \(K_t\) goes to \(\infty\)

\[ g(K_t) = \frac{K_{t+1}}{K_t} = s \left(\frac{A_tL_t}{K_t}\right)^{1-\alpha} + 1-\delta \]

It tends to \(+\infty\) as \(K_t \rightarrow 0^+\), and \(0^+\) as \(K_t \rightarrow +\infty\)

  1. Show that there is a unique steady state level of capital \(k^*\) and solve for it

Plugging \(K = K_{t+1} = K_t\) and solving for it in the equation above yields:

\[ k^* = AL \left(\frac{s}{\delta}\right)^{\frac{1}{1-\alpha}} \]

  1. How does the \(k^*\) vary with \(s\), \(\delta\), \(A\), \(L\) ? Interpret

Capital accumulation is facilitated by a high saving rate \(s\) , a low depreciation rate \(\delta\) and by high levels of effective labor \(AL\) (additional units of capital are highly productive up to a higher level of \(K\) )

4. Population and technology growth

We now assume that population grows at constant rate \(n\), and technology at rate \(g\):

\[L_{t+1} = (1+n) L_t\]

\[A_{t+1} = (1+g) A_t\]

We respectively denote \(y_t=Y_t/L_t\) and \(k_t=K_t/L_t\), the per-worker production and capital

We further define \(\hat{y_t}=\frac{Y_t}{A_t L_t}\) and \(\hat{k_t}=\frac{K_t}{A_tL_t}\) the production and capital per effective unit

  1. Dividing the capital accumulation equation by \(A_tL_t\), show that

    \[\hat{k_{t+1}} (1+n)(1+g)= (1-\delta)\hat{k_t} + s F(\frac{K_t}{A_tL_t},1)\]

Dividing the capital accumulation by \(A_tL_t\) yields:

\[ \frac{K_{t+1}}{A_tL_t}= (1-\delta)\hat{k_t} + s \frac{F(K_t,A_tL_t)}{A_tL_t} \]

Noticing that the left hand side is \(\frac{K_{t+1}}{A_{t+1}L_{t+1}} (1+n)(1+g)\) and that \(\frac{1}{A_tL_t}F(K_t,A_tL_t) = F(\frac{K_t}{A_tL_t},1) = F(\hat{k}_t,1)\) because of constant returns to scale yields the result.

  1. Show that there is a steady state level of capital per effective unit \(\hat{k^*}\) equal to:

\[ \hat{k^*} = \left(\frac{s}{(1+n)(1+g)-1+\delta}\right)^{\frac{1}{1-\alpha}} \]

Plugging \(\hat{k} = \hat{k}_{t+1} = \hat{k}_t\) and solving for it in the equation above yields the result.

Noticing that \(\hat{y} = \hat{k}^\alpha\) and plugging \(\hat{k}^*\) yields steady state per effective unit output:

\[ \hat{y^*} = \left(\frac{s}{(1+n)(1+g)-1+\delta}\right)^{\frac{\alpha}{1-\alpha}} \]

  1. Show that per capita outcomes \(y\) and \(k\) are on a balanced growth path. What is their respective growth rates?

\(k_t = A_t \hat{k}_t\) and \(y_t = A_t \hat{y}_t\). Plugging steady state levels of \(\hat{k}^*\) and \(\hat{y}^*\), then computing the equilibrium growth rate of capital and output yields:

\[ \frac{k_{t+1}}{k_t} = \frac{y_{t+1}}{y_t} = (1+g) \]

5. Discussion

  1. What makes this model a simplification of reality?
  2. Why cannot it explain all differences in output growth and per capita level across economies?

Firms:

  • Imperfect competition and appropriated innovation (through patents) yields profits and investment into private R&D

  • Heterogeneous firms/sectors need various “qualities” of physical and human capital, and different “vintages” of technology (a technology can get outdated)

Households:

  • Optimize and decide endogenously upon their labor supply, saving rate, fertility, education level (human capital)

State:

  • affects available consumption through taxes and transfers

  • affects investment into public research and public education (technological progress and human capital)

  • provides institutions that affect firms’ and households’ decisions

Trade (open economy):

  • Affects net investment in the country, and available output for consumption through imports/exports

Link to Aghion’s seminar on the evolution of growth theory (in French with possible subtitles) Growth theory and policies - Philippe Aghion - Collège de France

A bit more of a self-promotion of his work, but also interesting: Inaugural lecture of the School of Management & Impact - Philippe Aghion - Sciences Po

Note: A useful (although far from mandatory for this class) handbook for undergraduate macroeconomics is Macroeconomics by N. Gregory Mankiw