Saddlepath Outcome in the Ramsey Model
- Ramsey model: the economy is always on a saddlepath (SS locus in diagram)
e.g. if the starting value of k is khigh the decision-maker’s best choice is chigh.
- once on SS the economy then moves along the saddlepath to the steady state (k*, c*).
- What is happening on the saddlepath?
Dynamics of c and k are consistent with intertemporal optimization:
- path satisfies (Euler equation): β U ' (c j+1)
U ' (c j)[ FK (k j+1 )+1−δ ]=1
- path satisfies the resource constraint: kj+1= F(kj) -kj - cj
- not on a path that ends up at an endpoint that is incompatible with optimal behavior (i.e. not at the ‘overinvestment’ outcome
c=0, kmax.
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- So the saddlepath is a sequence of optimal temporary equilibria leading to the steady state.
Real-Business-Cycle Dynamics in a Ramsey Model:
- Modern macro definition of a business cycle: “path followed by the economy during its adjustment back to equilibrium” after some "shock".
- Ramsey model: moves through a sequence of temporary equilibria in each of which decision-makers are behaving
optimally.
- suggests little role for policy to offset cycles (outcomes are always optimal!)
- We want to look at economic responses to such shocks more closely.
- What kinds of real shocks are possible in the Ramsey model?
- Productivity shock (production function changes)
- source: technological? organization/incentives/policy? pandemic?
- positive shocks: technological progress, organizational improvements?
- negative shocks: technology unlikely to deteriorate; negative organizational. policy or incentive changes?
- Other possible shocks with real effects in this version of the model? - changes in the depreciation rate ()
- changes in consumer tastes (form of the utility function or the value of ()
- Downturn in an economy (recession) could be due to: - negative productivity shock- increase in depreciation. - households discount future utility less heavily: could
lower c now but is it a recession?
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- Very different from Keynesian models which stress the role of a fall in aggregate demand.
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The Effect of a Permanent Productivity Shock:
- Positive Productivity shock: FK and F(k) higher at all k so production function F(k) shifts up and is steeper at each k.
- This will affect both the k=0 and c=0 loci in the diagram on p.1
k=0 locus: c = F(k) -k now F(k) has increased (locus is higher at each k)
c=0 locus: FK(k)=+ FK(k) has increased at each k so the value of k where FK=+ is
higher than before (locus shifts right)
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- Diagram showing the effect of the shock on the phase diagram:
- Pre-shock steady state: k0*, c0* (where Old k=0 and Old c=0 intersect, Pt. A)
- New post-shock steady state: k1*, c1* (where New k=0 and New c=0 intersect, i.e. Pt. B)
- Long-run (LR) effect of the productivity shock is the move from A to B:k0* rises to k1* and c0* rises to c1*
- How does the economy move to the new LR equilibrium?
- start at point A in diagram (the old steady state).
- shock hits, current k is fixed (determined by past decisions) but y=F(k) is higher.
- jump to the new saddlepath (pt. C) and then move along it until the economy reaches B.
- Diagram shows pt. C as above pt. A (if so ‘c’ rises when the shock hits).
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- This is not the only possibility:
Euler equation tells you that since FK has risen due to the shock:
β U ' (c t+1)U ' (c t)
[ FK (k t +1 )+1−δ ]>1 just after the shock hits
(it was equal pre-shock)
- Possible responses? two things are going on.
(1) the economy is richer now and in future due to the shock: c is likely to rise (income effect)
(2) FK is higher: the return to saving and investment is higher: so decrease c boost saving now to build more k (substitution effect)
- If (1) is stronger than (2) then pt. C is above A (so consumption rises immediately as in the diagram).
- If (2) is stronger than (1) then pt. C is below A (c falls immediately)
- After the jump to point C: c and k rise along the new saddlepath to the new LR-steady state equilibrium. Note that y is rising too (since y=F(k)).
i.e. economy is booming.
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- How do y, c, i and k evolve as the economy responds to the shock?
- diagram below shows a possibility (assumes pt. C is above A).
- time is on the horizontal axis in each panel, vertical axis is either y, c, i or k.
- shock hits in period tshock. Just before the shock hits the economy is at pt. A in each panel. Prior to that each variable is constant at its old steady state value (y0*,c0*,i0*,k0*). Note that we must have i0*=k0* since it is a steady state.
- when the shock hits the economy immediately jumps from pt. A in each panel to pt. C (note k0 can’t change immediately as
it reflects past decisions but due to the rise in ‘i’ when the shock hits k will rise in the next period.
- in subsequent periods k is rising which causes y=F(k) to rise further. c continues rising since y is higher and the return in
investment (FK) is falling as k rises. i starts to fall as more k drives down
the return on investment. This all continues until the new steady state value of each variable is attained (y1*,c1*,i1*,k1*). All are higher than in the old steady state.
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- Think about a negative productivity shock and try and draw diagrams showing how the economy would respond.
Depreciation Shocks: Changes to
- Shocks to have similar effects to changes in productivity.
- As for a productivity shock, a change in affects both the k=0 and c=0 loci.
k=0 locus: c = F(k) -k a change in changes the steady state level of
investment (k)
c=0 locus: FK(k)=+ a change in alters the value of k at which FK=+
- A fall in is much like a positive productivity shock. A rise in is much like a negative productivity shock.
- A possible cause of a recession? Falling y and c could be due to a negative productivity shock or a rise in (reverse of the case above)
Shocks to Time Preference: Changes in )
- Say the discount rate falls ( falls, rises): future utility gets more weight vs present utility
k=0 locus: c = F(k) -k not affected by a change in .
c locus: FK(k*)= shifts right (a lower means higher k*).
(see diagram next page)
- Old steady state equilibrium: c0*, k0*
- New steady state: c1*, k1* where the k locus intersects the new c=0 locus.
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- Long-run: move from old to new steady state so higher c* and k*.
- Immediate effect of the shock: a fall in c to the new saddlepath (new saddlepath goes through the new steady state)
(move from old steady state to point B).
- intuition? Future utility (and c) are worth more after rises: save and invest more now to boost future c.
- Just after the shock hits: β U ' (c t+1)
U ' (c t)[ FK (k t+1 )+1−δ ]>1
(it had been an equality but rose due to the shock)
- So to satisfy Euler condition the left hand side must fall: a rise in ct+1 or kt+1 and a fall in ct can do this.
- Subsequent periods move along the new saddlepath from B to new steady state (rising c, k).
-Note that y is rising as k rises along the saddlepath.
- New long-run equilibrium: higher k*, higher c*.
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(Romer Ch. 2 has a version of this)
- The effect of a fall in time preference is similar to the effect of the rise in the savings rate (s) in the Solow model.
- Solow: rise in s, raises k and y (also can raise c if you start below the Golden rule level of k)
-Ramsey model: k, y are higher in the new steady state (as is c)
- Why so similar? Ramsey model: saving is endogenous but is partly determined by time preference.
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Extending the Ramsey Model: Adding Labour to the Basic Model
- Working time (labour) is implicitly fixed in the basic model.
- To use the model to think about business cycles we want to be able to talk about changes in the number of people employed. We must add labour supply!
- Add leisure to the utility function: U(c) becomes U(c,l) where l = leisure time.
Define: Uc ≡ ∂U/∂c and Ul ≡∂U/∂l
Both c and l are goods so: Uc>0 and Ul>0
Diminishing marginal utility is assumed (second derivatives Ucc, Ull<0)
Special case assumed: cross-partial derivative Ucl=0 (c, l are not substitutes or complements: making the utility function
additively separable in c and l is a way of doing this)
- Time constraint:
- Person’s time is used for leisure (l) or work (n).
- Set total time = 1 so time constraint: n+l=1
- Production function: labour is now an input that can be varied yt= F(kt, nt)
define marginal products: Fk ≡ ∂F/∂k >0 Fn ≡ ∂F/∂n>0
diminishing returns in each input (second derivatives <0)
Fkn>0 (more of one input raises the marginal product of the other input)
Inada conditions hold (as in the model without labour)
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- Lagrangian for the maximization problem:
L=∑t=1
∞
{β t U (c t , lt)+λt ¿F(kt,nt) - ct - kt+1 +(1-kt] +t[1-nt-lt] }
- objective as in earlier problem except l enters U ;
- resource constraint is like that of earlier problem except n enters the production fn. F (as before t is the lagrange multiplier for the resource constraint in period t)
- there is also a time constraint in each period (t is the Lagrange multiplier for the time constraint for period t, i.e. value in terms of utility of extra time).
- Decision maker chooses c, k, l and n in each period. f.o.c.’s for each are:
cj: j Uc,j - j = 0
kj+1: j+1 [Fk,j+1 +1 - - j =0
lj: j Ul,j - j = 0
nj: j Fn,j - j =0
(notation – two subscripts now: e.g. Uc,j is marginal utility of c in period j)
- Typical condition for choice of leisure in any period j (combine f.o.c. for c, l and n):
Ul,j = Uc,j Fn,j
- marginal value of leisure (Ul)equals marginal value of output from extra work (Fn is the extra output, Uc Fn is the utility value of this output).
(could also state in terms of MRS: Ul,j /Uc,j = Fn,j )
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- As earlier an Euler equation can be derived as:
β U c , j+1
U c , j[F k , j+1+1−δ ]=1
- this uses the f.o.c.’s for cj, cj+1 and kj+1.
- in a steady state (c same across periods at c* so Uc,j+1=Uc,j) once again we have:
Fk = (this uses ≡1/(1+) )
- the resource constraint in the steady state (with k constant over time at k*) becomes: F(k*,n*)=c+k* or c*=F(k*,n*)-k*
- Long run, steady state solution k*, c* and n* can be found using:Fk(k*,n*)= c*=F(k*,n*)-k*Ul,t(c*,1-n*) = Uc,t(c*,1-n*) Fn,t(k*,n*)
- this all suggests that the results for the extended model are similar to
that without labour.
- The model with labour suggests an additional “real” cause of business cycles.
- the value the decision maker places on leisure could change. i.e. utility function changes in a way that changes the marginal
value of leisure vs. consumption.
- could a recession be spurred by a rise in the preference for leisure?e.g. value leisure more highly, work less (lower ‘n’), output
falls (less labour input), consumption falls.
- could policies that affect the labour-leisure choice cause this type of recession? e.g. taxes on wage income; income support for people who are not working might induce people to work less.
- Casey Mulligan takes this view:https://www.manhattan-institute.org/html/redistribution-recession-how-labor-market-
distortions-contracted-economy-6148.html- Others think this is silly : ‘great vacation’ model of recessions.
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Centralized or One-Decision Maker version of the Ramsey Model Concluded:
- A basic Dynamic General Equilibrium model of a macroeconomy:
- inter-temporal (dynamic);
- optimal decision-making incorporated: rational decisions;
- consumption vs. savings and investment tradeoff lies at its heart;
- sources of fluctuations:
- changes in inter-temporal utility function (discount rate, taste for consumption vs. leisure)
- changes in the resource constraint (productivity shocks, depreciation changes)
- Ramsey model introduces the basic mechanics of the DSGE approach.
- Questions?
- One decision-maker, everyone the same, no markets: how useful is this?
- A planners problem: characterizes the optimal outcome
- The outcome is Pareto efficient: the decision-maker has made their best choice so the result can’t be improved upon.
- Microeconomics theory’s ‘Fundamental Theorems of Welfare Economics imply that this could be achieved by competitive markets (see our earlier notes on Robinson Crusoe model).
(we will look at this further pp. 14-17)
- Business cycles: are the kinds of shocks that cause fluctuations in this model sufficient to explain business cycles?
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Ramsey Model as a Model with Competitive Markets
- Recall from our discussion of simple models the “Fundamental Theorems of Welfare Economics”
(1) competitive outcomes are Pareto efficient;
(2) any Pareto efficient outcome can be supported by a competitive equilibrium.
Pareto efficiency: a set of outcomes is Pareto efficient if it cannot be changed in a way that will make one person better off without
making someone else worse off.
- The Ramsey solution above is Pareto efficient: maximized well-being of the representative agent so can’t change the outcome and make the agent better off.
- The second theorem tells us that a competitive outcome can achieve the same outcome as the Planner’s problem.
- Consequence: the Planner’s problem can serve as a shortcut for solving for the competitive outcome.
- Competitive markets version of the model:
- markets for consumption goods, inputs and financial assets.
- individuals are price takers (too small to affect prices) but their collective behavior determines prices. (Prices settle at levels where
supply=demand)
- buyers and sellers make decisions to maximize their well-being given market prices;
- Underlying this is the idea that the decision-maker in the model above represented the aggregate behavior of many similar decision makers.
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- Individual decision-makers:
- Households deciding how much to consume or save (borrow); how much to work etc.
- Business firms deciding how much to produce, how much capital and labour to use in production etc.
- Household problem (from notes on consumption: but in Ramsey T=∞):
- take the version with per period budget constraints:
Ct+ At+1= Yt + (1+rt)At
A= assets, Y=income, C=consumption, r=interest rate
(to make timing like Ramsey I am calling At assets at start of period t. In the consumption notes it was At-1)
with the Lagrangian from the notes on the household problem is:
L=∑t=1
T
βt U (C t )+∑t =1
T
λt (Y t+(1+rt ) A t−Ct−At+1 )
Maximize by choice of C and A in each period (as before).
- we want to show that this is the same problem as the Ramsey model provided that markets are competitive.
- in the Ramsey model this is a general equilibrium problem and we so need to say something about what determines Y and what A is.
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- Let income in each period be labour income: Yt = wtnt
where w = wage and n=labour supply (hours worked).
- The households own the business firms.
- they own the economy`s capital (k) – this is the asset in this economy.
At=kt (financial assets? what one household holds as an asset is another household’s debt – these
cancel outeconomy-wide)
- What about profit income (call it )? it could be included as part of Y:
Yt= wtnt+ t
we assume competitive markets with constant returns to scale production (like in Solow). This means t =0.
All of a firm’s revenues are paid out as wage income to workers or capital income to the owners of the capital.
- The household budget constraint:
Ct+ At+1= Yt + (1+rt)At
is then (using At=kt, Yt=wtnt and defining ct=Ct):
ct+ kt+1 = wtnt + (1+rt)kt
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- Our earlier models of a profit maximizing competitive firm gave first order conditions for hiring labour and capital in some period t:
ptFn = wt (Fn≡∂F/∂n)ptFk = rKt (Fk≡∂F/∂k, rK rental price of K)
and note that rK= user cost of capital = pK(r+-pK/pK) where pK is the purchase price of a piece of capital.
- in the Ramsey model the prices of a unit of output (p) and the purchase price of capital (pK) have both been set equal to 1, i.e. all values
are measured in real terms.
- so the first order conditions for optimal hiring become:Fn = wt
Fk = rKt = (r+) where r+ is the user cost in this special case (with pK=1 and pK=0) and so r=Fk-
- Use these results to replace ‘w’ and ‘r’ in the budget constraint:
ct+ kt+1 = wtnt + (1+rt)kt
ct+ kt+1 = Fnnt + kt +( Fk -)kt
or: Fnnt + Fkkt = ct+ kt+1-(1-)kt
Now: F(kt,nt) = Fn nt + Fkkt
which is true for a constant returns to scale production function! (see note on p. 18)
So we have:
F(kt,nt) = ct+ kt+1-(1-)kt
In other words when markets are competitive, the constraint faced by the consumer is the same as the Dynamic Resource Constraint in the earlier version of the model.
- So the household is solving the same problem as the planner in the Ramsey model when markets are competitive.
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Appendix: Constant Returns to Scale and Fnn+Fkk = F(k,n)
With constant returns: Y=F(k,n) = n F(k/n,1)
Given this: Fn=∂F/∂n = F(k/n,1)-kFk/n
Using this expression for Fn:
Fnn+Fkk = [F(k,1)-kFk/n]∙n +Fkk
= nF(k,1) = F(k,n)
Appendix: McCandless and Equivalence of Competitive and Planner’s Model
- An alternative way of setting up the problem? (see McCandless Ch.3)
- p. 39 McCandless does a version of the Robinson Crusoe or Planner’s problem with variable labour supply.
- like above maximizes utility s.t. economy-wide budget constraint, capital accumulation equation and a labour-
leisure time constraint.
- p. 43 does a version for a competitive economy.- many identical households: units chosen so the entire population is
1 and so sum of individual variables equals aggregates.
Constraints (notation difference: his r is our rK):ct = wtnt + rKtkt - It individual budget constraint (has labour
income and income from owning firm that is not reinvested in the firm: rKk-I)
wt = Fn(Kt,Nt) profit maximization foc for n rKt = Fk(Kt,Nt) profit maximization foc for kkt+1 = (1-)kt+It capital accumulation equation
i.e. has built competitive behavior into the problem by requiring marginal products to equal w and r and has recognized
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the general equilibrium aspects by linking capital income to ownership of the firm and capital accumulation.
- using foc’s from the competitive firm’s problem as constraints is a common way of turning the problem into a competitive
problem.
- McCandless combines some of the constraints (substitutes for w and r in the budget constraint)
- McCandless shows that this gives the same result as the Planner’s problem.
(a good assignment question?)
(McCandless’ version vs. the earlier version? notice that if you use the capital accumulation constraint to replace I in his budget constraint you end up with our constraint
ct = wtnt + rKtkt - It use kt+1 = (1-)kt+It
ct = wtnt + rKtkt-kt+1+(1-)kt
ct + kt+1 = wtnt + (rKt-)kt+ kt then use rK=(r+)ct + kt+1 = wtnt + rtkt+ kt which was our constraint )
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