Problem Set 2

1. Economic Growth, Inflation, and Population Growth

Consider a small model of economic growth, inflation, and population growth. Let 𝑌𝑡 denote nominal GDP, and 𝑦𝑡 nominal GDP per capita. 𝑁𝑡 denotes population in time 𝑡 and 𝑃𝑡 the price level in 𝑡. The economy has an initial GDP per capita endowment of 𝑦0. Assume that population grows with a rate of 𝑛 = 0.02, and the inflation rate is π = 0.12

Now assume the central bank is able to reduce inflation to the level of price stability as defined by the ECB at π = 0.02. Calculate again the annual nominal growth rate that keeps real GDP per capita constant

1. Economic Growth, Inflation, and Population Growth

Consider a small model of economic growth, inflation, and population growth. Let 𝑌𝑡 denote nominal GDP, and 𝑦𝑡 nominal GDP per capita. 𝑁𝑡 denotes population in time 𝑡 and 𝑃𝑡 the price level in 𝑡. The economy has an initial GDP per capita endowment of 𝑦0. Assume that population grows with a rate of 𝑛 = 0.02, and the inflation rate is π = 0.12

What is the avegare annual nominal GDP growth rate 𝛾 that is required to keep real GDP per capita constant over ten years?

1. Economic Growth, Inflation, and Population Growth

Consider a small model of economic growth, inflation, and population growth. Let 𝑌𝑡 denote nominal GDP, and 𝑦𝑡 nominal GDP per capita. 𝑁𝑡 denotes population in time 𝑡 and 𝑃𝑡 the price level in 𝑡. The economy has an initial GDP per capita endowment of 𝑦0. Assume that population grows with a rate of 𝑛 = 0.02, and the inflation rate is π = 0.12

Calculate the annual GDP growth rate 𝛾 that is required to keep real GDP per capita constant

1. Economic Growth, Inflation, and Population Growth

Consider a small model of economic growth, inflation, and population growth. Let 𝑌𝑡 denote nominal GDP, and 𝑦𝑡 nominal GDP per capita. 𝑁𝑡 denotes population in time 𝑡 and 𝑃𝑡 the price level in 𝑡. The economy has an initial GDP per capita endowment of 𝑦0. Assume that population grows with a rate of 𝑛 = 0.02, and the inflation rate is π = 0.12

Now assume additionally that population growth is decreasing to 𝑛 = 0.01. What is the level of 𝛾 required to keep GDP per capita constant, with the inflation rate of π = 0.02?

2. The Harrod-Domar Model

Within the Harrod-Domar Model, economic growth results from the process of capital formation. The capital formation process is given by 𝐾𝑡+1 = (1 − 𝛿)𝐾𝑡 + 𝑆𝑡, where 𝐾𝑡+1 and 𝐾𝑡 denote capital in periods 𝑡 + 1 and 𝑡 respectively, 𝑆𝑡 is gross savings in period 𝑡, and 𝛿 is the rate of depreciation of capital. It is assumed that the capital-output ratio 𝜃 = 𝐾𝑡 / 𝑌𝑡 as well as the savings rate 𝑠 = 𝑆𝑡 / 𝑌𝑡 are a constant fraction of GDP. Output grows at a constant rate of 𝛾, so that 𝑌𝑡+1 = (1 + 𝛾)𝑌𝑡. Population is denoted by 𝑁𝑡 and grows at a constant rate of 𝑛.

Derive the growth rate of GDP per capita within this model. Interpret your results.

Hint: Start with the capital formation process

2. The Harrod-Domar Model

Within the Harrod-Domar Model, economic growth results from the process of capital formation. The capital formation process is given by 𝐾𝑡+1 = (1 − 𝛿)𝐾𝑡 + 𝑆𝑡, where 𝐾𝑡+1 and 𝐾𝑡 denote capital in periods 𝑡 + 1 and 𝑡 respectively, 𝑆𝑡 is gross savings in period 𝑡, and 𝛿 is the rate of depreciation of capital. It is assumed that the capital-output ratio 𝜃 = 𝐾𝑡 / 𝑌𝑡 as well as the savings rate 𝑠 = 𝑆𝑡 / 𝑌𝑡 are a constant fraction of GDP. Output grows at a constant rate of 𝛾, so that 𝑌𝑡+1 = (1 + 𝛾)𝑌𝑡. Population is denoted by 𝑁𝑡 and grows at a constant rate of 𝑛.

What are the practical implications of the model for development planning? Briefly interpret each right-hand-side variable

2. The Harrod-Domar Model

Within the Harrod-Domar Model, economic growth results from the process of capital formation. The capital formation process is given by 𝐾𝑡+1 = (1 − 𝛿)𝐾𝑡 + 𝑆𝑡, where 𝐾𝑡+1 and 𝐾𝑡 denote capital in periods 𝑡 + 1 and 𝑡 respectively, 𝑆𝑡 is gross savings in period 𝑡, and 𝛿 is the rate of depreciation of capital. It is assumed that the capital-output ratio 𝜃 = 𝐾𝑡 / 𝑌𝑡 as well as the savings rate 𝑠 = 𝑆𝑡 / 𝑌𝑡 are a constant fraction of GDP. Output grows at a constant rate of 𝛾, so that 𝑌𝑡+1 = (1 + 𝛾)𝑌𝑡. Population is denoted by 𝑁𝑡 and grows at a constant rate of 𝑛.

Assume that the government of country X intends to triple its GDP per capita within the next fifteen years. According to the Harrod-Domar model, how large should the savings rate s be if population growth is 𝑛 = 0.01, depreciation is 𝛿 = 0.05, and capital-output ratio is 𝜃 = 4?

2. The Harrod-Domar Model

Within the Harrod-Domar Model, economic growth results from the process of capital formation. The capital formation process is given by 𝐾𝑡+1 = (1 − 𝛿)𝐾𝑡 + 𝑆𝑡, where 𝐾𝑡+1 and 𝐾𝑡 denote capital in periods 𝑡 + 1 and 𝑡 respectively, 𝑆𝑡 is gross savings in period 𝑡, and 𝛿 is the rate of depreciation of capital. It is assumed that the capital-output ratio 𝜃 = 𝐾𝑡 / 𝑌𝑡 as well as the savings rate 𝑠 = 𝑆𝑡 / 𝑌𝑡 are a constant fraction of GDP. Output grows at a constant rate of 𝛾, so that 𝑌𝑡+1 = (1 + 𝛾)𝑌𝑡. Population is denoted by 𝑁𝑡 and grows at a constant rate of 𝑛.

The government of country X realized that its current savings rate is just 𝑠 = 0.25, but it still wants to reach its target of tripling its GDP per capita within the next fifteen years. What is the size of the annual financing gap? Briefly explain the government’s options to close it.

3. The Harris-Todaro Model

Consider an economy that consists of two sectors, a traditional sector (𝑇) and a modern one (𝑀). Workers can move freely from one sector to the other. They do this by comparing their salary in the traditional sector (𝑤𝑇) to their expected salary in the modern sector (𝑤), which depends on the probability of finding a job in the urban sector. Both sectors produce the same consumption good 𝐶, albeit with different levels of technology. The total labor force in the economy is labelled as 𝐿 = 𝐿𝑇 + 𝐿𝑀. The production function of both sectors is 𝐶𝑖 = 𝜂𝐿𝛼𝑖 . For the modern sector 𝜂 = 1 and for the traditional sector it holds that 0 < 𝜂 < 1.

State the maximization problem of each sector and derive the first-order conditions with respect to the optimal choice of labor for each sector.

3. The Harris-Todaro Model

Consider an economy that consists of two sectors, a traditional sector (𝑇) and a modern one (𝑀). Workers can move freely from one sector to the other. They do this by comparing their salary in the traditional sector (𝑤𝑇) to their expected salary in the modern sector (𝑤), which depends on the probability of finding a job in the urban sector. Both sectors produce the same consumption good 𝐶, albeit with different levels of technology. The total labor force in the economy is labelled as 𝐿 = 𝐿𝑇 + 𝐿𝑀. The production function of both sectors is 𝐶𝑖 = 𝜂𝐿𝛼𝑖 . For the modern sector 𝜂 = 1 and for the traditional sector it holds that 0 < 𝜂 < 1.

Assume that the probability of getting a job in the modern sector is defined as 𝐿𝑀 / 𝐿−𝐿𝑇. What condition has to be fulfilled in terms of wages for migration from the traditional sector to the modern one to take place?

3. The Harris-Todaro Model

Consider an economy that consists of two sectors, a traditional sector (𝑇) and a modern one (𝑀). Workers can move freely from one sector to the other. They do this by comparing their salary in the traditional sector (𝑤𝑇) to their expected salary in the modern sector (𝑤), which depends on the probability of finding a job in the urban sector. Both sectors produce the same consumption good 𝐶, albeit with different levels of technology. The total labor force in the economy is labelled as 𝐿 = 𝐿𝑇 + 𝐿𝑀. The production function of both sectors is 𝐶𝑖 = 𝜂𝐿𝛼𝑖 . For the modern sector 𝜂 = 1 and for the traditional sector it holds that 0 < 𝜂 < 1.

Assume that 𝐿 = 1, 𝑤 = 1, 𝛼 = 1/2, and 𝜂 = 1/2. Calculate the equilibrium levels of employment, output, and unemployment for each sector.

3. The Harris-Todaro Model

Consider an economy that consists of two sectors, a traditional sector (𝑇) and a modern one (𝑀). Workers can move freely from one sector to the other. They do this by comparing their salary in the traditional sector (𝑤𝑇) to their expected salary in the modern sector (𝑤), which depends on the probability of finding a job in the urban sector. Both sectors produce the same consumption good 𝐶, albeit with different levels of technology. The total labor force in the economy is labelled as 𝐿 = 𝐿𝑇 + 𝐿𝑀. The production function of both sectors is 𝐶𝑖 = 𝜂𝐿𝛼𝑖 . For the modern sector 𝜂 = 1 and for the traditional sector it holds that 0 < 𝜂 < 1.

Calculate the Pareto efficient allocation of 𝐿𝑇 and 𝐿𝑀.

3. The Harris-Todaro Model

Consider an economy that consists of two sectors, a traditional sector (𝑇) and a modern one (𝑀). Workers can move freely from one sector to the other. They do this by comparing their salary in the traditional sector (𝑤𝑇) to their expected salary in the modern sector (𝑤), which depends on the probability of finding a job in the urban sector. Both sectors produce the same consumption good 𝐶, albeit with different levels of technology. The total labor force in the economy is labelled as 𝐿 = 𝐿𝑇 + 𝐿𝑀. The production function of both sectors is 𝐶𝑖 = 𝜂𝐿𝛼𝑖 . For the modern sector 𝜂 = 1 and for the traditional sector it holds that 0 < 𝜂 < 1.

Intuitively, how could the Pareto efficient allocation of 𝐿𝑇 and 𝐿𝑀 be achieved?