Calculus problems discussion

Calculus homework problems
Calculus homework problems. Please help me to solve questions that are shown on the photos. I need correct answers only.

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Make-Up Problem Set November 11, 2020 Due Date: November 24 1. Leverage: For the following question, assume the that bank’s balance sheet has only 3 categories: assets, liabilities, and capital. Suppose a bank has $5,000,000 in capital, and that it’s leverage ratio is 20. (a) Draw the bank’s balance sheet. How much assets and liabilities does this bank have? (b) What is the bank’s capital ratio? (c) Suppose that after one week, the bank’s assets depreciate by 3.5%. How much capital does the bank have left, and is it solvent? If the bank is insolvent, by how much is it insolvent? (d) Suppose instead that after one week, the banks assets depreciate by 6%. How much capital does the bank have left, and is it solvent? If the bank is insolvent, by how much is it insolvent? (e) Describe the relationship between leverage, expected prots, and – nancial fragility. You may nd it helpful to answer this question by using examples of balance sheets with dierent leverage ratios, although this is not required for a complete answer. (1-2 sentences is ne). 1 2. Okun’s Law: Assume Okun’s law is given by: ut−ut−1 = −0.4(gy−0.03). Suppose you are the head of the Council of Economic Advisors, and are at a dinner with the President. The President asks you to roughly calculate the following on the back of your napkin: (a) The President remarks that due to unfavorable economic conditions, growth this year will only be 1.5%. i. By how much will the unemployment rate increase or decrease? ii. Suppose instead that the President said that growth this year would be 4%. By how much will the unemployment rate increase or decrease? (b) The President mentions that unemployment is too high, and needs to be decreased by roughly 0.8 percentage points. i. What level of growth would be needed to reduce the unemployment by this amount? ii. Suppose that the best estimates place growth this year at 2%, and the multiplier for scal policy at 1.5. How much scal stimulus (as a percentage of GDP) would be required to bring unemployment down to the desired level? (c) What are the two reasons listed in your textbook that growth needs to be equal to 3% to have no change in unemployment? 3. Deriving the Slope of the IS Curve with a Balanced Budget Requirement: 1) Y = C(Y − T) + + I(Y, i) +− + G, with T = T(Y ) + , 0 < TY < 1, and G exogenous. 2) Y = C(Y − T) + + I(Y, i) +− + G , with T = T(Y ) + , 0 < TY < 1, but now G = T (i.e. there is a balanced budget requirement). (a) For economy one, use the total derivative to derive an expression for di dY . (b) For economy two, use the total derivative to derive an expression for di dY . (c) Is the slope of the IS curve for economy one equal to, steeper than, or atter than the slope for economy two? How do you know? (d) What is the intuition for your result in part iii? (1-2 sentences is ne). 4. Suppose now that the production function is given by: Y = 3K1/4N3/4 (a) Show that this production function characterized by constant returns to scale. 2 (b) Show that there are decreasing returns to capital. (c) Show that there are decreasing returns to labor. (d) Transform the production function into the relation between output per worker and capital per worker. (e) For a given savings rate, s, and depreciation rate, δ, give an expression for capital per worker in the steady state. (f) Give an expression for output per worker in the steady-state (g) Solve for the steady state level of output per worker when s = .032 and δ = .08 (h) Suppose the depreciation rate remains constant at δ = .08 , while the saving rate is reduced by half, to s = 0.16. What is the new steady-state output per worker? 5. Derivation of the Phillips Curve One version of the Phillips Curve is written: π − π e = m + z − αu (a) Find the natural rate of unemployment, un (b) Use the expression for un you derived above to rewrite the Phillips Curve as: π − π e = −α(u − un) i.e., in terms of the deviation of unemployment from the natural rate of unemployment. (c) According to our production function, Y = L(1 − u), so Yn = L(1 − un). Use these denitions to write u − un in terms of Y − Yn and L (i.e. u − un on the left hand-side, and all other variables on the right). (d) Substitute the expression for u − un you found in part c into the expression for the Phillips Curve in part b, to rewrite the Phillips Curve in terms of the output gap, Y − Yn, as it is found in chapter 9: π − π e = α L (Y − Yn) 6. Begin with the graph of medium run equilibrium. (a) What four conditions characterize medium run equilibrium? Write these conditions in words and math. Describe the intuition for these conditions (b) Beginning in medium run equilibrium, use a graph to show the eects of the following on 1) the output gap, 2) the deviation of ination from expected: i. An increase in the mark-up. ii. An increase in z, the catch-all term for labor market conditions. 3 iii. An increase in the IS curve. A decrease in the IS curve. iv. An increase in the LM curve. A decrease in the LM curve. v. An increase in the natural rate of unemployment. A decrease in the natural rate of unemployment. vi. An increase in potential output. A decrease in potential output. 4

Problem Set 5
November 12, 2020
Due Date: Wed, Nov 18

1. Suppose that the production function is given by:
   Y = .5K1/2N1/2
   (a) Derive the steady state levels of output per worker and capital per
   worker in terms of the saving rate, s, and the depreciation rate δ.
   (b) Derive the equation for the steady state level of consumption per
   worker in terms of s and δ.
   (c) What level of saving maximizes consumption in the steady state?
2. Suppose now that the production function is given by:
   Y = KαN1−α
   and assume α = 1/3
   (a) Is this production function characterized by constant returns to scale?
   Explain
   (b) Are there decreasing returns to capital?
   (c) Are there decreasing returns to labor?
   (d) Transform the production function into the relation between output
   per worker and capital per worker.
   (e) For a given savings rate, s, and depreciation rate, δ, give an expression for capital per worker in the steady state.
   (f) Give an expression for output per worker in the steady-state
   (g) Solve for the steady state level of output per worker when s = .32
   and δ = .08
   (h) Suppose the depreciation rate remains constant at δ = .08 , while
   the saving rate is reduced by half, to s = 0.16. What is the new
   steady-state output per worker?
3. Suppose now that the economy is given by: Y = K1/2
   (AN)
   1/2
   , the savings
   rate is 16%, the depreciation rate is 10%, the work force grows at 2% per
   year, and the rate of technological progress is 4% per year.
   1
   (a) Find the steady state values of the variables listed in i-v
   i. capital stock per eective worker
   ii. output per eective worker
   iii. growth rate of output per eective worker
   iv. growth rate of output per worker
   v. the growth rate of output
   (b) Suppose the rate of technological progress doubles to 8% per year.
   What are the new values of the above variables?
   2

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