ECON6002 Macroeconomics Analysis 1 Week 2: The Ramsey Model Nadine Yamout University of Sydney Semester 2, 2021 Class Outline 1. The Ramsey model 2. Households dynamic optimization problem 3. Model dynamics and steady state 4. Using the model 5. Summary Readings: Romer, Chapter 2.1-2.7 The Ramsey Model Similar to the Solow-Swan model, but saving rate is endogenous Uses microfoundations to motivate household and firm decisions about consumption and capital accumulation Allows welfare analysis of policy (e.g., should government encourage the golden-rule saving rate?) Technically complicated model, but still intuitive In an obituary for Ramsey, who died at age 26, John Maynard Keynes described Ramseys article about optimal saving as: ...one of the most remarkable contributions to mathematical economics ever made, both in respect of the intrinsic importance and difficulty of its subject, the power and elegance of the technical methods employed, and the clear purity of illumination with which the writers mind is felt by the reader to play about its subject. Question (pingo.coactum.de ? 544037) If there is a sudden change in the structure of the economy (e.g., a decline in the growth rate of technology), will households or firms adjust consumption or capital more quickly? A. Households will immediately adjust consumption B. Firms will immediately adjust capital Model Structure Microfoundations: profit maximizing firms facing perfectly competitive markets (price takers) households maximize utility that depends on consumption subject to an intertemporal budget constraint The technological side of the model is the same as that of the Solow model. The production function is F(K, AL) satisfies the same assumptions as in the Solow model. Technology and population growth remain exogenous; A grows at rate g and L grows at rate n. Capital accumulation, the real interest rate, consumption, and output are endogenous One good, closed economy, no unemployment, with and without government purchases, no capital depreciation for simplicity Firms Large number of identical firms with the aggregate production function Y (t) = F(K(t), A(t)L(t)) ? y(t) = f (k(t)). Firms hire workers and rent capital in competitive factor markets, and sell their output in a competitive output market. A input is freely available; that is, firms do not need to make any payments to use it. Firms maximize profits. They are owned by the households, so any profits they earn accrue to the households. Constant returns to scale and perfect competition (no barriers to entry) imply zero profits in equilibrium. Normalize price level to 1. Firms Behaviour At each point in time firms employ the stocks of labour and capital, pay them their marginal products, and sell the resulting output. In absence of depreciation, the real interest rate r(t) = ?F(K(t), A(t)L(t)) ?K(t) = f 0 (k(t)) The real wage W (t) = ?F(K(t), A(t)L(t)) ?L(t) = A(t)[f (k(t)) - k(t)f 0 (k(t))] Households Large number of identical households. The size of each household grows at rate n. Each member of the household supplies labour at every point in time and rents capital to firms. The household divides its income (from the labour and capital) at each point in time between consumption and saving. Households are infinitely lived and maximize lifetime (discounted) utility subject to an in intertemporal budget constraint Households Behaviour Household utility is Z 8 t=0 e -?t u(C(t))L(t) H dt ? is a discount rate, C(t) is consumption for each identical agent, L(t) is total no. of workers in economy, and H is no. of households Felicity (a.k.a. instantaneous utility) function has explicit CRRA form: u(C(t)) = C(t) 1-? 1 - ? ? > 0 Constant intertemporal elasticity of substitution: 1/? (and constant relative risk aversion -Cu00(C)/u 0 (C) = ?) When ? is smaller, marginal utility changes more slowly, and so household is more willing to allow consumption to vary over time. Functional form is convenient and consistent with balanced growth u(C) = ln C as ? ? 1 (corresponds to zero interest elasticity for C(t) due to offsetting income and substitution effects) Intertemporal Household Budget Constraint Present value of lifetime consumption cannot exceed initial wealth plus the present value of lifetime labour income: Household has L(t)/H members ? labour income at t is W (t)L(t)/H, Consumption expenditures are C(t)L(t)/H, initial wealth is 1/H of total wealth at time 0, or K(0)/H. Intertemporal budget constraint: Z 8 t=0 e -R(t)C(t) L(t) H dt = K(0) H + Z 8 t=0 e -R(t)W (t) L(t) H dt Present value calculation using R(t) = R t t=0 r(t )dt allows for time-varying real interest rates, but assumes perfect foresight Intensive Form Utility c(t) consumption per unit of effective labour ? C(t) = A(t)c(t). C(t) 1-? 1 - ? = [A(t)c(t)]1-? 1 - ? = [A(0)e gtc(t)]1-? 1 - ? = A(0)1-? e (1-?)gt c(t) 1-? 1 - ? Household objective function: U = Z 8 t=0 e -?t C(t) 1-? 1 - ? L(t) H dt = Z 8 t=0 e -?t A(0)1-? e (1-?)gt c(t) 1-? 1 - ? L(0)e nt H dt = A(0)1-? L(0) H Z 8 t=0 e -(?-n-(1-?)g)t c(t) 1-? 1 - ? = B Z 8 t=0 e -t c(t) 1-? 1 - ? B = A(0)1-?L(0)/H and = ? - n - (1 - ?)g, with > 0 to ensure lifetime utility is finite Budget Constraint in Intensive Form Again, the budget constraint is: Z 8 t=0 e -R(t)C(t) L(t) H dt = K(0) H + Z 8 t=0 e -R(t)W (t) L(t) H dt Total consumption: C(t) L(t) H = c(t) A(t)L(t) H Initial capital holdings: K(0) H = k(0) A(0)L(0) H Total labour income: W (t) L(t) H = w(t) A(t)L(t) H Z 8 t=0 e -R(t) c(t) A(t)L(t) H dt = k(0)A(0)L(0) H + Z 8 t=0 e -R(t)w(t) A(t)L(t) H dt Using the fact A(t)L(t) = A(0)L(0)e (n+g)t : Z 8 t=0 e -R(t) e (n+g)t c(t)dt = k(0) + Z 8 t=0 e -R(t) e (n+g)tw(t)dt Constrained Optimisation The households problem is to choose the path of c(t) to maximize lifetime utility subject to the budget constraint. Set up Lagrangian in terms of intensive (per unit of effective labour) measures: L = B Z 8 t=0 e -t c(t) 1-? 1 - ? dt + ? k(0) + Z 8 t=0 e -R(t) e (n+g)tw(t)dt - Z 8 t=0 e -R(t) e (n+g)t c(t)dt Differentiate L with respect to c(t) and ? and set to 0 to get first-order conditions (FOCs) of constrained optimization problem Equivalent to Hamiltonian optimal-control solution in this relatively simple setting. First-Order Conditions The FOC with respect to ? is just the binding intertemporal budget constraint: Z 8 t=0 e -R(t) e (n+g)t c(t)dt = k(0) + Z 8 t=0 e -R(t) e (n+g)tw(t)dt For c(t), the FOC is Be-t c(t) -? = ?e -R(t) e (n+g)t The FOCs describe the behaviour of the optimizing households. Interpreting the Behaviour of Households FOC : Be-t c(t) -? = ?e -R(t) e (n+g)t Take natural logs of FOC for c(t) and noting definition of R(t): ln B - t - ? ln c(t) = ln ? - Z t t=0 r(t )dt + (n + g)t Given this holds for all t, take derivatives of both sides with respect to t to get Euler equation (note: dlnc(t) dt = c? c ): - - ? c?(t) c(t) = -r(t) + n + g ? c?(t) c(t) = r(t) - n - g - ? ? C?(t) C(t) = r(t) - ? ? Note consumption per worker C(t) = c(t)A(t) Intuition of the Euler Equation C?(t) C(t) = r(t) - ? ? If real interest rate r(t) is higher than households discount rate ?, they will choose a path of increasing consumption per worker C?(t)/C(t) at rate that depends on intertemporal elasticity of substitution 1/? That is, households are relatively patient compared to the interest rate, so they will delay consumption today to have more in the future However, if the discount rate is higher than the real interest rate (i.e., r(t) - ? < 0), households will choose a path of decreasing consumption over time, reflecting their relative impatience A higher intertemporal elasticity of substitution 1/? (smaller ?) corresponds to more sensitivity of consumption growth to the real interest rate Small Group Discussion #1 Groups of 3-5 in breakout rooms on Zoom 1. If the interest rate increased relative to your discount rate, how would you change your current and future consumption? 2. What factors would contribute your decision about consumption today versus in the future? Have someone from the group type at least one answer to each question in the Ed Week 2 - Small Group Discussion #1 thread. Dynamics of c In equilibrium, r(t) = f 0 (k) (competitive factor markets) and Euler equation holds for economy as a whole, not just each household Thus, given = ? - n - (1 - ?)g, the endogenous law of motion for aggregate consumption per unit of effective labour is c?(t) c(t) = f 0 (k) - ? - ?g ? c? = 0 for k * such that f 0 (k * ) = ? + ?g Vertical line for ?c = 0 in c vs. k space c? > 0 ? f 0 (k) > ? + ?g ? k < k * c? < 0 ? f 0 (k) < ? + ?g ? k > k * Consumption (per unit of effective labour) increases when marginal product of capital f 0 (k) is high (i.e., k < k * ) and vice versa Adjustment of c given k Dynamics of k Same as Solow model Actual investment is income minus consumption f (k(t)) - c(t) and break-even investment is (n + g)k(t) given no depreciation Thus, ?k(t) = f (k(t)) - c(t) - (n + g)k(t) ?k = 0 when c = f (k) - (n + g)k Consumption would be maximized at Golden Rule level of capital, but lower otherwise Hump shaped curve around Golden Rule level of capital for ?k = 0 in c vs. k space (see picture) ?k > 0 if extra saving c < f (k) - (n + g)k and ?k < 0 if inadequate saving c > f (k) - (n + g)k Capital accumulation occurs when consumption is low and vice versa Adjustment of k given c Steady State Steady state in general equilibrium occurs when both ?c = 0 and ?k = 0 As with Solow-Swan model, steady state is unique (see phase diagram) k * is lower than kGR because f 0 (k * ) = ? + ?g, f 0 (kGR) = n + g, and > 0, where = ? - n - (1 - ?)g > 0 ? ? + ?g > n + g ? k * < kGR given f 0 (k * ) > f 0 (kGR) and f 00(k) < 0 Note: > 0 condition ensures finite utility How will the economy evolve? Depends on k and c Transition Dynamics Capital evolves gradually, but consumption is a jump variable Depending on initial endowment of capital k(0), households choose unique initial consumption c(0) to set path for consumption to adjust optimally towards steady state and satisfy the budget constraint: Z 8 t=0 e -R(t) e (n+g)t c(t)dt = k(0) + Z 8 t=0 e -R(t) e (n+g)tw(t)dt Other values of consumption would lead to a divergence away from steady state and so are not chosen (see picture) The set of initial cs for different ks form the saddle path that determines the transition dynamics of the model economy (see picture) Dont fall off the saddle path!! What happens for different initial levels of consumption? The Saddle Path Transition Dynamics Balanced Growth Path The long-run balanced growth path (once the economy has converged to Point E) is the same as in the Solow model. Because k, y, and c are constant in steady state, so is the steady-state saving rate s * = (y * - c * )/y * Capital K(t), output Y (t) and consumption C(t)L(t) all grow at rate g + n. Capital per worker K(t)/L(t), output per worker Y (t)/L(t) and consumption per worker C(t) all grow at rate g. The Social Optimum and the Golden Rule Level of Capital Difference between the balanced growth paths of the Solow and Ramsey models is that a balanced growth path with a capital stock above the golden-rule level is not possible in the Ramsey model. In the Solow model, a high saving rate causes the economy to reach a balanced growth path that involves higher consumption. In the Ramsey model, saving is derived from the behaviour of households whose utility depends on their consumption ? no equilibrium where higher consumption can be attained in every period (trade-off between consumption today and in future). If the initial capital stock exceeds the golden rule level ? initial consumption is above the level needed to keep k constant ? ?k < 0 ? k gradually approaches k * , which is below the golden rule level. Because k * is the optimal level of k for the economy to converge to, it is known as the modified golden rule capital stock. Small Group Discussion #2 Groups of 3-5 in breakout rooms on Zoom 1. Do you think transition dynamics will be faster or slower compared to the Solow model with a fixed saving rate during the transition? Why? 2. How will the discount rate ? and the IES 1/? affect how close steady-state capital k * is to the Golden Rule level kGR? Have someone from the group type at least one answer to each question in the Ed Week 2 - Small Group Discussion #2 thread. What If Households Become More Patient? The equivalent thought-experiment to increasing the saving rate in the Solow model is to assume households suddenly become more patient ? ? (? s * ?) A change in ? only affects ?c = 0 line in phase diagram shifts to the right given f 0 (k * ) = ? + ?g and f 00(k) < 0 Because capital accumulation takes time, consumption immediately drops to put the economy on the new saddle path k and c rise over time to higher steady-state values than when households were less patient Only difference from Solow model is a time-varying saving rate along the transition path Households become more patient Speed of Convergence? Again, to think about quantitative speed of convergence, linearize ?k and ?c around steady state using Taylor series approximation: ?k " ? ?k ?k k=k * # (k - k * ) + " ? ?k ?c c=c * # (c - c * ) c? ?c? ?k k=k * (k - k * ) + ?c? ?c c=c * (c - c * ) Solving involves a lot of uninteresting algebraic manipulations (see Romer textbook) Given a Cobb-Douglas productions function with a = 1/3, ? = 4%, n = 2%, g = 1%, and ? = 1 implies convergence rate of 5.4% versus 2% for Solow model with same parameters Faster convergence in this case is due to endogenous saving rate that responds to the real interest rate along the saddle path. The Effects of Government Purchases Assume, for simplicity, government purchases affect household utility in an additively separate way Also, assume government purchases are financed by lump-sum taxes (although results same if debt-financed under Ricardian equivalence) Let G(t) denote government purchases per unit of effective labour Only two changes to the model: 1. Capital accumulation depends on saving out of disposable income: k?(t) = f (k(t))-G(t) - c(t) - (n + g)k(t) 2. Present value of consumption in IBC depends on present value of disposable income: Z 8 t=0 e -R(t) e (n+g)t c(t)dt = k(0) + Z 8 t=0 e -R(t) e (n+g)t [w(t)-G(t)] dt Permanent Versus Temporary Change in Government Purchases A permanent increase in government spending shifts the ?k = 0 curve down (see phase diagram) A temporary increase causes a partial drop in consumption to a level that will result in the economy being on the original saddle path when the government spending program ends (see pictures) Very different than case of a constant saving rate, i.e. Solow model implicitly assumes Keynesian consumption function with constant marginal propensity to consume Consumption falls by same fixed fraction of disposable income regardless of whether change was permanent or temporary Permanent Increase in G Temporary Increase in G Summary Ramsey model provides microfoundations for understanding houseshold and firm decisions related to economic growth. Ramsey model will be the building block for modern business cycle models with microfoundations. But these business cycle models will also introduce deviations from perfect competition and other frictions. Meanwhile, Ramsey model still does not explain the Solow residual that accounts for the bulk of economic growth. Next time: Endogenous Growth Theory (Romer, Chapter 3)

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