Problem Set 3

Briefly explain the general theories of balanced and unbalanced growth.

Briefly explain the idea of a Big Push.

Briefly explain the general idea of a poverty trap. Illustrate it with an adequate figure.

2. The O-Ring Theory of Development

Following Kremer (1993)1, consider a firm conducting a production process consisting of 𝑛 tasks. For simplicity, we assume that one worker is assigned per task. Furthermore, suppose a continuum of workers who earn a wage 𝑤(𝑞𝑖) that depends on the respective level of worker quality, 𝑞𝑖 ∈ [0,1]. The workers supply labor fully inelastically and the distribution of skill follows a specific function 𝜙(𝑞). In case all tasks are performed perfectly (𝑞 = 1), the firm’s output is equal to 𝐵. The firm’s production function is:

How does this production function differ from standard production functions found in literature?

2. The O-Ring Theory of Development

Following Kremer (1993)1, consider a firm conducting a production process consisting of 𝑛 tasks. For simplicity, we assume that one worker is assigned per task. Furthermore, suppose a continuum of workers who earn a wage 𝑤(𝑞𝑖) that depends on the respective level of worker quality, 𝑞𝑖 ∈ [0,1]. The workers supply labor fully inelastically and the distribution of skill follows a specific function 𝜙(𝑞). In case all tasks are performed perfectly (𝑞 = 1), the firm’s output is equal to 𝐵. The firm’s production function is:

State the firm’s maximization problem and derive the first-order conditions with respect to the worker quality 𝑞𝑖. Explain the economic intuition of your results.

2. The O-Ring Theory of Development

Following Kremer (1993)1, consider a firm conducting a production process consisting of 𝑛 tasks. For simplicity, we assume that one worker is assigned per task. Furthermore, suppose a continuum of workers who earn a wage 𝑤(𝑞𝑖) that depends on the respective level of worker quality, 𝑞𝑖 ∈ [0,1]. The workers supply labor fully inelastically and the distribution of skill follows a specific function 𝜙(𝑞). In case all tasks are performed perfectly (𝑞 = 1), the firm’s output is equal to 𝐵. The firm’s production function is:

Formulate the quality-clustering theorem and prove it

2. The O-Ring Theory of Development

Following Kremer (1993)1, consider a firm conducting a production process consisting of 𝑛 tasks. For simplicity, we assume that one worker is assigned per task. Furthermore, suppose a continuum of workers who earn a wage 𝑤(𝑞𝑖) that depends on the respective level of worker quality, 𝑞𝑖 ∈ [0,1]. The workers supply labor fully inelastically and the distribution of skill follows a specific function 𝜙(𝑞). In case all tasks are performed perfectly (𝑞 = 1), the firm’s output is equal to 𝐵. The firm’s production function is:

Use the quality-clustering theorem to determine the equilibrium wage schedule.

2. The O-Ring Theory of Development

Following Kremer (1993)1, consider a firm conducting a production process consisting of 𝑛 tasks. For simplicity, we assume that one worker is assigned per task. Furthermore, suppose a continuum of workers who earn a wage 𝑤(𝑞𝑖) that depends on the respective level of worker quality, 𝑞𝑖 ∈ [0,1]. The workers supply labor fully inelastically and the distribution of skill follows a specific function 𝜙(𝑞). In case all tasks are performed perfectly (𝑞 = 1), the firm’s output is equal to 𝐵. The firm’s production function is:

Which source of underdevelopment is explained by the model? What development policy conclusions can be drawn?

3. Industrialization and the Big Push

Consider an economy consisting of 𝑘 sectors, with each sector producing one particular output good. The only factor of production is labor, labelled 𝐿𝑖 , and output is denoted by 𝑄𝑖, with 𝑖 denoting the sector (𝑖 = 1, . . . , 𝑘). In each sector 𝑖, the production technology can be traditional or industrialized. The traditional technology is 𝑄𝑖 = 𝐿𝑖. Sectors with the traditional technology face perfect competition, and firms can exit and enter markets at no cost. The industrialized technology is 𝑄𝑖 = 𝛼𝐿𝑖, with 𝛼 > 1. Industrialization requires a fixed investment cost 𝐹. The number of sectors that use the industrialized technology is denoted by 𝑛. Due to an externality, investment cost 𝐹 decreases in the number of sectors that have already industrialized in the past: 𝐹 = 𝐹(𝑛) with 𝜕𝐹(𝑛) / 𝜕𝑛 < 0. The output price in sector 𝑖 is 𝑃𝑖. The wage rate (𝑤) is normalized to 1.

State the profit functions for both types of technology. What can we say about the size of the profits, respectively?

3. Industrialization and the Big Push

Consider an economy consisting of 𝑘 sectors, with each sector producing one particular output good. The only factor of production is labor, labelled 𝐿𝑖 , and output is denoted by 𝑄𝑖, with 𝑖 denoting the sector (𝑖 = 1, . . . , 𝑘). In each sector 𝑖, the production technology can be traditional or industrialized. The traditional technology is 𝑄𝑖 = 𝐿𝑖. Sectors with the traditional technology face perfect competition, and firms can exit and enter markets at no cost. The industrialized technology is 𝑄𝑖 = 𝛼𝐿𝑖, with 𝛼 > 1. Industrialization requires a fixed investment cost 𝐹. The number of sectors that use the industrialized technology is denoted by 𝑛. Due to an externality, investment cost 𝐹 decreases in the number of sectors that have already industrialized in the past: 𝐹 = 𝐹(𝑛) with 𝜕𝐹(𝑛) / 𝜕𝑛 < 0. The output price in sector 𝑖 is 𝑃𝑖. The wage rate (𝑤) is normalized to 1.

Let there be a representative consumer with utility function 𝑢 = Π𝑖𝑘=1𝑥𝑖 , who cannot affect prices (price taker) and owns all firms and thus earns all profits Π̅ in the economy. The representative consumer supplies labor 𝐿 to all sectors inelastically. Deduce the consumer’s goods demand in each sector, respectively.

3. Industrialization and the Big Push

Consider an economy consisting of 𝑘 sectors, with each sector producing one particular output good. The only factor of production is labor, labelled 𝐿𝑖 , and output is denoted by 𝑄𝑖, with 𝑖 denoting the sector (𝑖 = 1, . . . , 𝑘). In each sector 𝑖, the production technology can be traditional or industrialized. The traditional technology is 𝑄𝑖 = 𝐿𝑖. Sectors with the traditional technology face perfect competition, and firms can exit and enter markets at no cost. The industrialized technology is 𝑄𝑖 = 𝛼𝐿𝑖, with 𝛼 > 1. Industrialization requires a fixed investment cost 𝐹. The number of sectors that use the industrialized technology is denoted by 𝑛. Due to an externality, investment cost 𝐹 decreases in the number of sectors that have already industrialized in the past: 𝐹 = 𝐹(𝑛) with 𝜕𝐹(𝑛) / 𝜕𝑛 < 0. The output price in sector 𝑖 is 𝑃𝑖. The wage rate (𝑤) is normalized to 1.

Calculate the price levels 𝑃𝑖 and the equilibrium level of output in each sector

3. Industrialization and the Big Push

Consider an economy consisting of 𝑘 sectors, with each sector producing one particular output good. The only factor of production is labor, labelled 𝐿𝑖 , and output is denoted by 𝑄𝑖, with 𝑖 denoting the sector (𝑖 = 1, . . . , 𝑘). In each sector 𝑖, the production technology can be traditional or industrialized. The traditional technology is 𝑄𝑖 = 𝐿𝑖. Sectors with the traditional technology face perfect competition, and firms can exit and enter markets at no cost. The industrialized technology is 𝑄𝑖 = 𝛼𝐿𝑖, with 𝛼 > 1. Industrialization requires a fixed investment cost 𝐹. The number of sectors that use the industrialized technology is denoted by 𝑛. Due to an externality, investment cost 𝐹 decreases in the number of sectors that have already industrialized in the past: 𝐹 = 𝐹(𝑛) with 𝜕𝐹(𝑛) / 𝜕𝑛 < 0. The output price in sector 𝑖 is 𝑃𝑖. The wage rate (𝑤) is normalized to 1.

Determine the profit of an industrialized firm. What is the macroeconomic level of profits in the economy?

3. Industrialization and the Big Push

Consider an economy consisting of 𝑘 sectors, with each sector producing one particular output good. The only factor of production is labor, labelled 𝐿𝑖 , and output is denoted by 𝑄𝑖, with 𝑖 denoting the sector (𝑖 = 1, . . . , 𝑘). In each sector 𝑖, the production technology can be traditional or industrialized. The traditional technology is 𝑄𝑖 = 𝐿𝑖. Sectors with the traditional technology face perfect competition, and firms can exit and enter markets at no cost. The industrialized technology is 𝑄𝑖 = 𝛼𝐿𝑖, with 𝛼 > 1. Industrialization requires a fixed investment cost 𝐹. The number of sectors that use the industrialized technology is denoted by 𝑛. Due to an externality, investment cost 𝐹 decreases in the number of sectors that have already industrialized in the past: 𝐹 = 𝐹(𝑛) with 𝜕𝐹(𝑛) / 𝜕𝑛 < 0. The output price in sector 𝑖 is 𝑃𝑖. The wage rate (𝑤) is normalized to 1.

What is the optimal technology choice for the firms? What are the determinants of this decision?

3. Industrialization and the Big Push

Consider an economy consisting of 𝑘 sectors, with each sector producing one particular output good. The only factor of production is labor, labelled 𝐿𝑖 , and output is denoted by 𝑄𝑖, with 𝑖 denoting the sector (𝑖 = 1, . . . , 𝑘). In each sector 𝑖, the production technology can be traditional or industrialized. The traditional technology is 𝑄𝑖 = 𝐿𝑖. Sectors with the traditional technology face perfect competition, and firms can exit and enter markets at no cost. The industrialized technology is 𝑄𝑖 = 𝛼𝐿𝑖, with 𝛼 > 1. Industrialization requires a fixed investment cost 𝐹. The number of sectors that use the industrialized technology is denoted by 𝑛. Due to an externality, investment cost 𝐹 decreases in the number of sectors that have already industrialized in the past: 𝐹 = 𝐹(𝑛) with 𝜕𝐹(𝑛) / 𝜕𝑛 < 0. The output price in sector 𝑖 is 𝑃𝑖. The wage rate (𝑤) is normalized to 1.

Describe the low-income trap in the model. What do we learn for development policy?