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: Return to scale

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

  1. \(y_1 = 10x^2y^2\)
  2. \(y_2 = \frac{1}{2}x^{1/3}y^{1/2}\)
  3. \(y_3 = x + 2y\)
  4. \(y_4 = \ln(xy)\)

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
  2. Show that the production function has constant return to scale
  3. Show that marginal productivity is positive for capital and labour
  4. Show that marginal productivity is decreasing for capital and labour
  5. Write the down the profit maximization program (maximizing profit with respect to K and L) and derive its first order conditions
  6. Show that all output is needed to pay capital and labor

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 \(\infty\)
  2. Show that there is a unique steady state level of capital \(k^*\) and solve for it
  3. How does the \(k^*\) vary with \(s\), \(\delta\), \(A\), \(L\) ? Interpret

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

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

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

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

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

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?