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Intermediate Macro Assignment 1: J une 24, 201 9. 1 Submission Deadline: Ju ne 28, 201 9 or before . (Total Points 36) Instructions: – Clearly label the diagrams you use to answer questions. – Clearly show the workings for each question. 1. Consider an economy described by the following equations: Y = C + I + G, Y = K1/3 L2/3 , = 1000 and = 1000 , G = 300 , T = 200 , C = 250 + 0 .5(Y – T) – 50 r, I = 300 – 50 r. a) With investment I on the x -axis and r on the y -axis, plot the investment function. (2 points) b) Solve for private savings and national savings as a function of r. Does national savings increase as r increases? Explain. (1 point ) c) Solve for consumption, private saving, investment and interest rate in the equilibrium. (3 points) d) Now suppose G increase to 350 while taxes remain unch anged. Solve for private saving, public saving and interest rate in the equilibri um. Is there evidence of crowding out ? Explain. (2 points) e) Now suppose G = 300 and taxes, T, increase to 300. Does equilibrium interest rate and investment increase or decrease? If more investment is good for long -run growth in the economy, should taxes be increased or decreased? Explain. (2 points) 2. A Non -economic Example of Steady State: There is a well -known relationship between the amount of calor ies a person consumes and burns, and how much the person weighs. Consider the following relationship: Suppose calories consumed de pends positively on the weight of a person (increases as weight increases) and is given by calories consumed Cc = 14+ 0.02 W, where W is the weight of the person in kilograms. Calories burned also rises with weight, that is Cb = 2 + 0 .18 W. The relation between change in weight, ∆W, Cc and Cb is given by: ∆W = Cc – Cb. a) On a graph where calories consumed and burned is on the y -axis and weight is on the x- axis, plot Cc and Cb. (3 points) b) Defi ne the steady state as ∆W = 0, that is weight does not change. Compute the steady state weight and indicate on the graph. (1 point ) Intermediate Macro Assignment 1: J une 24, 201 9. 2 c) Suppose a person weighs less than the steady state weight. Will this person be gaining or losing weight? Explain in words using the graph. (1 point ) 3. Solow Model with no growth in population or technology: For country A the aggregate produc tion function is given by Y = KαL1-α, where 0 < α < 1 is known as the `capital intensity’. The fraction of output saved and invested is s = 0 .25 and the depreciation rate is δ= 0.01. a) Argue that the aggregate production function displays constant returns to scale. (2 points) b) Suppose α = 1 /3. In per worker terms, derive the capital accumulation function. Compute the steady state levels of capital per worker and output per worker. (3 points) c) Suppose α = 1 /3 and for year 1 capital per worker is g iven by k = 8. In a table as illustrated below show capital per worker ( k), output per worker ( y) and change in capital stock per worker ( ∆k) over time. Continue this table up to 10 years. (6 points) Year k y ∆k 1 8 2 0.42 2 8.42 3 i. Calculate the growth rate of y between year 1 and year 2. ii. Calculate the growth rate of y between year 9 and year 10. iii. Compare the growth rates in parts i and ii. What can we conclude about rate of growth of y as the economy approaches its steady state? Expl ain the reason for this change in growth rate. d) The marginal product of capital for the aggregate production function is given by MPK = αkα-1, where k is capital per worker and α = 1 /3 . (5 points) i. Does the MPK for the aggregate production function decrease as capit al per worker increases? Explain your answer. ii. Calculate the golden rule savings rate and golden rule stock of capital per worker. iii. Suppose the government aims to maximize consumption in the steady state. Given that s = 0 .25, should the government promote sa vings in the economy? Explain. e) Country B has the same aggregate production function as country A. It also has the same rate of depreciation for capital ( δ= 0.01 ). However, output per worker in country B is one -third the output per worker in country A. That is yA/yB = 3. Suppose both economies are at their steady state level of output per worker. (5 points) i. Calculate the savings rate for country B. ii. For country B calculate the steady state capital stock per worker, consumption per worker and investment per worker. iii. List two reasons for why savings rates could diff er between countries. Intermediate Macro Assignment 1: J une 24, 201 9. 3 4. BONUS QUESTION Solow Model with growth in population and technology: An economy is at its long run steady state equilibri um. An earthquake destroys one -fifth of the economy’s capital stock and reduces its labour force by 10%. However, the level of technology, E, population growth rate and the growth rate of technology remain the same as before. (Hint: Use diagrams to answer the following question .) a) Describe what happens in the short run to stock of ca pital per e ffective worker ( ̃) and output per e ffective worker ( ̃). Using a diagram with time on the x -axis illustrate the transition of ̃ from the initial steady state to the new steady state. (5 points ) b) Due to the earthquake, does the steady state capital stock per e ffective worker increase or decrease? Explain. (2 points )