This part considers a modified version of the Solow growth model. Suppose the production function is given by F(K,bN) = Kª(bN)!-ª where b is the labour augmenting technology, which grows at a rate f, i.e., b+1 = (1+ f)bt. For simplicity, assume that the total factor productivity z = 1, and the population is constant, i.e., N, = N for all t. The rest of the model is the same as in the standard Solow model in the textbook. Especially, the aggregate capital stock evolves according to Kt41 = I + (1 – d)K1. And assume that the economy is still closed, and there is no government. For any aggregate variable X, let the lower case letter æ be the variable per effective unit of worker; that is r = . Show that the production technology specified above satisfies the assumption of con- stant returns to scale. ; List all the equilibrium conditions of this model. Using the equilibrium conditions you listed above, write down an expression that describes the evolution of the aggregate capital stock, Kį over time. Briefly explain why this expression is not particularly convenient for our analysis of the model?

This part considers a modified version of the Solow growth model. Suppose the production function is given by F(K,bN) = Kª(bN)!-ª where b is the labour augmenting technology, which grows at a rate f, i.e., b+1 = (1+ f)bt. For simplicity, assume that the total factor productivity z = 1, and the population is constant, i.e., N, = N for all t. The rest of the model is the same as in the standard Solow model in the textbook. Especially, the aggregate capital stock evolves according to Kt41 = I + (1 – d)K1. And assume that the economy is still closed, and there is no government. For any aggregate variable X, let the lower case letter æ be the variable per effective unit of worker; that is r = . Show that the production technology specified above satisfies the assumption of con- stant returns to scale. ; List all the equilibrium conditions of this model. Using the equilibrium conditions you listed above, write down an expression that describes the evolution of the aggregate capital stock, Kį over time. Briefly explain why this expression is not particularly convenient for our analysis of the model?

Exploring Economics

8th Edition

ISBN:9781544336329

Author:Robert L. Sexton

Publisher:Robert L. Sexton

Chapter20: Economic Growth In The Global Economy

Section: Chapter Questions

Problem 5P

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Hello, please help me to solve these questions.Consider the Solow growth model with technological progress at the rate g, population growth at the rate n, and capital depreciation rate at the rate δ. The savings rate is denoted by s and the production function is given by:Y = Kα (AL)1-α , 0 < α < 1.Y is aggregate output, K is aggregate capital, L is aggregate labour, A is technology and AL is effective labour. (a) Let ݇k = K/AL which denotes capital per unit of effective labour. Obtain the production function in terms of capital per unit of effective labour. Explain the properties this production function satisfies. (b) Derive the key equation that governs the evolution of capital per unit of effective labour in this Solow model. Provide the steady state value of capital and output in per unit of effective labour terms. What are the growth rates of capital per unit of labour and output per unit of labour in the steady state? (c) Analyse the effect of a decrease in the…
Suppose that we modify the Solow growth model by allowing long-run technological progress. That is, suppose that z = 1 for convenience and that there is labor-augmenting technological progress, with a production function Y =F(K,bN) where b denotes the number of units of "human capital" per worker, and bN is "efficiency units" of labor. Letting b' denote future human capital per worker, assume that b' = (1 + f ) b, where f is the growth rate in human capital. (c) In the real world, we usually consider education level as a proxy to human capital. To examine the theory, what suggestions can you make to growth economists? What are factors other than education can you think of that contribute to human capital?
Consider the following Solow growth model in which households save a constant fraction of their income. Let N be the population (also the labor force) in the current period. Assume that the population follows N′ = (1+n)N where N′ is the population in the future period, and n is the net population growth rate. Assume that the output is produced according to the production function Y = zF(K, N), where z is the total factor productivity, K is capital stock, and F(K, N) exhibits constant returns to scale. Capital depreciates at the rate d where 0 < d < 1. The capital stock changes over time according to K′ = (1 − d)K + I where I is the investment level. (a) Derive the equation that determines the future capital-per-worker in competitive equilibrium. Here is some additional information for parts (5b), (5c), and (5d). Suppose that the economy is initially in steady state, and experiences a natural disaster (e.g. the recent quake and tsunami in Japan in 2011) that destroys some of the…
Consider our graph of the basic Solow growth model. On the graph above: y represents real output (or income) per worker; y=F(k) is the production function; k is the capital stock per worker; s is the savings rate; δ is the rate of depreciation of capital; ‘i’ represents business investment (purchases of capital) per worker); ‘LF’ stands for Loanable Funds. (For purposed of intuition, think of capital as ‘machines.’) If we started out with a capital (per worker) stock lower than the steady-state stock ( , above), we would expect to see which of the following happen over time? Group of answer choices A) Positive growth rates while the capital stock increases. B) Negative growth rates while the capital stock increases. C) Negative growth rates while the capital stock decreases. D) Positive growth rates while the capital stock stays less than the steady-state level. E) Positive growth rates while the capital stock decreases.
Consider the basic Solow model with no population growth and no technological progress and a production function of the form F (K, H ), where H denotes the efficiency units of labor (human capital) given by where N is the set of all individuals in the population, and hi is the human capital of individual i. Assume that H is fixed. Suppose there are no human capital externalities and factor markets are competitive. (a) Calculate the steady-state equilibrium of this economy. (b) Prove that if 10% higher h at the individual level is associated with a% higher earnings, then a 10% increase in the country’s stock of human capital H will lead to a% increase in steadystate output. Compare this result to the immediate impact of an unanticipated 10% increase in H (i.e., consider the impact of a 10% increase in H with the stock of capital unchanged).
In the Solow growth model, suppose that the per-worker production function is given by y=zk2/3 . The saving rate is s, depreciation rate is d, and population growth rate is n. Calculate the per capita capital (k) and output per worker (y) in the steady state.
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The Solow model without exogenous productivity growth predicts that rich countries with more capital will grow faster than poor countries with less capital, assuming other economic conditions are equal. Is this statement true or false? Explain.
Consider the endogenous growth model AK, in which the production function is given by Y = AK. Suppose s denotes the saving rate; that δ represents the depreciation rate; and that the variable that represents the population and that grows at the rate n. Calculate the growth rate of capital per capita in the same way as for the Solow model and, from there, solve the differential equation to obtain the capital per capita (denoted by k) as a function of time.
Consider the Solow model extended with human capital. Suppose the production function is Cobb Douglas. Find the levels of physical and human capital in the steady state. Describe the joint dynamics of the two forms of capital if the economy is not at the steady state.
Explain how to derive the investment per efficiency unit of labour curve in the Solow growth model.
Derive the equilibrium law of motion for capital per worker in the Solow growth model (equation 7-19). State which equation you start with, and the operation you perform at each step. 2. Graph this equation in the space of capital tomorrow on the y-axis and capital today on the x-axis, and explain how you identify the steady-state level of capital per capita from the graph