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Question 1(4×10pts) Describe and graph how each of the following affect the k? = 0 and ?= 0 locus

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Question 1(4×10pts)

Describe and graph how each of the following affect the k? = 0 and ?= 0 locus. How does c and k

react immediately after change and what is the new steady state? Note: k and c will denote per

intensive unit of labor (same as k? and ? in the lecture slides).

a) Permanent rise in productivity growth rate g.

b) Rise in the preference for today’s consumption θ.

c) Proportional downward shift of production function f(k).

d) Decrease in the depreciation rate of capital δ.

Question 2(5pts+10pts+3×5pts+10pts+5pts+5pts+10pts)

Consider an economy with infinitely lived representative households which provide labor

services in exchange for wages, receive interest income on assets, purchase goods for

consumption and save by accumulating additional assets. We will modify here the standard

Ramsey model by assuming that government purchases affect utility from private consumption

and that government purchases and private consumption are perfect substitutes. Thus, the

representative household maximizes its lifetime welfare:

subject to its flow budget constraint and the No-Ponzi-Game condition, where n is the rate of

population growth, θ > 0, ρ > 0 and ρ > n. Assume further that the government purchases per

capita are gt = Gt/Lt, which are financed by a constant tax on consumption 1 > τc > 0, and the

government budget is balanced. The productive sector of the economy has competitive firms

which produce goods, pay wages for labor input and make rental payments for capital inputs.

The firms have neoclassical production function, expressed in per capita terms yt = Akα

t where 0

< α < 1 and capital depreciates at the rate δ > 0.

a) Specify the household’s dynamic optimization problem.

b) Derive the first order conditions of the household’s optimization problem.

c) Obtain the Euler equation.

d) Write down government’s budget constraint, the government spending per capita, and ?.

e) Rewrite the Euler equation in terms of c, r, θ, and ρ. How does the tax affect the

consumption choice?

f) Write down and solve the problem of a profit-maximizing representative firm. Using the

results above specify the competitive market equilibrium.

g) Derive the conditions for the steady-state level of capital and consumption per capita and

draw the phase diagram.

h) Find the value of k* for α =0.5, A=4, δ=0.4, and ρ=0.6.

i) Assume that the economy is initially at a steady state with k* and c* > 0. What are the

effects of a temporary increase in government purchases on the paths of consumption,

capital and interest rate (draw their behavior over time).

ECON 501: Lecture 1 Introduction Instructor: Jaime Meza-Cordero, Ph.D. Outline • What is Macroeconomics? • Long term and short term analysis: Growth and Business Cycles • Calculation of Growth Rates • Distribution of Income and Inequality Macroeconomics • Study of the economy as a whole, where macroeconomists try both to explain economic events (positive) and to device policies to improve or enhance economic performance (normative). • Studies how decisions of households and firms aggregate into the whole economy. Macroeconomic Analysis • Long term: What are the factors behind the differences in economic growth, and how can we control them? – Known as Growth Theory. • Short term: Why do countries observe periods of recessions and depressions and how can government reduce the severity of these episodes? – Known as Business Cycle Theory. • Other areas of analysis: Unemployment, inflation, government intervention, regional integration, institutions. Modeling • Models are simplifications of reality that are useful to understand relationships and explain how changes in exogenous variables affect endogenous variables. – Exogenous: Those that the model take as given. – Endogenous: Those that the model aims to explain. • Models define agents (Households, governments, firms, banks, etc.) and define their decision problems (consumption, work, taxes, profits, etc.) Economic Growth • Before the industrial revolution, the entire world was poor. Subsistence agriculture was the main sector. • After technological advances and significant increases in productivity, some economies have experienced sustained growth. • Growth is linked to, but doesn’t necessarily imply, improved living standards (examples: sweatshops, pollution, deforestation). Economic Growth • Variations in GDP and GDP per-capita are the most commonly used indicators of growth. • There are great global disparities in production and growth rates. Penn World Tables • The Penn World Tables (PWT) version 9.1 is a database with information on relative levels of income, output, input and productivity, covering 182 countries between 1950 and 2017. Access to the data is provided at: https://www.rug.nl/ggdc/productivity/pwt/ GDP and GDP per-capita Growth Rates • 2020 GDP growth rate = [(GDP2020 – GDP2019) / GDP2019] * 100 • 2020 GDP per capita growth rate = [(GDPpc2020 – GDPpc2019) / GDPpc2019] * 100 CLASS EXERCISE • Using the PWT, choose a country (can be your country of birth or the last country you traveled to). 1. Calculate the latest available GDP and GDPpc growth rates of this country. 2. Calculate the growth rates for the last decade. Predicting Future Production • Let y0 be the GDPpc at year 0 (or the base year), yT the GDPpc at year T, and x the average annual growth rate over that period (not in percentage points): yT = (1 + x)T y0 • Taking logarithms: ln yT − ln y0 = T ln(1 + x) ≈ Tx, or x ≈ (ln yT − ln y0)/T CLASS EXERCISE yT = (1 + x)T y0 • Using the growth rate of the last decade, predict the GDP in 2030 for the country you selected. Global Income Distribution • In 1960: ? Richest country was Switzerland ($15, 000) ? Poorest country was Tanzania ($381) ? In general, the wealthiest countries were OECD + Latin America (Venezuela, Argentina), while the poorest countries were in Africa (Tanzania, Uganda) and Asia (China, India, Indonesia) • In 2000: ? Richest country - Luxembourg ($44, 000) ? Poorest - Tanzania ($482) ? In general, the wealthiest countries were OECD + East Asia (Taiwan, Japan, Singapore), while the poorest countries: subSaharan Africa (Tanzania, Uganda) and Latin America and Asia. Global Income Distribution • During this period: ? Growth miracles: Singapore (6.2%), South Korea (5.9%), Hong Kong (5.4%), Thailand, Japan (after WWII), China, Ireland. ? Growth disasters: sub-Saharan Africa (Niger, Angola, Madagascar, Nigeria, Rwanda) + Latin America (Venezuela, Bolivia, Peru, Argentina). Global Income Distribution https://ourworldindata.org/economic-growth In GPD per capita, pick: Angola Argentina China India South Korea Luxembourg Singapore Switzerland Tanzania Uganda USA Venezuela Growth Convergence • Macroeconomic models show how countries that have exhausted the gains from capital accumulation (OECD countries) depend on technological progress for growth. • This means that countries that have low levels of capital per capita would be able to grow faster by increasing their stock of capital. Growth Convergence • Less developed countries have lower levels of capital, with a higher marginal productivity. • As we will see in the Solow model, increasing the capital level will significantly increase the output of these countries until convergence. • However, less developed countries have lower levels of national savings and weak institutions, which has historically limited capital accumulation. Growth Convergence • A possibility to accelerate the convergence could come from external funding (international multilateral organizations). • There is little evidence of funding from these organizations significantly increasing capital accumulation and growth, and instead has been commonly directed to address accumulated debt resulting from fiscal deficits. • Better controls and behavioral changes will be necessary to increase savings rates. Growth Modeling • All agents meet at the markets for goods and inputs (labor and capital market), where equilibrium prices and quantities are determined. • Households: • owners of all inputs and assets in economy • # of children + work (how many hours) vs leisure decision determines the Lt (work force) • consumption (Ct) vs savings (= investment) decision determines the Kt (capital) Growth Modeling • Firms: • hire people (Lt) and capital (Kt) to produce good (Yt) by production technology (F (·)) • have access to knowledge (At) that makes the production more effective • Other institutions: introduction depends on what issue to be studied • Government (taxation, social security) • Central bank (monetary creation) Final Thoughts • Moving forward we will be addressing economic growth through models that treat the savings rate as exogenous (given) as well as being decided within the households’ optimization problem. • The later part of the course will focus on business cycles and macroeconomic policy to address current challenges. ECON 501: Lecture 2 Long-Term Growth: The Solow-Swan Model Instructor: Jaime Meza-Cordero, Ph.D. Outline • • • • • • Long Term Growth Analysis Solow-Swan Model Assumptions Model Framework Model Dynamics and Growth Path Convergence and Steady State Savings and Consumption Long-Term Growth • Countries have grown at different rates since the 1800’s. • It is important to understand what determines this growth and if economies converge to a steady growth level. • We start by studying the framework of the Solow-Swan model. Solow-Swan Model • Assumptions: – Savings are taken as exogenous and a constant fraction 0 < s < 1 of output. – Labor force and knowledge grow at given exogenous rates, n and g, respectively. – HHs and firms optimization decisions have already been made. Solow-Swan Model Assumptions Neoclassical production function: • Y (t) = F (K(t), A(t)L(t)) capital K(t) - durable physical inputs (machines, buildings, computers, etc.) –labor L(t) - number of workers and the amount of time they work. –knowledge A(t) - the effectiveness of production. Solow-Swan Model Assumptions Neoclassical production function: • Y (t) = F (K(t), A(t)L(t)) • Constant returns to scale (CRS) in capital and effective labor: • F (cK, cAL) = cF (K, AL) for all c ≥ 0 – No gains from further specialization – Inputs such as land or natural resources are ignored. Solow-Swan Model Assumptions Production function can be written in intensive form (per effective worker). • Denote: – k = K/AL , y = Y/AL , f (k) = F (k,1) • Production function becomes: – y = Y/AL = F(K,AL)/AL = F(K/AL,1) = F(k,1) = f(k) Solow-Swan Model Assumptions • Positive and diminishing returns to inputs ∂? • >0, ∂? ∂? • >0, ∂? ∂2? • 2 =n ∂? ? ? ∂?(?) ?? ? A?(t) = = gA(t) => =g ∂? ? ? Solow Model Dynamics • Reminder of Chain, Product, and Quotient Rules: Solow Model Dynamics Capital: • Output is divided between consumption C(t) and savings S(t). • Savings are a constant fraction (s) of output and they are immediately used as investment I(t) into new capital: S(t) = sY(t) = I(t). • Existing capital depreciates over time at the rate δ. Solow Model Dynamics Capital: • The change in the capital stock is then the balance between new investment and depreciation: K?(t) = I(t) − δK(t) = sY(t) − δK(t) Solow Model Dynamics Change in the capital stock: - Rewriting to intensive units: a) divide by AL, b) express in terms of k? Solow Model Dynamics • Combining the previous two equations, we obtain the main relationship derived from the model: k?(t) = sf (k(t)) - [δ + n + g]k(t) • This relationship implies that the change in the capital per effective worker equals the new investment minus the break-even investment. Solow Model Dynamics The evolution of capital per effective unit of labor depends on two components: 1. sf(k(t)) savings from output per effective worker. 2. [δ +n+g]k(t) investment needed to keep the level of capital per effective worker, which decreases due to depreciation (δ) and the growing quantity of effective labor (n + g). Graphical Representation Solow Model Dynamics • The intersection between the new investment and the break-even investment is the actual investment, which we will call k*. At this point we reach a steady state or stable equilibrium. • If the economy starts at a point when k0 and the economy will accumulate capital. CLASS EXERCISE • Show graphically and explain what would happen to an economy for the case when k>k*. Balanced Growth Path • In equilibrium: – Capital per effective worker converges to k∗, remains constant with no growth. – By assumption, labor L grows at rate n and technology A grows at rate g. – Capital stock K grows at rate (n + g). – Output Y = F(K,AL) is CRS. K and AL both grow at rate (n+g) then Y grows at rate (n+g). – Output per worker Y/L as well as capital per worker K/L grow at rate g. Balanced Growth Path • Implications: – An economy will converge to a balanced growth path. – Economies with low initial levels of capital will increase their stock, while countries with too much capital will lose stock as they will face too much depreciation. – In this steady state the key variables will grow at constant rates (as in the previous slide). Changes to the Savings Rate • Assumed as exogenous and fixed between 0 and 1. • Behavioral or policy changes could change the value of the savings rate. • What would happen to an economy in a steady state if there is a permanent increase in the savings rate? Graphical Representation ? Changes to the Savings Rate • New investment is now greater than breakeven investment and the net investment k?>0. The level of k will rise continuously to a new equilibrium level k*new. • In the steady state, Y/L grows at rate g. However, in the transition to new steady state, the growth rate of k is positive and therefore also growth rate of Y/K jumps up, until gradually returning to g. Effects on Consumption • Consumption will ultimately be the indicator of well-being in this model. • Everything not invested is therefore consumed: c∗ = (1 − s)f(k∗) • When the savings rate increases, capital reacts slowly, so consumption will initially be reduced. • As the capital level increases, raising production, consumption will begin to increase. Effects on Consumption • A key question then becomes if the households are better off at the new level of consumption or before the change in the savings rate. • The answer depends on the steady state level: c* = f (k*) - [δ + n + g]k* ?? ?? • Since > 0, then the effect on consumption will depend on the term inside the bracket. Effects on Consumption • [f ’(k∗)−(n+g+δ)] > 0 when the marginal product of additional capital created from the new savings is greater than the break-even level. • This added production will translate into higher levels of consumption. • The opposite will happen if new capital is too expensive to sustain. • The maximum consumption level achievable given sGR is knows as golden rule of capital. CLASS EXERCISE • Show graphically both cases described in the previous slide, plus a level of capital at which consumption is maximized (Golden Rule). Golden Rule Level of Capital • There exists a savings rate that maximizes the level of consumption. • The level of k associated with this level of consumption is known as the golden rule level of capital. • This level can be calculated through a first order condition of c with respect to k. c = f (k) - [δ + n + g]k ECON 501: Lecture 3 Long-Term Growth: The Ramsey-Kass-Koopmans Model Instructor: Jaime Meza-Cordero, Ph.D. Outline • • • • Assumptions Behavior of Households Behavior of Firms Competitive Market Equilibrium Ramsey Model Assumptions • The Solow model assumes an exogenous savings rate that has already been fixed by the households. • The Ramsey-Kass-Koopmans model adds additional insights by explicitly stating the household’s optimization problem and deriving from it what the savings rate would be. Ramsey Model Assumptions • We will assume infinitely lived households (a continuum of generations), which decide how much to consume and how much to save for the future. • These households directly (or indirectly) rent capital purchased with their savings and sell their labor to firms. • Firms are owned by households and operate in competitive markets. Ramsey Model Assumptions • Firms: • Have access to a production technology Y=F(K,AL) - which satisfies the same assumptions as in the Solow model. • Knowledge grows at rate g (exogenous) and A(0) is normalized to 1: A?(t) = gA(t) and A(t) = egtA(0) = egt Ramsey Model Assumptions • Households: • Provide labor to firms in exchange for real wage. • Save money to accumulate assets, which are translated into capital that is rented by a firm and paid back with interests. • Population grows at rate n, L(t) is the population size at time t and L(0) is normalized to 1: L?(t) = nL(t) and L(t) = entL(0) = ent Ramsey Model Assumptions • Utility maximization for current household members and all future descendants: • Individual consumption c(t) = ? ? ? Ramsey Model Assumptions • ent accounts for the growing population, since u[c(t)]L(t) = u[c(t)]ent • e−ρt accounts for time preference (the future’s utility is less valuable today’s). Ramsey Model Assumptions • We assume a functional form of utility called constant relative risk aversion (CRRA): • Where θ > 0 and satisfies the conditions: • and u’’(c) = −θc(t)−θ−1 < 0 , such that it is increasing and concave, guaranteeing a well-defined solution. Ramsey Model Assumptions *Notation note for this model: • Now lower cased variables are per capita/worker values, (c(t) above), while intensive (per effective worker) values will be denoted with hats (kˆ(t)). Ramsey Model Assumptions • Assets: • The returns to assets is competitive and defined as r(t). • a(t) represents the asset holdings per person. • The capital income per person is r(t)a(t). • Household’s capital income at time t is L(t)r(t)a(t). Ramsey Model Assumptions • Each household member inelastically supplies 1 unit of time for wage w(t). – household’s labor income at time t is L(t)w(t). • The household income is the sum of the labor income and the capital income. • Households use this income to consume C(t) and purchase additional assets (rent capital). Ramsey Model Assumptions • Household budget constraint at time t: • Where we have the total change of holdings A?(t) and we are interested in calculating the change per person a?(t) = [A (t)?/L(t)]. CLASS EXERCISE • Combine the equations below to come up with a single expression for the per capita asset accumulation (budget constraint) Ramsey Model Assumptions • Combining these two equations and rearranging, we obtain the budget constraint in per capita terms: a?(t) = w(t) + [r(t) − n]a(t) − c(t) • A final assumption that we will make is that no household cannot borrow unlimitedly (no Ponzi games). Firm Behavior • Capital and output in intensive form: kˆ = K/AL, yˆ = Y/AL = f(kˆ) • Capital rental rate = R. • Total cost of capital rental = RK. • Capital depreciates at rate δ. • Rate of return from capital to households (owners of the capital) = r = R-δ • Firms pay wage w to the labor force hired. Firm Behavior • Firms maximize profit in each period. • The firm’s profit ? is: • At point in time t, AL is fixed (everyone employed). Firm Behavior • Firms optimize the level of effective capital rented kˆ: • Therefore, the optimal choice of effective capital that the firm rents takes place when f ’(kˆ) = r + δ. Firm Behavior • Labor is paid its marginal product: w • Therefore: w = A[−f ’(kˆ)kˆ + f(kˆ)] = [f(kˆ) −f ’(kˆ)kˆ]egt Firm Behavior • RECAP. In the competitive equilibrium, both factors of production are paid their marginal products: r = f ’(kˆ) − δ w = [f(kˆ) −f ’(kˆ)kˆ]egt Household Behavior • Households maximize their utility subject to a budget constraint, debt limit, initial stock of assets a(0), and non-negative consumption choice. Therefore: Household Behavior • The optimal solution can be obtained through a present-value Hamiltonian: H = u[c(t)]e−(ρ−n)t + µ(t)[w(t) + [r(t) − n]a(t) − c(t)] • First order conditions: Household Behavior • The next step is to differentiate µ from equation (i) with respect to time: µ? = u’’(c)c?e−(ρ−n)t − u’(c)(ρ − n)e−(ρ−n)t • Combining (i) and (ii): Household Behavior • This is known as the Euler equation, which describes how households optimize current and future consumption. • Households face a trade-off between positive returns on savings r and time preference. CLASS EXERCISE • Using the Euler equation above, describe the consumption choice c(t) when r = ?. Interpret. CLASS EXERCISE SOLUTION • When r = ?, c?/c = 0. • Therefore, households will have a fixed consumption profile c(t) = c for all t. CLASS EXERCISE • Use the CRRA utility function to calculate the consumption profile c?/c in the Euler Equation. Household Behavior • Using the assumed CRRA utility function: • The intertemporal elasticity of substitution is: • Therefore, the Euler e...