The Two-Sector Model of General Equilibrium

Harry G. Johnson

• Publisher: Routledge
• Year: 2021
• ISBN:
  9781000421248
• Edition: 1
• Topic: Economics
• Subtopic: Economic Theory
• Index: Economics

Originally published in 1971, this book presents in a lucid form the basic model of distribution in a two-sector general equilibrium system. While this model has been used by many economists, this was the first synoptic exposition of it to become readily available to students. The first part develops the two-sector model and its properties, using the geometrical tools of international trade theory. The second applies the model to some standard problems in the theory of income distribution, including the economics of redistributive taxes and subsidies, of trade union organization, and of minimum wage laws. The third part converts the model into a growth model and develops the conditions for convergence on a steady-state growth path and for the maximization of consumption per head at all points of time.

Appendix A

Mathematical Notes on the One-Sector Model

(a) The Hicknan Model of Distribution

Let x = f(a, b) be the aggregate production function where x is aggregate output and a and b are the quantities of the two factors.

The aggregate production function is assumed to be linear homogeneous (subject to constant returns to scale). This implies

$x={\mathit{\text{af}}}_a+{\mathit{\text{bf}}}_b(1)$

${\displaystyle f_{\mathit{\text{ab}}}=-\frac{a}{b}{f}_{\mathit{\text{aa}}}=-\frac{b}{a}{f}_{\mathit{\text{bb}}}}(2)$

${\displaystyle \sigma =\frac{f_a{f}_b}{{\mathit{\text{xf}}}_{\mathit{\text{ab}}}}}(3)$

where subscripts denote differentiation of the production function by the subscripted argument or arguments, and σ is the elasticity of substitution. (For proofs see R. G. D. Allen, Mathematical Analysis for Economists, pages 317 and 343.) By the usual assumptions f_ab > 0, and f_aa, f_bb < 0.

Define K_a = af_a, K_b = bf_b, the absolute shares of the factors, and k_a = af_a/x, k_b = bf_b/x, the relative shares of the factors,

${\displaystyle \begin{array}{ccc}\frac{{\mathit{\text{dK}}}_a}{\mathit{\text{da}}}& =& f_a+{\mathit{\text{af}}}_{\mathit{\text{aa}}}\\ & =& f_a(1-\frac{{\mathit{\text{bf}}}_{\mathit{\text{ab}}}}{f_a})\hfill \\ & =& f_a(1-\frac{{\mathit{\text{bf}}}_b}{x}·\frac{1}{\sigma })\hfill \\ & =& f_a\left(\frac{\sigma -k_b}{\sigma }\right)\hfill \\ & =& f_a\left(\frac{k_a-(1-\sigma )}{\sigma }\right)\hfill \end{array}}(4)$

${\displaystyle \frac{{\mathit{\text{dK}}}_a}{\mathit{\text{db}}}={\mathit{\text{af}}}_{\mathit{\text{ab}}}=\frac{1}{\sigma }\frac{{\mathit{\text{af}}}_a{f}_b}{x}}(5)$

By (5) an increase in the quantity of one factor must always increase the absolute share of the other. By (4) an increase in the quantity of one factor will increase that factor’s absolute share unless the elasticity of substitution is smaller than the relative share of the other factor, or (alternatively) unless the elasticity of substitution is below unity by more than the relative share of the augmented factor.

${\displaystyle \begin{array}{ccc}\frac{{\mathit{\text{dk}}}_a}{\mathit{\text{da}}}& =& \frac{1}{x}(x(f_a+{\mathit{\text{af}}}_{\mathit{\text{aa}}})-{\mathit{\text{af...}}}_{}\hfill \end{array}}$