Following the report you have prepared for the Treasury, one of the comments you received is that you are not allowing technology to growth over time, so you have decided to prepare a new report allowing for that possibility. You can assume that the growth level of technology is g> 0, and continue making the same assumptions on population growth you made before. When these assumptions are made, the capital accumulation equation (in per effective labour terms) assumes the following format: Ak₂ = sy₁ −(n+g+ó)k, where s is the savings rate and 6 is the capital depreciation rate. ke = K₁/(A.Lt) and yt = Y₁/(A+Lt). In your report you need to answer the following questions: 1. Find the steady-state level of income per capita assuming the production function follows the standard Cobb- Douglas production function. Show your work, step-by-step.

1. Consider Avataria, which can be described by the Solow model with the population growth. Its production function Y(K,L) = 2K¹2 L¹/2 Its investment rate is 40%, its depreciation rate is 10%, and its population growth rate is 10%. Calculate the capital-labor ratio, output per worker, and consumption per worker in the steady state. Assume that now the population growth slows down to 4%. What happens with the output per worker and consumption per worker in the new steady state? Calculate the exact values. (a) (b)

9 Based on the production function Y = AKaG1-a (with G representing public expenditure, and the remaining variables and parameters representing usual values), show, using the necessary assumptions the value of the tax amount that maximizes the economic growth rate. Then, represent graphically the situation.

Suppose that the populist leader of our imaginary country increases the savings rate from 13% to 15%, i.e., (s = 0.15), what is the new steady state level of capital per worker? [Note: Assume that the other parameters n and d remain unchanged.]

A population grows according to an exponential growth model. The initial population is  P₀=15, and the growth rate is r=0.35. Then: P₁= P₂= Find an explicit formula for Pₙ. Your formula should involve n. Pₙ= Use your formula to find P₁₂ P₁₂= Give all answers accurate to at least one decimal place

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A CES production function with physical and human capital Consider the CES production function in terms of physical capital, K, and human capital, H: where 0 a. Set up the Hamiltonian and find the first-order conditions. b. What is the optimal relation between K and H? Substitute this relation into the given production function to get a relation between Y and K. What does this “reduced-form” production function look like? c. What is the steady-state value of the ratio of physical to human capital, (K/H)∗? d. Describe the behavior of the economy over time if the initial condition is such that K(0)/H(0)? e. Suppose that the inequality restrictions IK ≥ 0 and IH ≥ 0 apply. How do these constraints affect the dynamics if the economy begins with K(0)/H(0)∗?

Consider an economy with fixed land T that is constant over time. Production is given by: AKTBL1-a-ß Y = Population grows at a rate n and productivity A grows at a rate equal to g. That is, Ĺ = nL and À = gA. Capital accumulates as Ø = I – SK with I = yY. These are assumptions typical of the K Solow model.

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Consider an economy that exhibits both population growth (L grows at rate n) and technological progress (A grows at rate a) described by the production function, Y = F(K,AL) = Kª(AL)¹-α| Here K is capital and Y is output. (a) Show that this production function exhibits constant returns to scale. What is the per-effective-worker production function? (c) Find expressions for the steady-state capital-output ratio, capital stock per effective worker, and output per effective worker, as a function of the saving rate (s), the depreciation rate (8), the population growth rate (n), the rate of technological progress (a), and the coefficient a. (You may assume the condition that capital per effective worker evolves according to Ak = sf (k) − (a +n+8)k.) (d) Show that at the Golden Rule steady state the saving rate for this economy is equal to the parameter a.