Given a closed economy where there is no public sector. Production in the economy can be described by the following production function: Y = F (K, AL) = K^α (AL)^(1−α) where Y is the productive capacity of the economy, K is the capital stock, L is the labor force, and A is knowledge. Think of AL as a single factor of production where knowledge and the amount of labor are multiplied together. Let's call the multiplier (AL) the efficiency of labor. It is given that A = 1.5 and saving is a fixed percentage of production, or s = 33%. The capital stock shrinks by 3% per year, but the population growth is zero. Finally, α = 0.4. Answer the following questions based on the above criteria. (a) Show mathematically that the marginal productivity of labor and capital is positive but diminishing. Explain in words and with a picture what the term positive but diminishing marginal productivity means. (b) Show mathematically that saving and investment are equivalent in a closed economy (c) Show mathematically that the production function satisfies the assumption of constant economies of scale. Explain what fixed economies of scale mean.
Given a closed economy where there is no public sector. Production in the economy can be described by the following production function:
Y = F (K, AL) = K^α (AL)^(1−α)
where Y is the productive capacity of the economy, K is the capital stock, L is the labor force, and A is knowledge. Think of AL as a single factor of production where knowledge and the amount of labor are multiplied together. Let's call the multiplier (AL) the efficiency of labor. It is given that A = 1.5 and saving is a fixed percentage of production, or s = 33%. The capital stock shrinks by 3% per year, but the population growth is zero. Finally, α = 0.4.
Answer the following questions based on the above criteria.
(a) Show mathematically that the marginal productivity of labor and capital is positive but diminishing. Explain in words and with a picture what the term positive but diminishing marginal productivity means.
(b) Show mathematically that saving and investment are equivalent in a closed economy
(c) Show mathematically that the production function satisfies the assumption of constant economies of scale. Explain what fixed economies of scale mean.
(d) Plot the production function as a function of labor efficiency. Denote sizes in proportion to labor efficiency with lowercase letters, e.g. k = K/AL . Hint: You should end up with y = k^α.
(e) What is the capital stock per effective labor force in equilibrium in the economy? Show your balance of the model graphically.
(f) Assume k=15 initially. What will be the
(g) Show the development of the capital stock per effective labor force in the first two years and their position at the end of year two. What does the growth say about the nation's position compared to other comparable nations that are balanced? Why does the economy grow or shrink while other comparable nations remain stagnant? What effects are at work here?
(h) Assume that the economy has now reached equilibrium. Now the savings rate increases to 50%. What will be the growth rate per effective labor force in the first year of the change? Show the effect on the balance of the model graphically and explain in words.
(i) Now the savings rate is again 33%, but instead knowledge in the economy increases by a third. What will be the
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