Cross-Country Income Differences is Because of What?

Another lecture note from macroeconomic class.

Question: why the log GDP per capita are widely different across countries and that they doesn’t seem to converge?

Assuming Cobb-Douglas, data analysis says that over 60% of cross-country income dispersion is attributable to TFP differences, not input differences. Countries are not poor simply because they lack capital or education — they are poor because they use their inputs far less productively. Technology/efficiency differences ($A$) of the production function are the dominant explanation.

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Misallocation Verification

Consider a static economy with two sectors $i\in \lbrace M, A\rbrace$ (agriculture and manufacturing) both with CRS Cobb-Douglas production function $Y_i =F_i(K_i, L_i) = A_i K_i^{\alpha_i}L_i^{1 - \alpha_i}$. Consumers with convex utility function $U(c_M, c_A)$. The the planner’s problem:

$$ \begin{align*} & \max_{\lbrace K_i, L_i, c_i\rbrace} U(c_M, c_A)\cr & \text{s.t.}\quad c_i = A_i K_i^{\alpha_i}L_i^{1 - \alpha_i}, \forall i\cr & \qquad K_M + K_A = K,\cr & \qquad L_M + L_A = L. \end{align*} $$

FOC implies

$$ \frac{{\partial U}/{\partial c_M}}{\partial U /\partial c_A}\cdot \frac{\partial F_M/\partial L_M}{\partial F_A/\partial L_A}=1 \tag{1} $$

Remark. Substitute the resource constraints $K_A = K - K_M$, $L_A = L - L_M$ and the production functions $c_i = F_i$ directly into the objective to reduce the planner’s problem to an unconstrained maximization over $(K_M, L_M)$. The FOC with respect to $L_M$ applies the chain rule through both channels of utility — the gain from raising $c_M$ and the cost of lowering $c_A$ — yields $(\partial U/\partial c_M)(\partial F_M/\partial L_M) = (\partial U/\partial c_A)(\partial F_A/\partial L_A)$, which in ratio form is equation (1).

The condition admits a clean interpretation: the MRS between the two goods must exactly equal the ratio of marginal products of labor across sectors, so that no reallocation of a single worker can improve welfare — the demand-side valuation and the supply-side transformation rate are in perfect balance.

Consumer’s optimization

$$ \begin{align*} & \max_{\lbrace c_i\rbrace} U(c_M, c_A)\cr & \text{s.t.}\quad c_A p_A + c_M p_M \le B \end{align*} $$

implies

$$ \frac{{\partial U}/{\partial c_M}}{\partial U /\partial c_A} = \frac{p_M}{p_A}\tag{2}. $$

The production functions’ HD1 structure implies

$$ \frac{\partial F_M/\partial L_M}{\partial F_A/\partial L_A} = \frac{(1 - \alpha_M) Y_M/L_M}{(1 - \alpha_A) Y_A/L_A}.\tag{3} $$

So [ (2) + (3) ] + (1) implies

$$ \frac{p_M Y_M/L_M}{p_AY_A/L_A} = \frac{1 - \alpha_A}{1 - \alpha_M}.\tag{4} $$

$Y_i, p_i, L_i$ are production, price, labor ratio of each sector which can be get from data.

$\alpha_i$ can be get from the firm’s problem:

$$ \begin{align*} & \max_{K, L} \; AK^\alpha L^{1-\alpha} - rK - wL\cr FOC \Rightarrow & \begin{cases} \frac{\partial Y}{\partial K} = \alpha A K^{\alpha-1} L^{1-\alpha} = r\cr \frac{\partial Y}{\partial L} = (1-\alpha) A K^{\alpha} L^{-\alpha} = w \end{cases} \end{align*} $$

Some algebras on the FOC plus using the fact that $Y = A K^\alpha L^{1 - \alpha}$ would imply

$${\frac{wL}{Y}} = 1 - \alpha \qquad \text{and} \qquad \underbrace{\frac{rK}{Y}}_{\text{capital share}} = \alpha$$

This is a special and convenient property of Cobb-Douglas that really, does not hold for any general production functions…

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So from the data, we can verify whether an economy that is assumed to satisfy Cobb Douglas is efficient.