Problem Set #1: Exogenous Growth Models

Size: px
Start display at page:

Download "Problem Set #1: Exogenous Growth Models"

Transcription

1 University of Warwick EC9A2 Advanced Macroeconomic Analysis Problem Set #1: Exogenous Growth Models Jorge F. Chavez December 3, 2012 Question 1 Production is given by: Y t F (K t, L t ) = AK α t L 1 α t where L t+1 = (1 + n)l t and α (0, 1). a. Show that F exhibits a constant return to scale technology Solution. This implies showing that F : R 2 R is homogeneous of degree 1. That s it, we need to show that for λ > 0, F (λk t, λl t ) = λf (K t, L t ). Then: F (λk t, λl t ) = A(λK t ) α (λl t ) 1 α = λak α t L 1 α t = λf (K t, L t ) b. Express output as a function of the capital labor ratio k t = K t /L t. Solution. Use the fact that F is homogeneous of degree 1: ( ) α F (K t, L t ) = AKt α L 1 α Kt t = L t A = L t Akt α ; L t c. Find the wage rate per worker and the rental rate per capital. Solution. Profit maximization (either by a social planner or by the representative firm in the economy) yields: w t = F (K t, L t ) L t R t = F (K t, L t ) K t = (1 α) AK α t L α t = (1 α) Ak α t = αakt α 1 L 1 α t = αakt α 1 Note that this assumes that the good acts here as a numeraire (its price is set to 1). 1

2 d. Find the dynamical system (describing the evolution of k t over time) under the assumption that the saving rate is s (0, 1) and the depreciation rate is δ (0, 1]. Solution. We need to characterize the path of {k t } t=0 given s (0, 1) and δ (0, 1]. That is, we need to find the non-linear difference equation k t+1 = ϕ(k t ). Start from the law of motion of the aggregate stock of capital: K t+1 = sy t + (1 δ)k t Put everything in per-worker terms by dividing both sides by L t+1 : k t+1 = K t+1 L t+1 = s Y t L t + (1 δ) K t L t L t+1 L t L t L t+1 = sakα t + (1 δ) k t 1 + n ϕ (k t ) (1) where I am using y t = f (k t ) = Ak α t. Sometimes it useful to re-express this condition in terms of k t+1. To do so subtract k t from both sides of (1) to get: k t+1 = sakα t (n + δ) k t 1 + n e. What is the growth rate of k t, γ kt (k t+1 k t ) /k t? Solution. γ kt = k t+1 k t k t = sakα 1 t (δ + n) 1 + n f. Find the steady state level of the stock of capital per worker k, income per worker y and consumption per capita c. Solution. In steady-state kt+1 ss = kt ss = k, which implies that γ k = 0. Then from the expression for γ kt : k = ( ) 1/(1 α) sa δ + n Income per worker is ( ) α y = f (k) = Ak α = A 1 1 α s 1 α δ + n Finally, to get consumption per worker in steady state note that in general y t = c t + i t, i t = sy t and c t = (1 s)y t. Then, in steady state: ( ) α c = (1 s) y = (1 s) A 1 1 α s 1 α δ + n Jorge F. Chávez 2

3 g. What is the Golden Rule value of k? (k in steady state s.th. the consumption in steady state is maximized?) Solution. The Golden Rule value of k t is the stock of capital per worker that maximizes consumption in steady-state. Recall that in general: c t = (1 s)y t = f(k t ) sf(k t ) Note that there is no way to maximize consumption in all states as consumption is a function of f(k t ) which is unbounded (we can only get the corner solution k t = 0 if we look for a stationary point). However, the steady-state condition is: sf (k) }{{} Investment per capita = (n + δ) k }{{} Part of k that is lost because of pop. growth & depreciation (2) which is true for any steady-state (any setting in which k ss t+1 = k ss t = k). Then because c t = f(k t ) sf(k t ), in steady state: c = f(k) (n + δ)k which now will accept an interior solution. The FONC for the maximization problem is: c k = f (k) (n + δ) = 0 f ( k GR) = n + δ For y = Ak α we can get a closed form solution for k GR : f ( k GR) = αa ( k GR) α 1 = n + δ k GR = ( ) 1/(1 α) αa n + δ (3) The concept of the Golden Rule for the case of the Solow model is illustrated in figure 1. There you can see three alternative saving rates, and you can visualize the Golden Rule condition f ( k GR) = n + δ: the slope of f( ) must be equal to the slope of the (n + δ)k line. h. What saving rate is needed to yield the Golden Rule? Solution. Comparing equation (3) with the expression for capital in steady state it is straightforward to see that the saving rate that allows the economy to reach k GR is s GR = α. Alternatively recall that f (k GR ) = n + δ and k t f (k t )/f(k t ) = α. Then: k GR f ( k GR) f (k GR ) = kgr (n + δ) f (k GR ) = α But then the Golden Rule level of capital also satisfies the more general condition for all steady-states. Combining that condition (2) with the above one we can see that s GR = α. i. Find the elasticity of y with respect to s (in steady-state). Can observed differences in saving rates explain the observed differences in income per-capita across the world? Jorge F. Chávez 3

4 Figure 1: The Golden Rule f(k ss ) c 1 (n+δ)k ss c 2 c GR s 2 f(k ss ) s 1 f(k ss ) s GR f(k GR ) k 2 k GR k 1 k ss Solution. Recall that: ( ) α/(1 α) sa y = Ak α = A n + δ Taking logs 1 : ln y = α 1 α ln A + α ln s α }{{} ε ys Suppose that α = 1/3 as it is usually found in empirical studies and there is a difference in saving rates of 3 times across countries (300%). Then the elasticity will be α/(1 α) = 0.5 which implies that the difference in y according to this model (with Cobb-Douglas technology) should be % = 150%. However the observed difference is nearly 20 times (this comes from the lecture notes). j. Find the dynamical system describing the evolution of y t under the assumption of full depreciation δ = 1. Solution. With δ = 1 the law of motion of the stock of capital per worker is just: 1 To see this: k t+1 = sakα t 1 + n log y log y log x = y y x log x x y y x x = % y % x Jorge F. Chávez 4

5 Note that the numerator is investment in per capita terms. Then, this expression says that the new capital stock (which is completely renewed each period) is lower than investment per worker, due to population growth. 2 We want to characterize {y t } t=0 by analyzing a non-linear difference equation y t+1 = φ(y t ). Recall that y t = Akt α. Then: ( ) sak y t+1 = Akt+1 α α α ( ) α = A t syt = A φ (y t ) 1 + n 1 + n k. Suppose you estimate a regression in which ln ( yt+1 i t) /yi is on the left hand side (i.e. you estimate the log of γy i t + 1 across countries, where i indexes countries) and ln s i, ln ( 1 + n i) and ln yt i are on the right hand side. According to the dynamical system you defined in question j, what would be the coefficients on your explanatory variables? How would you interpret these coefficients? Is there β convergence? How would you interpret the constant term? Solution. Consider the regression: ( y i ) ln t+1 = ln A + γ 1 ln s i + γ 2 ln ( 1 + n i) + γ 3 ln yt i + ε it y i t From the model we can get an estimable form by taking logs to the law of motion of y t+1 when δ = 0: ln y i t+1 = ln A + α ln s α ln (1 + n) + α ln y i t Then by subtracting lny i t from both sides of the equation above we can see that γ 0 = ln A, γ 1 = α, γ 2 = α and γ 3 = (1 α). 2 Investment occurs in t while the new capital stock will be readily available in t + 1. Jorge F. Chávez 5

6 Question 2 Output per worker is an increasing and concave function of capital per worker, given by: y t = f(k t ). Output is divided between labor income and capital income according to their marginal productivity. Namely, f (k t ) [ f (k t ) f (k t ) k t ] + f (k t ) k t = w t + r t k t Suppose that the rate of saving from wage income is s w [0, 1] and the rate of saving from capital income is s r [0, 1]. Therefore, total saving are given by s t = s w [ f (k t ) f (k t ) k t ] + sr f (k t ) k t Population and technology are constant and the rate of capital depreciation δ (0, 1) a. Derive the dynamical system governing the evolution of capital per capita: k t+1 = ϕ(k t ). Solution. Recall that the law of motion of the stock of capital per capita comes from the law of motion of the aggregate stock per capita: K t+1 = I t + (1 δ)k t where I t denotes aggregate investment. Because there are two distinct saving rates, we can think of aggregate savings to be determined by an average saving rate ( s) which is a weighted sum of s w and s r : K t+1 = sy t + (1 δ)k t Now, because there is no population growth, if we divide everything by L we get: k t+1 = s t + (1 δ)k t where s t = sf(k t ) = s w [f(k t ) f (k t )k t ] + s r f (k t )k t. Then: k t+1 = s w [f(k t ) f (k t )k t ] + s r f (k t )k t + (1 δ)k t = s w f(k t ) + (s r s w ) f (k t )k t ϕ(k t ) Note that if s w = s r, then the expression for φ(k t ) is the same we had before with a single saving rate. b. Suppose there exist a range of k t where f (k) = 0 (the third derivative is zero). Find a condition on the saving rates s w and s r such that the dynamical system k t+1 = ϕ(k t ) is convex (ϕ (k t ) > 0 in the range of f (k t ) = 0. Solution. First let s get the derivatives: ϕ (k t ) = s w f (k t ) + (s r s w ) [f (k t ) k t + f (k t )] + (1 δ) ϕ (k t ) = s w f (k t ) + (s r s w ) [f (k t ) k t + f (k t )] Let k be the maximum stock of capital per worker that can be attained by economy and assume that Jorge F. Chávez 6

7 k 1 k 2 [0, k] such that k t [k 1, k 2 ], f (k t ) = 0. Then, for those values of k t, using (4): ϕ (k t ) kt [k 1,k 2] = s w f (k t ) + (s r s w ) f (k t ) = (2s r s w ) f (k t ) Therefore, for ϕ(k t ) > 0 we need 2s r < s w and strict concavity in the range [k 1, k 2 ] (concavity is not enough) of f. 3 Suppose now that s r = 0 and f(k t ) = ln (1 + k t ) c. Derive the dynamical system k t+1 = ϕ(k t ). Solution. Now savings come only from wage income: k t+1 = s w f (k t ) s w f (k t ) k t + (1 δ) k t ϕ(k t ) Replacing the functional form for f( ) we get that f (k t ) = 1/(1 + k t ). Then: k t+1 = s w log (1 + k t ) s w + (1 δ) k t ϕ (k t ) 1 + k t k t d. Find ϕ (k t ), kt 0 ϕ (k t ), kt + ϕ (k t ), and ϕ (k t ). Solution. (i) First derivative ϕ (k t ) = = ( ) s w s w (1 + k t ) k t 1 + k t (1 + k t ) 2 + (1 δ) s w k t 2 + (1 δ) (1 + k t ) (ii) Limit when k t 0 k t 0 ϕ (k t ) = 1 δ (iii) Limit when k t + s w k t k t + ϕ (k t ) = k t (1 δ) (1 + k t ) = 1 δ Note that the it of the first term of the RHS is not undetermined (no need to apply L Hospital rule) 3 Strict concavity implies that f (k t) < 0 Jorge F. Chávez 7

8 (iv) Second derivative: ϕ (k t ) = s w [ (1 + k t ) 2 2 (1 + k t ) k t (1 + k t ) 4 [ ] = s w 1 + 2k t + kt 2 2k t 2kt 2 (1 + k t ) 4 ( ) = s w 1 kt 2 (1 + k t ) 4 e. Is the trivial steady state, k = 0, locally stable? Explain. Solution. To analyze stability of the trivial steady-state (k = 0), we need to evaluate the it of the first derivative of ϕ (k t ): k ϕ (k t ) = (1 δ) (0, 1) t 0 Therefore k = 0 is locally stable. Remark 1. Recall that for a non-linear difference equation like x t+1 = g(x t ) with fixed-point or steady state x ss4. Then: If f (x ss ) < 1 then x ss is locally asymptotically stable. If f (x ss ) > 1 then x ss is locally asymptotically instable. Remark 2. Recall that in the standard case: k t+1 = sf(k t ) + (1 δ)k t Then, to analyze the stability of k = 0 we need: k = 0 ϕ (0) = s f (0) + (1 δ) > 1 }{{} + which implies that in that case k = 0 is unstable. ] f. Find the range of k t in which the dynamical system is strictly convex. Solution. We want to find k 1 and k 2 such that k t [k 1, k 2 ], ϕ (k t ) > 0. Recall that: ϕ (k t ) = s w 1 k2 t (1 + k t ) 4 Now, setting ϕ (k t ) > 0 1 k 2 t > 0 k 2 t < 1 k t < 1 or k t > 1. But k t is the stock of physical capital per worker, so in principle k t 0. Thus, because ϕ (0) > 0, the range we were looking for is [0, 1]. g. Show that for δ = 0 a non-trivial steady state level of k does not exist (that is, explain why there exists no k > 0 such that k = ϕ( k)). Find the growth rate of k t (i.e., k t+1 /k t 1) as k t for δ = 0. 4 That is x ss = g(x ss ) Jorge F. Chávez 8

9 Solution. WTS if δ = 0 then a non-trivial steady state k > 0 for {k t }. Note that with δ = 0: k t 0 ϕ (k t ) = k t + ϕ (k t ) = 1 which means that this system does not have an interior steady-state. More formally: Claim 1. There is no non-zero fixed point for ϕ(k t ) (that is k > 0 s.th. k = ϕ(k)). Proof. Suppose not: k > 0 s.th k = ϕ(k). Then: ϕ (k t ) δ=0 = s w log (1 + k t ) s w + k t 1 + k t k t Then k > 0 must satisfy: k = ϕ (k) δ=0 = s w log (1 + k) sw k + k + k k Manipulating this expression a little bit we get to an expression that k > 0 must satisfy: log (1 + k) (1 + k) = k 1 + k = exp ( k ) 1 + k (4) Think about slopes. The LHS is a linear function with a constant slope equal( to 1. ) The RHS is an exponential function with slope < 1 for all values of k 0. 5 For 1 + k = exp to be true, the two functions must intersect at some point k > 0. They only intersect (in fact they are only tangent) at k = 0. This is a contradiction, which means that our starting premise was wrong. Q.E.D Finally, the growth rate of k t is: k 1+k γ kt = k t+1 k t log k t+1 log k t (5) k [ t log (1 + = s w kt ) 1 ] (6) k t 1 + k t 5 To see this note that the slope is ( ) k exp 1+k 1 = k (1 + k) 2 } {{ } <1 ( k exp ) < k } {{ } <1 Jorge F. Chávez 9

10 where the second equality comes from replacing k t+1 with the expression for ϕ(k t ). Taking its: [ log (1 + γ kt ) k t = k t k t sw 1 ] k t 1 + k t [ ] = s w log (1 + k t ) 1 k t k t k t 1 + k t [ ] 1 = s w 1+k t 1 k t 1 k t 1 + k t = 0 where again, the second-to-last equality applies L Hospital rule for undetermined its. Jorge F. Chávez 10

11 Question 3 Production if given by: Y t F (K t, L t ) = Kt α (B t L t ) 1 α where L t+1 = (1 + n)l t, B t+1 = (1 + g)b t and α (0, 1). a. Find the dynamical system describing the evolution of k t = K t / (B t L t ) (that is the stock of capital per-effective worker). Find the steady state level of k t. What is the growth rate of output per worker y t = Y t /L t in the steady state? What is the growth rate of capital per worker K t /L t = B t k t in the steady state? Solution. i. We want to find the non-linear difference equation k t+1 = φ(k t ) for k t K t /(B t L t ) Start from the law of motion of the aggregate stock of capital K t+1 = I t + (1 δ) K t. Recall that this is a closed economy and that there is a constant (and exogenous) saving rate s: S t = I t = sy t Putting everything in per-effective worker terms (by dividing by B t L t ): K t+1 = s F (K t, L t ) B t+1 L t+1 B t L t We get the non-linear difference equation: k t+1 = skα t + (1 δ) k t (1 + g) (1 + n) φ (k t) B t L t + (1 δ) K t B t L t B t+1 L t+1 B t L B t+1 L t+1 As before it is useful to also express the law of motion of k t in terms of k t+1 k t+1 k t. To do this subtract k t from both sides of φ(k t ): k t+1 = skα t + (1 δ) k t (1 + n + g + gn 1 + δ) k t (1 + g) (1 + n) = skα t (n + g + gn + δ) k t (1 + g) (1 + n) ii. Steady state for k t In steady-state kt+1 ss = kt ss = k. Then: k ss t = 0 sk α (n + g + gn + δ) k = 0 Solving for k: [ s k = (n + g + gn + δ) ] 1/(1 α) To check stability of the steady-state we need to check whether k t φ (k t ) = 1 δ (1 + g) (1 + n) < 1 k t φ (k t ) is less than unity: Jorge F. Chávez 11

12 Therefore, the steady-state k is stable. iii. Growth rate of output per worker y t = Y t /L t in steady state Note that: Y t = Kα t (B t L t ) 1 α ( ) α ( ) α Kt = B 1 α Kt t = B L t L t L t B t L t = kt α B t t Now, once the economy reaches an steady-state y ss t = k α B t Then, we can see: yt+1 ss yt ss 1 = B t+1 1 = g B t Alternatively we can take logs to y ss t = k α B t : log yt+1 ss log yt ss = log B t+1 log B t g iv. Growth rate of capital per worker K t /L t = B t k t, where k t = K t /(B t L t ). Same as above. First define the per-capita stock of capital k t = K t /L t : K t L t k t = k t B t Then get the growth rates directly or by the logarithm approximation: log k t+1 log k t = log B t+1 log B t g b. According to Solow s growth accounting, the growth rate in Total Factor Productivity (TFP) is calculated by: A t A t = y t y t α k t k t where y t Y t /L t, k t K t /L t, α is the elasticity of y t with respect to k t (which is equal to the share of capital in a competitive economy). Find TFP growth rate A t /A t in the steady state according to the model in part a. Solution. According to part (a) both y t = Y t /L t and k t = K t /L t will grow at a rate g in steady-state. Therefore: A t A t = y t y t = g αg α k t k t Jorge F. Chávez 12

Economic Growth: Lecture 2: The Solow Growth Model

Economic Growth: Lecture 2: The Solow Growth Model 14.452 Economic Growth: Lecture 2: The Solow Growth Model Daron Acemoglu MIT October 29, 2009. Daron Acemoglu (MIT) Economic Growth Lecture 2 October 29, 2009. 1 / 68 Transitional Dynamics in the Discrete

More information

Economic Growth. Spring 2013

Economic Growth. Spring 2013 Economic Growth Spring 2013 1 The Solow growth model Basic building blocks of the model A production function Y t = F (K t, L t, A t ) This is a hugely important concept Once we assume this, then we are

More information

14.452 Economic Growth: Lectures 2 and 3: The Solow Growth Model

14.452 Economic Growth: Lectures 2 and 3: The Solow Growth Model 14.452 Economic Growth: Lectures 2 and 3: The Solow Growth Model Daron Acemoglu MIT November 1 and 3, 2011. Daron Acemoglu (MIT) Economic Growth Lectures 2 and 3 November 1 and 3, 2011. 1 / 96 Solow Growth

More information

Macroeconomics Lecture 1: The Solow Growth Model

Macroeconomics Lecture 1: The Solow Growth Model Macroeconomics Lecture 1: The Solow Growth Model Richard G. Pierse 1 Introduction One of the most important long-run issues in macroeconomics is understanding growth. Why do economies grow and what determines

More information

Neoclassical growth theory

Neoclassical growth theory Chapter 1 Neoclassical growth theory 1.1 The Solow growth model The general questions of growth: What are the determinants of long-run economic growth? How can we explain the vast differences in both output

More information

Economic Growth I: Capital Accumulation and Population Growth

Economic Growth I: Capital Accumulation and Population Growth CHAPTER 8 : Capital Accumulation and Population Growth Modified for ECON 2204 by Bob Murphy 2016 Worth Publishers, all rights reserved IN THIS CHAPTER, YOU WILL LEARN: the closed economy Solow model how

More information

Universidad de Montevideo Macroeconomia II. The Ramsey-Cass-Koopmans Model

Universidad de Montevideo Macroeconomia II. The Ramsey-Cass-Koopmans Model Universidad de Montevideo Macroeconomia II Danilo R. Trupkin Class Notes (very preliminar) The Ramsey-Cass-Koopmans Model 1 Introduction One shortcoming of the Solow model is that the saving rate is exogenous

More information

Economics 304 Fall 2014

Economics 304 Fall 2014 Economics 304 Fall 014 Country-Analysis Project Part 4: Economic Growth Analysis Introduction In this part of the project, you will analyze the economic growth performance of your country over the available

More information

The Solow Model. Savings and Leakages from Per Capita Capital. (n+d)k. sk^alpha. k*: steady state 0 1 2.22 3 4. Per Capita Capital, k

The Solow Model. Savings and Leakages from Per Capita Capital. (n+d)k. sk^alpha. k*: steady state 0 1 2.22 3 4. Per Capita Capital, k Savings and Leakages from Per Capita Capital 0.1.2.3.4.5 The Solow Model (n+d)k sk^alpha k*: steady state 0 1 2.22 3 4 Per Capita Capital, k Pop. growth and depreciation Savings In the diagram... sy =

More information

Chapters 7 and 8 Solow Growth Model Basics

Chapters 7 and 8 Solow Growth Model Basics Chapters 7 and 8 Solow Growth Model Basics The Solow growth model breaks the growth of economies down into basics. It starts with our production function Y = F (K, L) and puts in per-worker terms. Y L

More information

CHAPTER 7 Economic Growth I

CHAPTER 7 Economic Growth I CHAPTER 7 Economic Growth I Questions for Review 1. In the Solow growth model, a high saving rate leads to a large steady-state capital stock and a high level of steady-state output. A low saving rate

More information

Review Questions - CHAPTER 8

Review Questions - CHAPTER 8 Review Questions - CHAPTER 8 1. The formula for steady-state consumption per worker (c*) as a function of output per worker and investment per worker is: A) c* = f(k*) δk*. B) c* = f(k*) + δk*. C) c* =

More information

Economic Growth: Theory and Empirics (2012) Problem set I

Economic Growth: Theory and Empirics (2012) Problem set I Economic Growth: Theory and Empirics (2012) Problem set I Due date: April 27, 2012 Problem 1 Consider a Solow model with given saving/investment rate s. Assume: Y t = K α t (A tl t ) 1 α 2) a constant

More information

VI. Real Business Cycles Models

VI. Real Business Cycles Models VI. Real Business Cycles Models Introduction Business cycle research studies the causes and consequences of the recurrent expansions and contractions in aggregate economic activity that occur in most industrialized

More information

Chapter 7: Economic Growth part 1

Chapter 7: Economic Growth part 1 Chapter 7: Economic Growth part 1 Learn the closed economy Solow model See how a country s standard of living depends on its saving and population growth rates Learn how to use the Golden Rule to find

More information

Economic Growth. (c) Copyright 1999 by Douglas H. Joines 1

Economic Growth. (c) Copyright 1999 by Douglas H. Joines 1 Economic Growth (c) Copyright 1999 by Douglas H. Joines 1 Module Objectives Know what determines the growth rates of aggregate and per capita GDP Distinguish factors that affect the economy s growth rate

More information

University of Saskatchewan Department of Economics Economics 414.3 Homework #1

University of Saskatchewan Department of Economics Economics 414.3 Homework #1 Homework #1 1. In 1900 GDP per capita in Japan (measured in 2000 dollars) was $1,433. In 2000 it was $26,375. (a) Calculate the growth rate of income per capita in Japan over this century. (b) Now suppose

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON4310 Intertemporal macroeconomics Date of exam: Thursday, November 27, 2008 Grades are given: December 19, 2008 Time for exam: 09:00 a.m. 12:00 noon

More information

The previous chapter introduced a number of basic facts and posed the main questions

The previous chapter introduced a number of basic facts and posed the main questions 2 The Solow Growth Model The previous chapter introduced a number of basic facts and posed the main questions concerning the sources of economic growth over time and the causes of differences in economic

More information

Handout on Growth Rates

Handout on Growth Rates Economics 504 Chris Georges Handout on Growth Rates Discrete Time Analysis: All macroeconomic data are recorded for discrete periods of time (e.g., quarters, years). Consequently, it is often useful to

More information

The Golden Rule. Where investment I is equal to the savings rate s times total production Y: So consumption per worker C/L is equal to:

The Golden Rule. Where investment I is equal to the savings rate s times total production Y: So consumption per worker C/L is equal to: The Golden Rule Choosing a National Savings Rate What can we say about economic policy and long-run growth? To keep matters simple, let us assume that the government can by proper fiscal and monetary policies

More information

Problem 1. Steady state values for two countries with different savings rates and population growth rates.

Problem 1. Steady state values for two countries with different savings rates and population growth rates. Mankiw, Chapter 8. Economic Growth II: Technology, Empirics and Policy Problem 1. Steady state values for two countries with different savings rates and population growth rates. To make the problem more

More information

MASTER IN ENGINEERING AND TECHNOLOGY MANAGEMENT

MASTER IN ENGINEERING AND TECHNOLOGY MANAGEMENT MASTER IN ENGINEERING AND TECHNOLOGY MANAGEMENT ECONOMICS OF GROWTH AND INNOVATION Lecture 1, January 23, 2004 Theories of Economic Growth 1. Introduction 2. Exogenous Growth The Solow Model Mandatory

More information

MACROECONOMICS SECTION

MACROECONOMICS SECTION MACROECONOMICS SECTION GENERAL TIPS Be sure every graph is carefully labeled and explained. Every answer must include a section that contains a response to WHY the result holds. Good resources include

More information

Lecture 14 More on Real Business Cycles. Noah Williams

Lecture 14 More on Real Business Cycles. Noah Williams Lecture 14 More on Real Business Cycles Noah Williams University of Wisconsin - Madison Economics 312 Optimality Conditions Euler equation under uncertainty: u C (C t, 1 N t) = βe t [u C (C t+1, 1 N t+1)

More information

ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE

ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.

More information

Preparation course MSc Business&Econonomics: Economic Growth

Preparation course MSc Business&Econonomics: Economic Growth Preparation course MSc Business&Econonomics: Economic Growth Tom-Reiel Heggedal Economics Department 2014 TRH (Institute) Solow model 2014 1 / 27 Theory and models Objective of this lecture: learn Solow

More information

Long Run Growth Solow s Neoclassical Growth Model

Long Run Growth Solow s Neoclassical Growth Model Long Run Growth Solow s Neoclassical Growth Model 1 Simple Growth Facts Growth in real GDP per capita is non trivial, but only really since Industrial Revolution Dispersion in real GDP per capita across

More information

Economic Growth. Chapter 11

Economic Growth. Chapter 11 Chapter 11 Economic Growth This chapter examines the determinants of economic growth. A startling fact about economic growth is the large variation in the growth experience of different countries in recent

More information

Finance 30220 Solutions to Problem Set #3. Year Real GDP Real Capital Employment

Finance 30220 Solutions to Problem Set #3. Year Real GDP Real Capital Employment Finance 00 Solutions to Problem Set # ) Consider the following data from the US economy. Year Real GDP Real Capital Employment Stock 980 5,80 7,446 90,800 990 7,646 8,564 09,5 Assume that production can

More information

The Phase Plane. Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations

The Phase Plane. Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations The Phase Plane Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations Phase Portraits of Linear Systems Consider a systems of linear differential

More information

Lecture 3: Growth with Overlapping Generations (Acemoglu 2009, Chapter 9, adapted from Zilibotti)

Lecture 3: Growth with Overlapping Generations (Acemoglu 2009, Chapter 9, adapted from Zilibotti) Lecture 3: Growth with Overlapping Generations (Acemoglu 2009, Chapter 9, adapted from Zilibotti) Kjetil Storesletten September 10, 2013 Kjetil Storesletten () Lecture 3 September 10, 2013 1 / 44 Growth

More information

Name: Final Exam Econ 219 Spring You can skip one multiple choice question. Indicate clearly which one

Name: Final Exam Econ 219 Spring You can skip one multiple choice question. Indicate clearly which one Name: Final Exam Econ 219 Spring 2005 This is a closed book exam. You are required to abide all the rules of the Student Conduct Code of the University of Connecticut. You can skip one multiple choice

More information

Note on growth and growth accounting

Note on growth and growth accounting CHAPTER 0 Note on growth and growth accounting 1. Growth and the growth rate In this section aspects of the mathematical concept of the rate of growth used in growth models and in the empirical analysis

More information

Hello, my name is Olga Michasova and I present the work The generalized model of economic growth with human capital accumulation.

Hello, my name is Olga Michasova and I present the work The generalized model of economic growth with human capital accumulation. Hello, my name is Olga Michasova and I present the work The generalized model of economic growth with human capital accumulation. 1 Without any doubts human capital is a key factor of economic growth because

More information

( ) = ( ) = ( + ) which means that both capital and output grow permanently at a constant rate

( ) = ( ) = ( + ) which means that both capital and output grow permanently at a constant rate 1 Endogenous Growth We present two models that are very popular in the, so-called, new growth theory literature. They represent economies where, notwithstanding the absence of exogenous technical progress,

More information

Lecture 2 Dynamic Equilibrium Models : Finite Periods

Lecture 2 Dynamic Equilibrium Models : Finite Periods Lecture 2 Dynamic Equilibrium Models : Finite Periods 1. Introduction In macroeconomics, we study the behavior of economy-wide aggregates e.g. GDP, savings, investment, employment and so on - and their

More information

Notes on the Theories of Growth

Notes on the Theories of Growth Notes on the Theories of Growth Economic Growth & Development These recitation notes cover basic concepts in economic growth theory relevant to the cases seen in class (Singapore, Indonesia, Japan, and

More information

Growth Accounting: Objective 2. Growth Theory: Review. Technology, F 4. Technical framework 3. Many factors play role to determine output in a country

Growth Accounting: Objective 2. Growth Theory: Review. Technology, F 4. Technical framework 3. Many factors play role to determine output in a country : Objective and Technical Framework : Objective 2 Growth Theory: Review Lecture 1, Exogenous Growth Economic Policy in Development 2, Part 2 April 20, 2007 Many factors play role to determine output in

More information

Name: Date: 3. Variables that a model tries to explain are called: A. endogenous. B. exogenous. C. market clearing. D. fixed.

Name: Date: 3. Variables that a model tries to explain are called: A. endogenous. B. exogenous. C. market clearing. D. fixed. Name: Date: 1 A measure of how fast prices are rising is called the: A growth rate of real GDP B inflation rate C unemployment rate D market-clearing rate 2 Compared with a recession, real GDP during a

More information

General Equilibrium Theory: Examples

General Equilibrium Theory: Examples General Equilibrium Theory: Examples 3 examples of GE: pure exchange (Edgeworth box) 1 producer - 1 consumer several producers and an example illustrating the limits of the partial equilibrium approach

More information

14.452 Economic Growth: Lecture 11, Technology Diffusion, Trade and World Growth

14.452 Economic Growth: Lecture 11, Technology Diffusion, Trade and World Growth 14.452 Economic Growth: Lecture 11, Technology Diffusion, Trade and World Growth Daron Acemoglu MIT December 2, 2014. Daron Acemoglu (MIT) Economic Growth Lecture 11 December 2, 2014. 1 / 43 Introduction

More information

Walrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.

Walrasian Demand. u(x) where B(p, w) = {x R n + : p x w}. Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian

More information

David N. Weil October 10, Lecture Notes in Macroeconomics

David N. Weil October 10, Lecture Notes in Macroeconomics David N. Weil October 10, 2006 Lecture Notes in Macroeconomics Growth, part 1 Miscellaneous Preliminaries What is a production function? A production function is a mathematical function that tells us how

More information

MA Macroeconomics 10. Growth Accounting

MA Macroeconomics 10. Growth Accounting MA Macroeconomics 10. Growth Accounting Karl Whelan School of Economics, UCD Autumn 2014 Karl Whelan (UCD) Growth Accounting Autumn 2014 1 / 20 Growth Accounting The final part of this course will focus

More information

Lecture Notes on Elasticity of Substitution

Lecture Notes on Elasticity of Substitution Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 210A March 3, 2011 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before

More information

Productioin OVERVIEW. WSG5 7/7/03 4:35 PM Page 63. Copyright 2003 by Academic Press. All rights of reproduction in any form reserved.

Productioin OVERVIEW. WSG5 7/7/03 4:35 PM Page 63. Copyright 2003 by Academic Press. All rights of reproduction in any form reserved. WSG5 7/7/03 4:35 PM Page 63 5 Productioin OVERVIEW This chapter reviews the general problem of transforming productive resources in goods and services for sale in the market. A production function is the

More information

Outline of model. Factors of production 1/23/2013. The production function: Y = F(K,L) ECON 3010 Intermediate Macroeconomics

Outline of model. Factors of production 1/23/2013. The production function: Y = F(K,L) ECON 3010 Intermediate Macroeconomics ECON 3010 Intermediate Macroeconomics Chapter 3 National Income: Where It Comes From and Where It Goes Outline of model A closed economy, market-clearing model Supply side factors of production determination

More information

First Welfare Theorem

First Welfare Theorem First Welfare Theorem Econ 2100 Fall 2015 Lecture 17, November 2 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem Last Class Definitions A feasible allocation (x, y) is Pareto

More information

1 National Income and Product Accounts

1 National Income and Product Accounts Espen Henriksen econ249 UCSB 1 National Income and Product Accounts 11 Gross Domestic Product (GDP) Can be measured in three different but equivalent ways: 1 Production Approach 2 Expenditure Approach

More information

The Cobb-Douglas Production Function

The Cobb-Douglas Production Function 171 10 The Cobb-Douglas Production Function This chapter describes in detail the most famous of all production functions used to represent production processes both in and out of agriculture. First used

More information

19 : Theory of Production

19 : Theory of Production 19 : Theory of Production 1 Recap from last session Long Run Production Analysis Return to Scale Isoquants, Isocost Choice of input combination Expansion path Economic Region of Production Session Outline

More information

Indifference Curves and the Marginal Rate of Substitution

Indifference Curves and the Marginal Rate of Substitution Introduction Introduction to Microeconomics Indifference Curves and the Marginal Rate of Substitution In microeconomics we study the decisions and allocative outcomes of firms, consumers, households and

More information

Review of Fundamental Mathematics

Review of Fundamental Mathematics Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

More information

Macroeconomics 2. Technological progress and growth: The general Solow model. Mirko Wiederholt. Goethe University Frankfurt.

Macroeconomics 2. Technological progress and growth: The general Solow model. Mirko Wiederholt. Goethe University Frankfurt. Macroeconomics 2 Technological progress and growth: The general Solow model Mirko Wiederholt Goethe University Frankfurt Lecture 3 irko Wiederholt (Goethe University Frankfurt) Macroeconomics 2 Lecture

More information

Name. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions.

Name. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Name Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Question 1. A firm has a production function F (x 1, x 2 ) = ( x 1 + x 2 ) 2. It is a price

More information

Critical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.

Critical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima. Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =

More information

Towards a Structuralist Interpretation of Saving, Investment and Current Account in Turkey

Towards a Structuralist Interpretation of Saving, Investment and Current Account in Turkey Towards a Structuralist Interpretation of Saving, Investment and Current Account in Turkey MURAT ÜNGÖR Central Bank of the Republic of Turkey http://www.muratungor.com/ April 2012 We live in the age of

More information

Endogenous Growth Theory

Endogenous Growth Theory Endogenous Growth Theory Motivation The Solow and Ramsey models o er valuable insights but have important limitations: Di erences in capital accummulation cannot satisfactorily account for the prevailing

More information

Lecture Notes on Elasticity of Substitution

Lecture Notes on Elasticity of Substitution Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 20A October 26, 205 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before

More information

The RBC methodology also comes down to two principles:

The RBC methodology also comes down to two principles: Chapter 5 Real business cycles 5.1 Real business cycles The most well known paper in the Real Business Cycles (RBC) literature is Kydland and Prescott (1982). That paper introduces both a specific theory

More information

Introduction to the Economic Growth course

Introduction to the Economic Growth course Economic Growth Lecture Note 1. 03.02.2011. Christian Groth Introduction to the Economic Growth course 1 Economic growth theory Economic growth theory is the study of what factors and mechanisms determine

More information

Agenda. Long-Run Economic Growth, Part 1. The Sources of Economic Growth. Long-Run Economic Growth. The Sources of Economic Growth

Agenda. Long-Run Economic Growth, Part 1. The Sources of Economic Growth. Long-Run Economic Growth. The Sources of Economic Growth Agenda The Sources of Economic Growth Long-Run Economic Growth, Part 1 Growth Dynamics: 8-1 8-2 Long-Run Economic Growth Countries have grown at very different rates over long spans of time. The Sources

More information

Estimation and Inference in Cointegration Models Economics 582

Estimation and Inference in Cointegration Models Economics 582 Estimation and Inference in Cointegration Models Economics 582 Eric Zivot May 17, 2012 Tests for Cointegration Let the ( 1) vector Y be (1). Recall, Y is cointegrated with 0 cointegrating vectors if there

More information

Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.

Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all. 1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.

More information

Nonlinear Regression Functions. SW Ch 8 1/54/

Nonlinear Regression Functions. SW Ch 8 1/54/ Nonlinear Regression Functions SW Ch 8 1/54/ The TestScore STR relation looks linear (maybe) SW Ch 8 2/54/ But the TestScore Income relation looks nonlinear... SW Ch 8 3/54/ Nonlinear Regression General

More information

Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

More information

EXOGENOUS GROWTH MODELS

EXOGENOUS GROWTH MODELS EXOGENOUS GROWTH MODELS Lorenza Rossi Goethe University 2011-2012 Course Outline FIRST PART - GROWTH THEORIES Exogenous Growth The Solow Model The Ramsey model and the Golden Rule Introduction to Endogenous

More information

The Delta Method and Applications

The Delta Method and Applications Chapter 5 The Delta Method and Applications 5.1 Linear approximations of functions In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X 1, X,... of independent and

More information

DEPARTMENT OF ECONOMICS. Unit ECON 12122 Introduction to Econometrics. Notes 4 2. R and F tests

DEPARTMENT OF ECONOMICS. Unit ECON 12122 Introduction to Econometrics. Notes 4 2. R and F tests DEPARTMENT OF ECONOMICS Unit ECON 11 Introduction to Econometrics Notes 4 R and F tests These notes provide a summary of the lectures. They are not a complete account of the unit material. You should also

More information

INTRODUCTION TO ADVANCED MACROECONOMICS Preliminary Exam with answers September 2014

INTRODUCTION TO ADVANCED MACROECONOMICS Preliminary Exam with answers September 2014 Duration: 120 min INTRODUCTION TO ADVANCED MACROECONOMICS Preliminary Exam with answers September 2014 Format of the mock examination Section A. Multiple Choice Questions (20 % of the total marks) Section

More information

This paper is not to be removed from the Examination Halls

This paper is not to be removed from the Examination Halls This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON EC2065 ZA BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences, the Diplomas

More information

14.452 Economic Growth: Lectures 6 and 7, Neoclassical Growth

14.452 Economic Growth: Lectures 6 and 7, Neoclassical Growth 14.452 Economic Growth: Lectures 6 and 7, Neoclassical Growth Daron Acemoglu MIT November 15 and 17, 211. Daron Acemoglu (MIT) Economic Growth Lectures 6 and 7 November 15 and 17, 211. 1 / 71 Introduction

More information

Readings. D Chapter 1. Lecture 2: Constrained Optimization. Cecilia Fieler. Example: Input Demand Functions. Consumer Problem

Readings. D Chapter 1. Lecture 2: Constrained Optimization. Cecilia Fieler. Example: Input Demand Functions. Consumer Problem Economics 245 January 17, 2012 : Example Readings D Chapter 1 : Example The FOCs are max p ( x 1 + x 2 ) w 1 x 1 w 2 x 2. x 1,x 2 0 p 2 x i w i = 0 for i = 1, 2. These are two equations in two unknowns,

More information

Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x) SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

More information

4. In the Solow model with technological progress, the steady state growth rate of total output is: A) 0. B) g. C) n. D) n + g.

4. In the Solow model with technological progress, the steady state growth rate of total output is: A) 0. B) g. C) n. D) n + g. 1. The rate of labor augmenting technological progress (g) is the growth rate of: A) labor. B) the efficiency of labor. C) capital. D) output. 2. In the Solow growth model with population growth and technological

More information

I. Basic concepts: Buoyancy and Elasticity II. Estimating Tax Elasticity III. From Mechanical Projection to Forecast

I. Basic concepts: Buoyancy and Elasticity II. Estimating Tax Elasticity III. From Mechanical Projection to Forecast Elements of Revenue Forecasting II: the Elasticity Approach and Projections of Revenue Components Fiscal Analysis and Forecasting Workshop Bangkok, Thailand June 16 27, 2014 Joshua Greene Consultant IMF-TAOLAM

More information

GROWTH, INCOME TAXES AND CONSUMPTION ASPIRATIONS

GROWTH, INCOME TAXES AND CONSUMPTION ASPIRATIONS GROWTH, INCOME TAXES AND CONSUMPTION ASPIRATIONS Gustavo A. Marrero Alfonso Novales y July 13, 2011 ABSTRACT: In a Barro-type economy with exogenous consumption aspirations, raising income taxes favors

More information

Written exam for the M. Sc. in Economics Summer Economic Growth (Videregående vækstteori) June 7, Four hours. No auxiliary material

Written exam for the M. Sc. in Economics Summer Economic Growth (Videregående vækstteori) June 7, Four hours. No auxiliary material Written exam for the M. Sc. in Economics Summer 2004 Economic Growth (Videregående vækstteori) June 7, 2004 Four hours. No auxiliary material To be answered in Danish or English 1 (The problem set is the

More information

Environmental problems and economic development in an endogenous fertility model

Environmental problems and economic development in an endogenous fertility model University of Heidelberg Department of Economics Discussion Paper Series No. 428 Environmental problems and economic development in an endogenous fertility model Frank Jöst, Martin Quaas and Johannes Schiller

More information

Calibration of Normalised CES Production Functions in Dynamic Models

Calibration of Normalised CES Production Functions in Dynamic Models Discussion Paper No. 06-078 Calibration of Normalised CES Production Functions in Dynamic Models Rainer Klump and Marianne Saam Discussion Paper No. 06-078 Calibration of Normalised CES Production Functions

More information

Constrained optimization.

Constrained optimization. ams/econ 11b supplementary notes ucsc Constrained optimization. c 2010, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values

More information

TESTING THE ONE-PART FRACTIONAL RESPONSE MODEL AGAINST AN ALTERNATIVE TWO-PART MODEL

TESTING THE ONE-PART FRACTIONAL RESPONSE MODEL AGAINST AN ALTERNATIVE TWO-PART MODEL TESTING THE ONE-PART FRACTIONAL RESPONSE MODEL AGAINST AN ALTERNATIVE TWO-PART MODEL HARALD OBERHOFER AND MICHAEL PFAFFERMAYR WORKING PAPER NO. 2011-01 Testing the One-Part Fractional Response Model against

More information

A Classical Monetary Model - Money in the Utility Function

A Classical Monetary Model - Money in the Utility Function A Classical Monetary Model - Money in the Utility Function Jarek Hurnik Department of Economics Lecture III Jarek Hurnik (Department of Economics) Monetary Economics 2012 1 / 24 Basic Facts So far, the

More information

Figure 4.1 Average Hours Worked per Person in the United States

Figure 4.1 Average Hours Worked per Person in the United States The Supply of Labor Figure 4.1 Average Hours Worked per Person in the United States 1 Table 4.1 Change in Hours Worked by Age: 1950 2000 4.1: Preferences 4.2: The Constraints 4.3: Optimal Choice I: Determination

More information

Chapter 3 A Classical Economic Model

Chapter 3 A Classical Economic Model Chapter 3 A Classical Economic Model what determines the economy s total output/income how the prices of the factors of production are determined how total income is distributed what determines the demand

More information

Lecture 1: OLG Models

Lecture 1: OLG Models Lecture : OLG Models J. Knowles February 28, 202 Over-Lapping Generations What the heck is OLG? Infinite succession of agents who live for two periods Each period there N t old agents and N t young agents

More information

6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium

6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium 6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline Pricing-Congestion Game Example Existence of a Mixed

More information

Structural Econometric Modeling in Industrial Organization Handout 1

Structural Econometric Modeling in Industrial Organization Handout 1 Structural Econometric Modeling in Industrial Organization Handout 1 Professor Matthijs Wildenbeest 16 May 2011 1 Reading Peter C. Reiss and Frank A. Wolak A. Structural Econometric Modeling: Rationales

More information

Solutions to Exercises in Introduction to Economic Growth (Second Edition)

Solutions to Exercises in Introduction to Economic Growth (Second Edition) Solutions to Exercises in Introduction to Economic Growth (Second Edition) Charles I. Jones (with Chao Wei and Jesse Czelusta) Department of Economics U.C. Berkeley Berkeley, CA 94720-3880 September 18,

More information

Equilibrium with Complete Markets

Equilibrium with Complete Markets Equilibrium with Complete Markets Jesús Fernández-Villaverde University of Pennsylvania February 12, 2016 Jesús Fernández-Villaverde (PENN) Equilibrium with Complete Markets February 12, 2016 1 / 24 Arrow-Debreu

More information

False_ If there are no fixed costs, then average cost cannot be higher than marginal cost for all output levels.

False_ If there are no fixed costs, then average cost cannot be higher than marginal cost for all output levels. LECTURE 10: SINGLE INPUT COST FUNCTIONS ANSWERS AND SOLUTIONS True/False Questions False_ When a firm is using only one input, the cost function is simply the price of that input times how many units of

More information

Agenda. Productivity, Output, and Employment, Part 1. The Production Function. The Production Function. The Production Function. The Demand for Labor

Agenda. Productivity, Output, and Employment, Part 1. The Production Function. The Production Function. The Production Function. The Demand for Labor Agenda Productivity, Output, and Employment, Part 1 3-1 3-2 A production function shows how businesses transform factors of production into output of goods and services through the applications of technology.

More information

E-322 Muhammad Rahman. Chapter 7: Part 2. Subbing (5) into (2): H b(1. capital is denoted as: 1

E-322 Muhammad Rahman. Chapter 7: Part 2. Subbing (5) into (2): H b(1. capital is denoted as: 1 hapter 7: Part 2 5. Definition of ompetitive Equilibrium ompetitive equilibrium is very easy to derive because: a. There is only one market where the consumption goods are traded for efficiency units of

More information

CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises

CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =

More information

2. Real Business Cycle Theory (June 25, 2013)

2. Real Business Cycle Theory (June 25, 2013) Prof. Dr. Thomas Steger Advanced Macroeconomics II Lecture SS 13 2. Real Business Cycle Theory (June 25, 2013) Introduction Simplistic RBC Model Simple stochastic growth model Baseline RBC model Introduction

More information

REVIEW OF MICROECONOMICS

REVIEW OF MICROECONOMICS ECO 352 Spring 2010 Precepts Weeks 1, 2 Feb. 1, 8 REVIEW OF MICROECONOMICS Concepts to be reviewed Budget constraint: graphical and algebraic representation Preferences, indifference curves. Utility function

More information

Chapter 12: Cost Curves

Chapter 12: Cost Curves Chapter 12: Cost Curves 12.1: Introduction In chapter 11 we found how to minimise the cost of producing any given level of output. This enables us to find the cheapest cost of producing any given level

More information

The Real Business Cycle Model

The Real Business Cycle Model The Real Business Cycle Model Ester Faia Goethe University Frankfurt Nov 2015 Ester Faia (Goethe University Frankfurt) RBC Nov 2015 1 / 27 Introduction The RBC model explains the co-movements in the uctuations

More information

Endogenous Growth Models

Endogenous Growth Models Endogenous Growth Models Lorenza Rossi Goethe University 2011-2012 Endogenous Growth Theory Neoclassical Exogenous Growth Models technological progress is the engine of growth technological improvements

More information