7 the depreciation line is a constant it does not

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Unformatted text preview: s what we call the savings function, sf (A, k ) k and the most important thing is that it will inherit the diminishing marginal returns from capital. • The negative one is what we call the depreciation line, n + δ Suppose that you want to plot a graph where the x-axis is capital per capita, and in the vertical axis you will measure both the savings line and the depreciation line. 7 • The depreciation line is a constant. It does not depend on capital. For any level of capital, n + δ is going to be the same. So, it is a horizontal line in our graph. • The savings funcion is slightly more complicated. But we can do it! – Suppose that capital goes to 0. You should realize that the function goes to ∞ (something divided by 0 explodes to infinity) – As capital gets bigger and bigger, the denominator increases but also the numerator! Which increase is going to dominate? Since f (A, k ) has diminishing returns to capital, there is a moment where the amount of capital is so big that the numerator increases by a very small amount. But the denominator continues increasing. Therefore, the function tends to 0. Then, we can plot it as follows, You should realize that there exists, FOR SURE, a point where both lines cross! This point is called the Steady State. And this implies that the growth rate of capital is 0!!! (Again, we will discuss the implications next week...) 4 Problem: Solow-Swan Model with a Cobb-Douglas function (similar to the one in the problem set, and similar to problems in past years exams) The mayor of Solowland has hired a sophomore Columbia Economics student in order to study how he can improve growth in his city. Solowland is the only city in the planet of Mars, so, they cannot trade anything with other cities or planets. The city produces wheat using a Cobb-Douglas production function, and the labor income share is half of the total income. The mayor is so nice, that he has decided he will not steal money from the citizens in order to create government spending; but, in exchange, all citizens know that they should keep 30% of their wheat every period, in order to produce more wheat in the next period. Due to the tough climatic conditions in Mars, capital depreciates at a rate of 9% every year. Additionaly, population in Solowland increases at a 1% rate every period. Our student takes as given that A=1. a) Which is the production function in Solowland? b) Which is the production function in per capita terms? 8 c) After a long unpaid research, the student estimates that Solowland has a capital per worker of 4 units. Do you think that the student will say that this measure will increase/decrease/not change in the next period? d) Which is the level of capital per capita in the steady-state? e) Which is the rate of growth of per capita income in the steady-state? Which is the level of per capita income in the steady state? Which is the rate of growth of the level of per capita income? 9...
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This note was uploaded on 11/18/2012 for the course ECON W3213 ECON W3213 taught by Professor Xaviersala-i-martin during the Spring '10 term at Columbia.

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