Question 1 [14 points] Consider an economy with n = 10 goods, with sector 1 producing good 1, sector 2 producing good 2 and so on. For the representative consumer of the economy, all 10 goods are weighted equally. The economy has L = 1000 units of labour which is supplied by the representative consumer.
Each sector has a competitive fringe of firms. Initially firms in all sectors use the traditional mode of production in which 1 unit of labour can produce 1 unit of good and the wage is 1.
There is a new technology (the modern mode) in which 1 unit of labour can produce t = 4 units of the good. The modern mode pays wage 1 + v = 2 (so v = 1).
(a) [6 points] First suppose no investment is required for the modern mode and once a sector adopts the modern mode, all firms in the competitive fringe of that sector have the modern mode.
In this case, determine the income of the economy and the demand for each good when 5 out of 10 sectors adopt the modern mode.
(b) [8 points] Now suppose the modern mode requires an initial employment of F = 35 units of labour. Assume that in each sector, there is a modern firm that can adopt the modern mode by making this initial employment of F units of labour. Because the wage in modern mode is 1 + v, the cost of investment is (1 + v)F [recall that we are given 1 + v = 2 and F = 35].
In this case we say a sector is “industrialized” if the modern firm of that sector makes the investment and adopts the modern mode. Once the modern firm in a sector adopts the modern mode, all other firms in the fringe are driven out of the market.
Denote b = 1 – (1 + v)/t. We know that when k sectors are industrialized, the profit net of investment cost of any modern firm that has adopted the modern mode is positive if and only if the expression gk below is positive.
gk = bL – n(1 + v)F + vFk
Using the properties of gk
(i) determine all equilibrium industrialization structures of the economy and
(ii) determine if there is a minimum critical level k* so that if there is investment by modern firms in k* sectors, modern firms in the remaining sectors will also have incentive to make investment; if there is such a k*, find its numerical value.
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Economic Analysis of Modern Mode Adoption in a Multi-Sector Economy
Economic Analysis of Modern Mode Adoption in a Multi-Sector Economy
In the given economic scenario, we explore the implications of adopting a modern mode of production across different sectors in a multi-sector economy. The adoption of the modern mode, which enhances labor productivity, has varying effects on income, demand for goods, and industrialization patterns within the economy.
Case without Investment Requirement for Modern Mode Adoption
(a) In this case, when 5 out of the 10 sectors adopt the modern mode with a productivity level of t = 4 units of good per unit of labor, the income of the economy can be determined as follows:
- Initially, each unit of labor produces 1 unit of good at a wage of 1.
- With the modern mode, each unit of labor produces 4 units of good at a wage of 2 (1 + v).
- Therefore, the total income of the economy increases as more sectors adopt the modern mode, leading to a higher overall demand for goods.
Case with Investment Requirement for Modern Mode Adoption
(b) When the modern mode adoption requires an initial employment of F = 35 units of labor and incurs an investment cost of (1 + v)F = 70 units, the industrialization dynamics of the economy change significantly. The profitability condition for modern firms adopting the modern mode in k sectors can be expressed as:
gk = bL - n(1 + v)F + vFk,
where b = 1 - (1 + v)/t.
(i) By analyzing the properties of gk, we can determine the equilibrium industrialization structures of the economy, i.e., the combinations of sectors that will adopt the modern mode to maximize profits. This analysis will provide insights into the optimal industrialization patterns within the economy.
(ii) Furthermore, we can ascertain if there exists a minimum critical level k* such that if k* sectors invest in the modern mode, it triggers a cascading effect leading to investment in the remaining sectors. By calculating the numerical value of k*, we can identify the threshold at which further industrialization becomes economically viable for all sectors.
In conclusion, the adoption of the modern mode of production in a multi-sector economy has far-reaching implications on income distribution, demand dynamics, and industrialization patterns. By examining both scenarios with and without investment requirements for modern mode adoption, we can gain valuable insights into optimizing economic outcomes and fostering sustainable growth across sectors.
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