Ramsey-Cass-Koopmans model.pptx

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1. Ramsey-Cass-Koopmans model Haile Girma (Assistant Professor) Department of Economics Salale University

2. Ramsey-Cass-Koopmans growth model ( consumption smoothening) In the Solow-Swan model , saving rate and, hence, the ratio of consumption to income are exogenous and constant . The overall amount of investment in the economy was still given by the saving of families , and that saving remained exogenous. Not useful to study how the economy reacted to changes in interest rates, tax rates, or other variables . We need a complete picture of the process of economic growth allow for the path of consumption and, hence, the saving rate to be determined by optimizing households and firms that interact on competitive markets. Households choose consumption and saving to maximize utility subject to an inter-temporal budget constraint Names: Frank Ramsey, Tjalling Koopmans, David Cass

3. Key idea : Replace ad hoc savings [consumption] function by forward-looking theory based utility maximization Specification of consumer behavior is a key element in the Ramsey growth model, as constructed by Ramsey (1928) and refined by Cass (1965) and Koopmans (1965). Hence the name Ramsey-Cass-Koopmans This model differs from the Solow-Swan growth model only in one crucial respect: It explicitly models the consumer side and endogenizes savings. In other words, it allows consumer optimization .

4. 1. Representative consumer (RC) Assumptions : Infinitely lived households; Identical households; each hh : h as the same preference parameters, faces the same wage rate (because all workers are equally productive), begins with the same assets per person , and has the same rate of population growth. Use of representative-agent framework , in which the equilibrium derives from the choices of a single household -heterogeneity issues !

5. A representative household with instantaneous utility function With properties: u(c(t)) is strictly increasing, concave, twice continuously differentiable positive but diminishing marginal felicity of consumption. u(c) satisfies Inada conditions: Labour supply is exogenous and grows exponentially (with initial labour equals 1): and All members of the household supply their labor inelastically . Each adult supplies inelastically one unit of labor services per unit of time.

6. Households hold assets in the form of ownership claims on capital or as loans. Negative loans represent debts . Households can lend to and borrow from other households, at interest rate, r (t) but the representative household will end up holding zero net loans in equilibrium. Households are competitive in that each takes as given the interest rate, r(t ) , and the wage rate , w(t) , paid per unit of labor services. Sources of income : interest income plus wage income Use of income : consumption plus savings [ asset accumulation]

7. The household is fully altruistic towards all of its future members , and always makes the allocations of consumption (among household members ) cooperatively. This implies that the objective function of each household at time t = 0, U(0 ), can be written as: Where , c(t ) is consumption per capita at time t, i.e. Each household member will have an equal consumption ρ is the subjective discount rate and is assumed to be the same across generations T he effective discount rate is ρ−n

8. Notice that: the household will receive a utility of u(c(t )) per household member at time t, or a total utility of Utility at time t is discounted back to time 0 with a discount rate of . We also assume throughout that Ensures that in the model without growth, discounted utility is finite. Otherwise, the utility function would have infinite value, and standard optimization techniques would not be useful in characterizing optimal plans. A positive value of ρ ( ρ >0) indicates parental “selfishness” Suppose that starting from a point at which the levels of consumption per person in each generation are the same. Then parents prefer a unit of their own consumption to a unit of their children’s consumption .

9. No technological progress Factor and product markets are competitive. Production possibilities set of the economy: Standard constant returns to scale and Inada assumptions still hold. Per capita production function f(.) Where

10. Competitive factor markets imply: And Households use the income that they do not consume to accumulate more assets Denote asset holdings of the representative household at time t by A(t ). Then, r(t ) is the risk-free market flow rate of return on assets, and w(t)L(t ) is the flow of labor income earnings of the household.

11. Defining per capita assets as: To get: Household assets can consist of capital stock, K(t ), which they rent to firms and government bonds, B(t ). With uncertainty, households would have a portfolio choice between K(t ) and riskless bonds . With incomplete markets, bonds allow households to smooth idiosyncratic shocks. But for now no need. Why? There is no government! Market clearing condition:

12. No uncertainty and depreciation rate of δ, the market rate of return on assets is : The Budget Constraint The differential equation: Is a flow constraint. It is just an identity; hhs could accumulate debt indefinitely If the household can borrow unlimited amount at market interest rate, it has an incentive to pursue a Ponzi-game. The household can borrow to finance current consumption and then use future borrowings to roll over the principal and pay all the interest.

13. In this case, the household’s debt grows forever at the rate of interest, r(t) . To rule out chain-letter possibilities, we assume that the credit market imposes a constraint on the amount of borrowing. The appropriate restriction turns out to be that the present value of assets must be asymptotically nonnegative: Consider the case of borrowing by households Infinite-lived households tend to accumulate debt by borrowing and never making payments for principal or interest.

14. Naturally, the credit market rules out this chain-letter finance schemes in which a household’s debt grows forever at the rate r or higher . In order to borrow on this perpetual basis , households would have to find willing lenders Other households that were willing to hold positive assets that grew at the rate r or higher . Households will be unwilling to absorb assets asymptotically at such a high rate . It would be suboptimal for households to accumulate positive assets forever at the rate r or higher, because utility would increase if these assets were instead consumed in finite time.

15. Household maximization Set up the current value of Hamiltonian function with state variable a, control variable c and current-value costate variable μ . It represents the value of an increment of income received at time t in units of utils at time 0. FOCs : ( i ) (ii)

16. The transversality condition is: What is transversality condition? The transversality condition for an infinite horizon dynamic optimization problem is the boundary condition determining a solution to the problem's first-order conditions together with the initial condition. The transversality condition requires the present value of the state variables to converge to zero as the planning horizon recedes towards infinity Intuition : The transversality condition ensures that the individual would never want to ‘die’ with positive wealth. An optimizing agents do not want to have any valuable assets left over at the end.

17. From (ii), we obtain, T he multiplier changes depending on whether the rate of return on assets is currently greater than or less than the discount rate of the household. The first necessary condition above implies that

18. The Euler Equation Differentiate equation (i) with respect to time and divide by), , we get the basic condition for choosing consumption over time: Upon substitution into (ii), we get the famous consumer Euler equation: Where is the elasticity of the marginal utility u’(c(t)).

19. Consumption will grow over time when the discount rate is less than the rate of return on assets . It also specifies the speed at which consumption will grow in response to a gap between this rate of return and the discount rate. Elasticity of marginal utility is the inverse of the intertemporal elasticity of substitution. The elasticity between the dates t and s > t is defined as: As s approaches t, we get:

20. Equilibrium Prices The market rate of return for consumers, r (t), is given by: Substituting this into the consumer’s problem, we have: It is simply the equilibrium version of the consumption growth equation.

21. Optimal Growth Capital and consumption path chosen by a benevolent social planner trying to achieve a Pareto optimal outcome . The optimal growth problem simply involves the maximization of the utility of the representative household subject to technology and feasibility constraints. Subject to and k (0) > 0.

22. S et up the current-value Hamiltonian: With state variable k, control variable c and current-value costate variable μ . The necessary conditions for an optimal path are:

23. It is straightforward to see that these optimality conditions imply: The transversality condition Both are identical with the previous results This establishes that the competitive equilibrium is a Pareto optimum The equilibrium is Pareto optimal and coincides with the optimal growth path maximizing the utility of the representative household.

24. Steady-State Equilibrium Characterize the steady-state equilibrium and optimal allocations A steady state equilibrium is an equilibrium path in which capital-labor ratio , consumption and output are constant . Since f(k ∗) > 0, we must have a capital-labor ratio k ∗ such that: The steady-state capital-labor ratio only as a function of the production function, the discount rate and the depreciation rate . This corresponds to the modified golden rule: The interest rate equals:

25. The modified golden rule involves a level of the capital stock that does not maximize steady-state consumption This is due to discounting (i.e. earlier consumption is preferred to later consumption). The objective is not to maximize steady state consumption rather giving higher weight to earlier consumption Given k ∗ , the steady-state consumption level is: Which is similar to the consumption level in the basic Solow model.

26. Transitional Dynamics Unlike the Solow-Swan model, equilibrium is determined by two differential equations: Moreover, we have an initial condition k(0 ) > 0, also a boundary condition at infinity: The intersection of and define the steady state (next slide). The former is vertical since a unique level of k* can keep per consumption constant ( ).

27. Dynamics of c since all households are the same, the evolution of C for the entire economy is: There are two ways for to be zero: ( i ) c(t)=0; corresponds to the horizontal axis (ii) which is a vertical line at k*. Ignore the first case, and focus on the second. This provide the optimal level of k, denoted by k*

28. When k exceeds k*, The opposite holds when k is less than k*. This is summarized in Figure 1 (next slide) c is rising if k<k* and declining if k>k*. The occurs at k=k* and c is constant for this value of k.

29. Figure 1: Dynamics of c

30. Dynamics of k The dynamics of the economy is given by: Notice that implies that Consumption equals the difference between actual output and break-even investment. c is increasing in k until The last expression gives the golden rule of capital per worker (k*) When exceeds that yields , k is decreasing, and vice versa .

31. Figure 2: Dynamics of k When k is large and break-even investment exceeds total output, for all positive values of c.

32. The dynamics of c and k: bringing the two together The arrows show direction of motion of c and k Consider the following: points to the left of and above The former is positive the latter is negative c is rising while k is falling On the curves, only one of c and k is changing. Example: on the line and above locus, c is constant and k is falling. At point E when holds, there is no movement.

33. Figure 3: Dynamics of c and k

34. The economy can converge to this steady state if it starts in two of the four quadrants in which the two schedules divide the space . Given this direction of movements, it is clear that there exists a unique stable arm, the one-dimensional manifold tending to the steady state. All points away from this stable arm diverge, and eventually reach zero consumption or zero capital stock

35. Consider the following: If initial consumption, c(0 ), started above this stable arm, say at c’(0 ), the capital stock would reach 0 in finite time, while consumption would remain positive . But this would violate feasibility . Therefore , initial values of consumption above this stable arm cannot be part of the equilibrium If the initial level of consumption were below it, for example, at c’’(0), consumption would reach zero.

36. Thus capital would accumulate continuously until the maximum level of capital (reached with zero consumption ) Continuous capital accumulation towards with no consumption would violate the transversality condition . There exists a unique equilibrium path starting from any k(0)>0 and converging to the unique steady-state (k ∗ , c ∗ ) with k ∗.