ITFD Growth and Development UPF

ITFD Growth and Development UPF 2009-2010
LECTURE SLIDES SET 3 Professor Antonio Ciccone I did almost the first 70 of these slides in 2 lectures (one session). BGSE Growth and Development, SLIDE SET 3

II. ECONOMIC GROWTH WITH ENDOGENOUS SAVINGS
BGSE Growth and Development, SLIDE SET 3

BGSE Growth and Development, 2009-10 SLIDE SET 3
1. Household savings behavior BGSE Growth and Development, SLIDE SET 3

1. “Keynesian theory” of savings and consumption
1. The Keynesian consumption (savings) function So far we assumed a “Keynesian” savings function where s is the marginal propensity to save. BGSE Growth and Development, SLIDE SET 3

Because of the BUDGET CONSTRAINT this implies the “Keynesian” consumption function where c is the marginal propensity to consume. BGSE Growth and Development, SLIDE SET 3

2. Limitations CONCEPTUAL The consumption behavior is assumed to be “mechanic” and “short-sighted”: Are households really only looking at CURRENT income when deciding consumption? Not really. Many households borrow from banks in order to be able to consume more today because they know they will be able to pay the money back in the future. If people save, presumably they are doing this for future consumption. Hence, savings is a FORWARD-LOOKING decision and must take into account what happens in the future. BGSE Growth and Development, SLIDE SET 3

Assuming savings as a function of current income therefore appears to contradict the use that households make of their savings. EMPIRICAL “Consumption smoothing:” Empirically, we observe that households smooth consumption. To put it differently, the income of households is often more volatile than their consumption. This suggests that households look forward and try to stabilize consumption (their standard of living) as much as they can. BGSE Growth and Development, SLIDE SET 3

FIGURE 1: CONSUMPTION SMOOTHING: A VOLATILE INCOME PATH HOUSEHOLD INCOME OF FARMER time BGSE Growth and Development, SLIDE SET 3

FIGURE 2: INCOME AND "KEYNESIAN CONSUMPTION" HOUSEHOLD INCOME OF FARMER HOUSEHOLD CONSUMPTION OF FARMER (“KEYNESIAN” theory) time BGSE Growth and Development, SLIDE SET 3

FIGURE 3: CONSUMPTION SMOOTHING HOUSEHOLD INCOME OF FARMER HOUSEHOLD CONSUMPTION OF FARMER (EMPIRICAL OBSERVATION) time BGSE Growth and Development, SLIDE SET 3

FIGURE 4: SAVINGS AND DIS-SAVINGS IN CONSUMPTION SMOOTHING MODELS HOUSEHOLD INCOME CONSUMPTION SMOOTHING DIS-SAVE TO MAINTAIN CONSUMPTION LEVELS SAVE FOR “RAINY DAYS” time BGSE Growth and Development, SLIDE SET 3

INTERESTINGLY: The Keynesian theory of consumption seems to do better at the aggregate level than at the level of individual households. For example: Keynesian theory does well in describing relationship between consumption and income of a country at different in different years Theory does also well in describing relationship between consumption and income across different countries BGSE Growth and Development, SLIDE SET 3

A PUZZLE? CONSUMPTION AGGREGATE LEVEL Germany 1980 Or Country 3 INDIVIDUAL HOUSE- HOLD LEVEL Ms B Ms D Mr A Mr C Germany 1960 Or Country 2 Germany 1950 Or Country 1 INCOME BGSE Growth and Development, SLIDE SET 3

2. The permanent income theory of consumption and savings
1. Basic idea and two-period model Households make consumption decisions: LOOKING FORWARD to future USING SAVINGS AND LOANS from BANKS to maintain their living standards STABLE in time to the extent possible BGSE Growth and Development, SLIDE SET 3

SIMPLEST POSSIBLE formal model (2 PERIODS) INGREDIENTS: Household lives 2 periods and tries to maximize INTERTEMPORAL utility Understands that will earn LABOR income Lw[0] in period 0 and Lw[1] in period 1 Starts with 0 WEALTH Can save and borrow from bank at interest rate r BGSE Growth and Development, SLIDE SET 3

MATHEMATICAL MAXIMIZATION PROBLEM: by choosing C0 and C1 subject to S=Lw0-C0 C1=Lw1+(1+r)S DISCOUNT APPLIED TO FUTURE UTILITY NOTE that S can be NEGATIVE (which means the household is BORROWING or DISSAVING) BGSE Growth and Development, SLIDE SET 3

MATHEMATICAL FORMULATION Maximize INTERTEMPORAL UTILITY by choosing C subject to INTERTEMPORAL BUDGET CONSTRAINT C1=Lw1+(1+r)S= Lw1+(1+r)(Lw0-C0) Elaborate on Budget COnstraint BGSE Growth and Development, SLIDE SET 3

INTERTEMPORAL BUDGET CONSTRAINT can also be written: IMPORTANT TERMINOLOGY: PERMANENT INCOME (PI) PRICE OF FUTURE CONSUMPTION RELATIVE TO CURRENT CONSUMPTION BGSE Growth and Development, SLIDE SET 3

GRAPHICALLY: INCOME LEVELS AND CONSUMTION C[1] Lw[1] Lw[0] C[0] BGSE Growth and Development, SLIDE SET 3

THE INTERTEMPORAL BUDGET CONSTRAINT C[1] Lw[1] 1+r Lw[0] C[0] BGSE Growth and Development, SLIDE SET 3

INTERTEMPORAL UTILITY MAXIMIZATION C[1] Lw[1] 1+r Lw[0] C[0] BGSE Growth and Development, SLIDE SET 3

C[1] Lw[1] C[1] 1+r C[0] Lw[0] C[0] BGSE Growth and Development, SLIDE SET 3

BORROWING FOR CURRENT CONSUMPTION C[1] Lw[1] REPAY C[1] BORROW 1+r C[0] Lw[0] C[0] BGSE Growth and Development, SLIDE SET 3

2. Closed form solution in a simple case SUPPOSE THAT INTEREST RATE is ZERO: r = 0 FUTURE UTILITY DISCOUNT is ZERO: MAXIMIZATION PROBLEM BECOMES: with respect to C subject to BGSE Growth and Development, SLIDE SET 3

FIRST ORDER MAXIMIZATION CONDITIONS: First-order conditions can be obtained from with respect to C0 where we have substituted the budget constraint. TAKE DERIVATIVE WITH RESPECT TO C[1] AND SET EQUAL ZERO: OR C1 BGSE Growth and Development, SLIDE SET 3

EQUALIZE MARGINAL UTILITY AT DIFFERENT POINTS IN TIME THIS IMPLIES  “PERFECT CONSUMPTION SMOOTHING” Using the INTERTEMPORAL BUDGET CONSTRAINT yields consumption as a function of PERMANENT INCOME BGSE Growth and Development, SLIDE SET 3

"CONSUMPTION FUNCTION" C[0] 0.5*Lw[0]+0.5*Lw[1] 0.5*Lw[1] Lw[0] BGSE Growth and Development, SLIDE SET 3

THE EFFECT OF AN INCREASE IN INITIAL-PERIOD INCOME ON C[0] “TEMPORARY” INCREASE IN INCOME C[0] 0.5*Lw[0]+0.5*Lw[1] 0.5*Lw[1] INCREASE In first-period income Lw[0] BGSE Growth and Development, SLIDE SET 3

THE EFFECT OF AN INCREASE IN INITIAL AND FUTURE INCOME “PERMANENT” INCREASE IN INCOME C[0] 0.5*Lw[0]+0.5*Lw[1] INCREASE Lw[1] INCREASE Lw[0] Lw[0] BGSE Growth and Development, SLIDE SET 3

DISCOUNTING OF FUTURE UTILITY, AND INTEREST MAXIMIZATION WITH DISCOUNTING&INTEREST with respect to C subject to INTERTEMPORAL BUDGET CONSTRAINT BGSE Growth and Development, SLIDE SET 3

FIRST-ORDER CONDITIONS “EFFECTIVE TIME DISCOUNTING”  CONSTANT CONSUMPTION DISCOUNTING OF FUTURE UTILITY AND POSTITIVE INTEREST RATE JUST OFFSET BGSE Growth and Development, SLIDE SET 3

UPWARD SLOPING CONSUMPTION PATHS IN TIME:  INCREASING CONSUMPTION OVER TIME POSITIVE INTEREST MORE THAN OFFSETS UTILITY DISCOUNTING DOWNWARD SLOPING CONSUMPTION PATHS IN TIME:  DECREASING CONSUMPTION OVER TIME UTILITY DISCOUNTING MORE THAN OFFSETS POSITIVE INTEREST (1-β)(1+r) > 1 (1-β)(1+r) < 1 BGSE Growth and Development, SLIDE SET 3

AN EXAMPLE Take the following utility function: FIRST-ORDER CONDITION BECOMES or BGSE Growth and Development, SLIDE SET 3

3. The case of 3 and more periods -- Timing -- Intertemporal budget constraint -- Optimality conditions -- Time consistency BGSE Growth and Development, SLIDE SET 3

PRESENT-VALUE INCOME AND CONSUMPTION C[0] C[1] C[2] YOU ARE HERE t=0 t=1 t=2 interest discounting interest discounting interest discounting Q[0] w[0]L w[1]L w[2]L - PERMANENT INCOME THERE IS A BIT OF A TIMING ISSUE HERE: Are you being paid salary beginning or end of period? HERE BEGINNING OF PERIOD! BUT BETTER END OF PERIOD!!!!!!!!!!!!!! NEEDS TO BE CHANGED. In thiS CASE Q refers to beginning of period. Put in graph of when INTERED IS PAID - PRESENT VALUE CONSUMPTION BGSE Growth and Development, SLIDE SET 3

INTERTEMPORAL BUDGET CONSTRAINT BGSE Growth and Development, SLIDE SET 3

BUDGET CONTRAINT AND TIME EVOLUTION OF WEALTH t=0 t=1 t=2 C[2] C[0] C[1] Q[0] w[0]L w[1]L w[2]L BGSE Growth and Development, SLIDE SET 3

INTERTEMPORAL BUDGET CONSTRAINT BGSE Growth and Development, SLIDE SET 3

OPTIMAL SOLUTION OF CONSUMPTION PROBLEM MAXIMIZE BETWEEN ADJACENT PERIODS plus BUDGET CONSTRAINT WITH EQUALITY BGSE Growth and Development, SLIDE SET 3

Shortest way from A to B? B A BGSE Growth and Development, SLIDE SET 3

Shortest way from A to B B A BGSE Growth and Development, SLIDE SET 3

Must be the shortest way between ANY two points
C D A BGSE Growth and Development, SLIDE SET 3

INFINITE HORIZON =TIME ZERO (PRESENT) VALUE OF 1 EURO PAID AT (end of) PERIOD t BGSE Growth and Development, SLIDE SET 3

INTERTEMPORAL BUDGET CONSTRAINT NO-PONZI-GAME condition BGSE Growth and Development, SLIDE SET 3

WHAT IF: NO PONZI GAME CONDITION VIOLATED? -e TIME T BGSE Growth and Development, SLIDE SET 3

INTERTEMPORAL BUDGET CONSTRAINT WITH EQUALITY NO-PONZI-GAME condition BGSE Growth and Development, SLIDE SET 3

WHAT IF: e TIME T BGSE Growth and Development, SLIDE SET 3

CAN INCREASE TIME-0 CONSUMPTION  CONSUMPTION PLAN NOT OPTIMAL!
NECESSARY FOR OPTIMALITY: BGSE Growth and Development, SLIDE SET 3

TIME CONSISTENCY of HOUSOLD CONSUMPTION PLANS BGSE Growth and Development, SLIDE SET 3

TIME 0 CONSUMPTION PLANS C[0] C[1] C[2] YOU ARE HERE t=0 t=1 t=2 interest discounting interest discounting interest discounting Q[0] w[0]L w[1]L w[2]L TIME 1 CONSUMPTION PLANS (NO NEW INFO) C[1] C[2] YOU ARE HERE THERE IS A BIT OF A TIMING ISSUE HERE: Are you being paid salary beginning or end of period? HERE BEGINNING OF PERIOD! BUT BETTER END OF PERIOD!!!!!!!!!!!!!! NEEDS TO BE CHANGED. In thiS CASE Q refers to beginning of period. Put in graph of when INTERED IS PAID t=0 t=1 t=2 interest discounting interest discounting Q[0] Q(1) w[1]L w[2]L BGSE Growth and Development, SLIDE SET 3

***** TIME CONSISTENCY ***** C[0] C[1] C[2] YOU ARE HERE t=0 t=1 t=2 interest discounting interest discounting interest discounting Q[0] w[0]L w[1]L w[2]L TIME 1 CONSUMPTION PLANS (NO NEW INFO) C[1] C[2] YOU ARE HERE THERE IS A BIT OF A TIMING ISSUE HERE: Are you being paid salary beginning or end of period? HERE BEGINNING OF PERIOD! BUT BETTER END OF PERIOD!!!!!!!!!!!!!! NEEDS TO BE CHANGED. In thiS CASE Q refers to beginning of period. Put in graph of when INTERED IS PAID t=0 t=1 t=2 interest discounting interest discounting Q(1) w[1]L w[2]L BGSE Growth and Development, SLIDE SET 3

3. Optimal consumption and savings in continuous time 1. Infinite horizon subject to = TIME ZERO (PRESENT) VALUE OF 1 EURO PAID AT TIME t Here still OK on budget constraint BGSE Growth and Development, SLIDE SET 3

2. Intertemporal budget constraint Wealth in discrete time Wealth in continuous time BGSE Growth and Development, SLIDE SET 3

Intertemporal budget constraint in continuous time satisfied with equality if BGSE Growth and Development, SLIDE SET 3

3. Interpretation of r and r
r is the interest rate that is received between two very close periods in time is the discount rate applied PER UNIT OF TIME between two very close periods in time TO SEE THAT is the discount rate applied PER UNIT OF TIME between two very close periods in time Note that the utility discount between period 0 and t is: BGSE Growth and Development, SLIDE SET 3

2) Hence the utility discount per unit of time is: 3) What is the limit as t0? Hopital’s rule yields BGSE Growth and Development, SLIDE SET 3

4. First-order condition where: is INTERTEMPORAL RATE OF TIME PREFERENCE and measures how IMPATIENT people are is the INTERTEMPORAL ELASTICITY OF SUBSTITUTION and measures how much future consumption increases when the interest rate goes up (how much people “respond to interest rates”) BGSE Growth and Development, SLIDE SET 3

CONSTANT CONSUMPTION IN TIME OPTIMAL CONSUMPTION PATH r = r C(t) C(0) TIME BGSE Growth and Development, SLIDE SET 3

INCREASING CONSUMPTION IN TIME OPTIMAL CONSUMPTION PATH r > r C(t) C(0) TIME BGSE Growth and Development, SLIDE SET 3

DEACREASING CONSUMPTION IN TIME OPTIMAL CONSUMPTION PATH r < r C(0) C(t) TIME BGSE Growth and Development, SLIDE SET 3

5. Closed form solution in special case ASSUME (consumers have an INFINITE HORIZON) SOLUTION CHARACERIZED BY  PEOPLE WANT CONSTANT CONSUMPTION OVER TIME (“PERFECT CONSUMPTION SMOOTHING” CASE) BGSE Growth and Development, SLIDE SET 3

THE INTERTEMPORAL BUDGET CONSTRAINT without initial wealth
HENCE BGSE Growth and Development, SLIDE SET 3

6. Deriving the continuous time first-order condition MAXIMIZATION BETWEEN ANY TWO PERIODS SEPARATED BY TIME x subject to = TOTAL SPENDING IN TWO PERIODS BGSE Growth and Development, SLIDE SET 3

Take the following utility function: with Take the following utility function: with Take the following utility function: BGSE Growth and Development, SLIDE SET 3

FIRST ORDER CONDITIONS FOR THE TWO PERIODS IN TIME making use of the utility function BGSE Growth and Development, SLIDE SET 3

REWRITING THIS CONDITIONS YIELDS subtracting 1 from both sides BGSE Growth and Development, SLIDE SET 3

DIVIDE BY x (the TIME BETWEEN THE TWO PERIODS) to get CONSUMPTION GROWTH PER UNIT OF TIME What happens when the two periods get closer and closer (x0)? BGSE Growth and Development, SLIDE SET 3

Apply Hopital’s rule BGSE Growth and Development, SLIDE SET 3

HENCE as two periods become VERY CLOSE WHICH IS WHAT WE WANTED TO SHOW BGSE Growth and Development, SLIDE SET 3

SUMMARIZING QUESTION: What characterizes the optimal consumption PATH that solves subject to BGSE Growth and Development, SLIDE SET 3

ANSWER: and or BGSE Growth and Development, SLIDE SET 3

2. The Ramsey-Cass-Koopmans model
BGSE Growth and Development, SLIDE SET 3

1. Equilibrium growth with infinite-horizon households We will now integrate a household that chooses consumption optimally over an infinite horizon in the Solow model. The results is often refereed to as the Cass-Koopmans model. The Cass-Koopmans model is exactly like the SOLOW MODEL only that the household does NOT behave mechanically but instead chooses consumption and savings to maximize: subject to where BGSE Growth and Development, SLIDE SET 3

In order to NOT complicate things too much we will simplify the model by assuming: no technological changes (i.e. a=0 in Solow model) no population growth (i.e. n=0 in Solow model) BGSE Growth and Development, SLIDE SET 3

1. Technology and the capital market WHAT WE CAN KEEP FROM THE SOLOW MODEL CONSTANT RETURNS PRODUCTION FUNCTION E(1) E(2) CAPITAL ACCUMULATION EQUATION E(3) PRODUCTION FUNCTION BGSE Growth and Development, SLIDE SET 3

CAPITAL MARKET EQUILIBRIUM E(4) E(5) BGSE Growth and Development, SLIDE SET 3

2. Household behaviour WHAT WE CANNOT KEEP IS INSTEAD: E(6) E(7) INTERTEMPORAL BUDGET CONSTRAINT WITH EQUALITY where c[t] is CONSUMPTION per PERSON BGSE Growth and Development, SLIDE SET 3

3. Dynamic equilibrium system WE WILL TRY TO CHARACTERIZE THE EQUILIBRIUM OF THIS ECONOMY IN TERMS OF THE EVOLUTION OF c and k. The goal is to reduce the equations above to a TWO-DIMENSIONAL DIFFERENTIAL EQUATION SYSTEM WHERE CHANGE in CONSUMPTION c=FUNCTION OF k and c CHANGE IN CAPITAL k=FUNCTION OF k and c (E6) and (E5) imply E(8) BGSE Growth and Development, SLIDE SET 3

(E3) and (E4) imply recall that there is NO population growth and therefore E(9) BGSE Growth and Development, SLIDE SET 3

SO WE HAVE OUR TWO EQUATIONS: BGSE Growth and Development, SLIDE SET 3

2. Equilibrium growth and optimality THESE CAN BE BEST ANALYZED IN A PHASE DIAGRAM Start with capital accumulation equation FIRST: Find ISOCLINE, which are the (c, k) combinations such that INTERPRETATION: capital per worker does NOT grow IF the economy consumes all of the output net of capital depreciation. In this case, investment is just enough to cover the depreciation of capital. BGSE Growth and Development, SLIDE SET 3

k-ISOCLINE c k-ISOCLINE: CAPITAL DOES NOT GROW k BGSE Growth and Development, SLIDE SET 3

CHANGES IN k in PHASE DIAGRAM c k-ISOCLINE: CAPITAL DOES NOT GROW k BGSE Growth and Development, SLIDE SET 3

Continue with the optimal consumption equation FIRST: Find ISOCLINE, which are the (c, k) combinations such that BGSE Growth and Development, SLIDE SET 3

c-ISOCLINE c-ISOCLINE: CONSUMPTION DOES NOT GROW c k* is the k such that f’(k)=d+r k BGSE Growth and Development, SLIDE SET 3

CHANGES IN c in PHASE DIAGRAM c-ISOCLINE: CONSUMPTION DOES NOT GROW c k* is the k such that f’(k)=d+r k BGSE Growth and Development, SLIDE SET 3

CHANGES IN c in PHASE DIAGRAM c c-ISOCLINE: CONSUMPTION DOES NOT GROW k* is the k such that f’(k)=d+r k BGSE Growth and Development, SLIDE SET 3

PUTTING CHANGES in k and c TOGETHER c c-ISOCLINE: NO CONSUMPTION GROWTH k-ISOCLINE: CAPITAL DOES NOT GROW k k* BGSE Growth and Development, SLIDE SET 3

c c-ISOCLINE: NO CONSUMPTION GROWTH k-ISOCLINE: NO CAPITAL GROWTH k k* BGSE Growth and Development, SLIDE SET 3

All these paths satisfy by construction: period-by-period consumer maximization capital market equilibrium They DO NOT necessarily satisfy constraints like: non-negative capital stock k[t]>=0 intertemporal budget constraint with EQUALITY BGSE Growth and Development, SLIDE SET 3

PATHS that violate NON-NEGATIVE capital stock (consume too much in beginning) c-ISOCLINE: NO CONSUMPTION GROWTH c k-ISOCLINE: NO CAPITAL GROWTH k(0) k k* BGSE Growth and Development, SLIDE SET 3

PATHS THAT DO NOT SATISFY BUDGET CONSTRAINT WITH EQUALITY (consume too little in beginning) c-ISOCLINE: NO CONSUMPTION GROWTH c k-ISOCLINE: NO CAPITAL GROWTH k_bar k(0) k k* BGSE Growth and Development, SLIDE SET 3

(1) Wealth=Capital Q(t)=K(t) or q(t)=k(t) (2) Intertemporal budget constraint with equality BGSE Growth and Development, SLIDE SET 3

PATHS THAT DO NOT SATISFY BUDGET CONSTRAINT WITH EQUALITY c-ISOCLINE: NO CONSUMPTION GROWTH c f’(k)-d=r=0 f(k)-dk k_bar k(0) k* k POSITIVE INTEREST NEGATIVE INTEREST RATE BGSE Growth and Development, SLIDE SET 3

NEGATIVE INTEREST RATE time t BGSE Growth and Development, SLIDE SET 3

PATHS THAT DO NO SATISFY BUDGET CONSTRAINT WITH EQUALITY c-ISOCLINE: NO CONSUMPTION GROWTH c YOU ARE NOT SPENDING ALL YOUR PERMANENT INCOME!!!!!!! k_bar k(0) k k* BGSE Growth and Development, SLIDE SET 3

EQUILIBRIUM (“SADDLE”) PATH c c-ISOCLINE: NO CONSUMPTION GROWTH k-ISOCLINE: NO CAPITAL GROWTH k(0) k k* BGSE Growth and Development, SLIDE SET 3

SADDLE PATH SATISFIES INTERTEMPORAL BUDGET CONSTRAINT WITH EQUALITY Capital market equilibrium: Income per worker=Labor income + Capital income: Hence: BGSE Growth and Development, SLIDE SET 3

Moreover: As: given that interest rates>0 for k<=k* BGSE Growth and Development, SLIDE SET 3

OPTIMALITY -- What would social planner do? - Social planner: dictator who decides allocation according to HH welfare subject to physical contraints BGSE Growth and Development, SLIDE SET 3

The GLOBALLY OPTIMAL PATH MUST SATISFY MRS=MRT (A) If not satisfied, the planner could increase utility between adjacent periods by either: -- consuming one unit less today, investing that unit, and consuming the resulting additional output tomorrow -- consuming one unit more today, invest one unit less today, and reducing future consumption accordingly BGSE Growth and Development, SLIDE SET 3

The GLOBALLY OPTIMAL PATH MUST SATISFY RESOURCE CONSTRAINT (B) To see why, suppose first that -- in this case the planner must be throwing away goods (investment goods) because the increase in the number of machines is LESS THAN the machines built less depreciation : BUT THROWING AWAY GOODS CANNO BE OPTIMAL!! Now suppose instead -- now the planner is a REAL MAGICIAN!! as the number of machines in the economy goes up by which is GREATER THAN machines built less depreciation BGSE Growth and Development, SLIDE SET 3

ALL THE PATHS THAT SATISFY CONDITIONS (A) and (B) c c-ISOCLINE: NO CONSUMPTION GROWTH k-ISOCLINE: NO CAPITAL GROWTH k k* BGSE Growth and Development, SLIDE SET 3

NOW NOTE: -- Starting the allocation by jumping ABOVE the SADDLE PATH CANNOT BE OPTIMAL because you end up violating the non-negativity constraint for capital -- Starting the allocation by jumping BELOW the SADDLE PATH CANNOT BE OPTIMAL either. The proof is to construct another path—that is clearly not optimal either—but that still is BETTER THAN the paths starting out below the saddle path. How to do that is explained on the next slides. BGSE Growth and Development, SLIDE SET 3

We are trying to show that the RED PATH CANNOT BE GLOBALLY OPTIMAL c-ISOCLINE: NO CONSUMPTION GROWTH c k-ISOCLINE: NO CAPITAL GROWTH k(0) k k* BGSE Growth and Development, SLIDE SET 3

CONSIDER THE ALTERNATIVE GREEN PATH, which: -- concides with RED PATH until k* is reached and then JUMPS UP to the green dot where is stay forever c-ISOCLINE: NO CONSUMPTION GROWTH c k-ISOCLINE: NO CAPITAL GROWTH k(0) k k* BGSE Growth and Development, SLIDE SET 3

-- The GREEN PATH CANNOT POSSIBLY BE OPTIMAL because consumption JUMPS and therefore the green path violates CONSUMPTION SMOOTHING, which was CONDITION A above. -- Still, the GREEN PATH is certaintly better than the RED PATH because it has the same consumption until k* and MORE consumption from there onwards!!! -- For all RED PATHS (that is, all paths starting below the saddle path), there is a GREEN PATH. So no paths starting below the saddle path can be optimal (despite the fact that it satisfies conditions A and B). BGSE Growth and Development, SLIDE SET 3

HENCE: The only path starting at k[0] that : -- satisfies CONDITIONS A and B, which are necessary for optimality -- satisfies non-negativity of capital -- satisfies that there is NO OTHER PATH we can construct that is better IS THE SADDLE PATH EQUILIBRIUM AND OPTIMAL ALLOCATIONS ARE EQUAL BGSE Growth and Development, SLIDE SET 3

OPTIMAL AND EQUILIBRIUM ALLOCATION c c-ISOCLINE: NO CONSUMPTION GROWTH k-ISOCLINE: NO CAPITAL GROWTH k(0) k k* BGSE Growth and Development, SLIDE SET 3

In the steady state: Savings rate is constant, just like in the Solow model But it is endogenous in the sense of depending on “fundamentals” like time preference etc. In the simplest case: S=I=dK Combined with: r+d=MPK and r=r BGSE Growth and Development, SLIDE SET 3

In the steady state with technological change:
With technological change and population growth: S=I=(d+a)K Growth of consumption=s(r-r)  What is the relationship between the SS savings rate S/Y and the rate of technological change a? BGSE Growth and Development, SLIDE SET 3

Comparative statics Greater impatience (discount rate)? (effects on income, capital, wages, interest rates) Capital income taxation? A temporary cut of lump-sum taxes? BGSE Growth and Development, SLIDE SET 3

ITFD Growth and Development UPF

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