ECN101: Intermediate Macroeconomic Theory TA Section

ECN101: Intermediate Macroeconomic Theory TA Section (jwjung@ucdavis.edu) Department of Economics, UC Davis October 27, 2014 Slides revised: October 27, 2014

Outline 1 Announcement 2 Review: Chapter 5 3 Homework Guide for HW #2 Problem 2 Problem 4 Problem 5 Problem 6 4 Review: Chapter 6 5 Homework Guide for HW #3 Problem 1 Problem 2 Problem 3 6 Take-Home Message

Review Sessions for Midterm Midterm (35%): Next Thursday, Nov. 6, in class Review sessions for midterm next week Mon.(Nov. 3), 6:10-8:00 p.m., Wellman 119: led by TA Seungduck Lee Tue.(Nov. 4), 6:10-8:00 p.m., Chemistry 179: led by TA Jae-Wook Jung FYI, my office hour: Tue. this week 3:10-5:10 p.m. at 139 SSH next week office hour: Mon. 3-5 p.m. for your HW #3 Other office hour: TA Seungduck Lee: Wed. 1-3 p.m. at 117 SSH Professor Geromichalos: Tue. 10:40-1:30 at 1102 SSH Piazza online Q&A forum is always available.

Solow Model The Steady State Solution of the Solow model The steady state means the time period which all the variables eventually approach constant levels of variables no matter where they begin. At the steady state, capital stock stays at constant, K t = K t+1 = K t+2 =. Solow model does not explain long-run economic growth. ( K t+1 = I t dk t = sy t dk t = 0) I = sy = dk

Solow Model The Steady State Solution of the Solow model Write K as a function of parameters/exogenous variables. From sy = dk, = K = ( ) ( Y s d = )Ā(K s d ) 1/3 (L ) 2/3 = (K ) 2/3 = = K = ( s Ā ) 3/2L d = ( s Ā ) (L d ) 2/3 ( s Ā ) 3/2 L d

Solow Model The Steady State Solution of the Solow model From the value of K, we can get Y = d s K = d ( s Ā ) 3/2 L = Ā s d 3/2( s d C = (1 s)y = (1 s)ā3/2( s d I = sy = ( sā)3/2 d 1/2 L L = L ) 1/2 L ) 1/2 L

Solow Model Transition Dynamics What happens if the parameters change? Compare new steady state K to original steady state K. ( ) s Ā d 3/2 L What if s goes down (saving policy), Y t, C t, I t, and K t? What if d goes down (smaller depreciation), What if Ā goes up (technological progress), What if L goes up (immigration), What if K0 goes down (destruction), Keep in mind the steady state solution of K = Solow model helps to explain some differences across countries.

Solow Model Transition Dynamics ( s )

Solow Model Transition Dynamics ( s )

Solow Model Transition Dynamics ( d )

Solow Model Transition Dynamics ( d )

Solow Model Transition Dynamics (Ā )

Solow Model Transition Dynamics (Ā )

Solow Model Transition Dynamics ( L )

Solow Model Transition Dynamics ( L )

Solow Model Transition Dynamics (Destruction of K )

Solow Model Transition Dynamics (Destruction of K )

Problem 2 HW #2 Problem 2 Consider the production model that we studied in class (Chapter 4) and assume that the production function is now given by Y = ĀK 3/4 L 1/4. Everything else in the model remains unchanged. a) Reproduce the analogue of Table 4.1 (5 equations and 5 unknowns) for this new economy. That is, state carefully what are your endogenous variables, parameters, and what equations you have in order to solve for equilibrium. b) Fully describe the solution of the model (in other words provide formulas that relate each endogenous variable of the model with the known parameters).

Problem 2 HW #2 Problem 2 c) Let Ā = 1, K = 100, L = 1000. What are the equilibrium values of the wage, the rental rate of capital, and total output? d) How does the equilibrium wage that you reported in (c) change if L = 1500 and everything else remains the same? Why? e) Again consider general values for Ā, K, L. What is the equilibrium value of output per person?

Problem 2 HW #2 Problem 2 We have an alternative production function Y = ĀK 3/4 L 1/4. The exponent of K means the contribution of capital on the production output, and this is the same as the portion of output distributed to capital or capital income share. From the profit maximization problem of a representative firm, max K,L Π = F (K, L) rk wl = ĀK 3/4 L 1/4 rk wl

Problem 2 HW #2 Problem 2 The necessary conditions for maximization problem, F.O.C. s: K : Ā 3 ( K ) 1/4 4 K 3/4 1 L 1/4 r = Ā3 3 Y r = 4 L 4 K r = 0 L : Ā 1 ( K ) 3/4 4 K 3/4 L 1/4 1 r = Ā1 1 Y w = 4 L 4 L w = 0 From these two conditions, we get the rule for hiring capital and the rule for hiring labor 3 Y 4 K = r, 1 Y 4 L = w

Problem 2 HW #2 Problem 2 From the market clearing condition which means (Demand) = (Supply) K = K, L = L From the production function, Y = ĀK 3/4 L 1/4 5 unknowns(endo. var.), Y, K, L, r, w and 5 equations. Parameters/exog. var.: Ā, K, L

Problem 2 HW #2 Problem 2 How to solve the model In the solution, or called equilibrium, these 5 equations should be satisfied by the solutions, the set of the optimal chosen prices, r, w and the optimal quantities Y, K, L The solution must be described with only parameters and exog. variables, here Ā, K, L

Problem 4 HW #2 Problem 4 Suppose a country enacts a tax policy that discourages investment. As a result, the value of the parameter s now goes to a smaller value s. a) Assuming that the economy starts at its initial steady state, use the Solow model to explain what happens to the economy (after the change of s) over time and in the long run. b) Draw a graph showing how output evolves over time (put Y t on the vertical axis and time on the horizontal axis). What happens to economic growth over time?

Problem 5 HW #2 Problem 5 Suppose the level of TFP in an economy rises permanently from Ā to Ā. a) Assume that the economy starts at the initial steady state. Use the Solow model to explain what happens to this economy (after the change of TFP) over time and in the long run. b) Draw a graph showing how output evolves over time, and explain what happens to per capita income.

Problem 5 HW #2 Problem 5 c) How is the response of the economy to the increase in TFP different from the economy s response to an increase in the investment rate (like the one of Problem 4 above)? (Hint: Think about what happens to consumption)

Problem 6 HW #2 Problem 6 In class we pointed out that, in the baseline Solow model, the variables K, Y remain constant in the long-run (that is, there is no long run growth since the economy settles at the steady-state). a) Verify that the variables y, C, c are also constant in the long-run. b) Now suppose that everything in the Solow model remains the same except one assumption: labor supply is not constant over time any more, but assume that it grows at a constant rate per period. Intuitively (no need to write down any equations), will the variables K, Y, y, C, c still reach a long run steady state? Can we define the steady state in aggregate variables? Yes, aggregate variables grow at the constant rate of g L since per capita variables stay constant and only labor force L t grows at the constant rate of g L. This supports a balanced growth path.

Romer Model Romer Model: Basic Difference from Solow Model Data: Most countries show continuous economic growth at positive growth rates. Solow model: exogenous growth model but no sustained long run growth Why no sustained growth in the long run? Due to diminishing MPK or constant returns to scale property of the Cobb-Douglass production function How can we solve this?

Romer Model Romer Model: Basic Difference from Solow Model The Romer model introduces nonrivalrous idea stock. Ideas (A) : infinitely accessible without competition = nonrivalrous Objects (K, L) : limited and finite = rivalrous Fixed research cost would never be recovered in P = MC economy. Problems with pure competition! = Introducing increasing returns to scale production function in ideas

Romer Model Romer Model: Basic Structure Unknowns/Endog. var s: Y t, A t, L yt, and L at Parameters/Exog. var s: z, L, l, and Ā0 Equations: Output prod. func. : Y t = A t L yt (1) Flow of knowledge : A t+1 = za t L at (2) Resource constraint: L yt + L at = L (3) Allocation of labor: L at = l L (4)

Romer Model Solving the Romer Model What are we solving for? To show sustained long run growth or positive growth rates in the long run First, note that (3) and (4) make L yt = (1 l) L. Then, combine with (1), Y t = A t (1 l) L Now, we can consider the growth rate of output according to the rule of growth rates. g(y t ) = g(a t ) + g(1 l) + g( L) = g(a t )

Romer Model Solving the Romer Model Recall g(x t ) X t+1 X t From (2) and (4), = X t+1 X t X t. g(a t ) = A t+1 A t = zl at = z l L > 0 Thus, g(y t ) = g(a t ) = z l L. i.e., The output and the idea stocks grow at the same positive constant rate of z l L. Balanced Growth (Path)! - All endogenous variables grow at a constant rate. If plugging A 0 into (2), we get series of A t s. Apply these to (1), then we also get series of Y t s.

Problem 1 HW #3 Problem 1 Suppose the economy is on a balanced growth path in the Romer model, and then, in the year 2030, research productivity rises permanently to z > z a) Solve for the new growth rate of knowledge and y t. b) Make a graph of y t over time (on a ratio scale). On a ratio scale, the slope of y t = its (constant) growth rate. c) Why might the parameter z increase in an economy? Build your own example. It could be anything to promote technology in the research sector.

Problem 2 HW #3 Problem 2 Consider Romer s growth model of Chapter 6 and let Ā0 = 100, l = 0.06, z = 1/3000, and L = 1000. a) What is the growth rate of output per person in this economy? b) What is the initial level of output per person? What is the output per person in 100 years? Use y t = Ā0(1 l)(1 + ḡ) t, equation (6.9) c) How do your answers in parts a, b change if the follwoing changes occur? A doubling of Ā0, a doubling of l, a doubling of the population L, and a doubling of z. (Note: Explain these changes one at a time). d) If you could advocate for one of the changes above, which one would you choose? Why?

Problem 3 HW #3 Problem 3 Consider the following simple variation of the Romer model. Y t = A 1/2 t L yt A t+1 A t = za t L at L yt + L at = L L at = l L In short, the only difference is the exponent of A t in the (final good) production function, so that now there are diminishing returns to ideas in that sector. a) Provide an economic interpretation for the first equation. Now idea exhibits diminishing returns. What does it mean? b) What is the growth rate of knowledge in this economy? c) What is the growth rate of output per person in this economy? First, get an equation similar to (6.9) Use useful tips for growth rates in Chapter 3.

Take-Home Message The rule of thumb for transition dynamics of the Solow model: Compare old and new steady state K. Which one is bigger than other? Change rapidly after shock, then slow down gradually. The Solow model can explain the difference of incomes across countries It cannot explain the long-run growth. The Romer model allows the growth of A All variables grow at the same rate, z l L in the long-run: Balanced Growth Path.