MACROECONOMICS BGSE/UPF

MACROECONOMICS BGSE/UPF 2008-2009
LECTURE SLIDES SET 3 Professor Antonio Ciccone I did almost the first 70 of these slides in 2 lectures (one session). BGSE/UPF, Macroeconomics, SLIDE SET 3

II. ECONOMIC GROWTH WITH ENDOGENOUS SAVINGS
BGSE/UPF, Macroeconomics, SLIDE SET 3

BGSE/UPF, Macroeconomics, 2008-09 SLIDE SET 3
1. Household savings behavior BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

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

FIGURE 1: CONSUMPTION SMOOTHING: A VOLATILE INCOME PATH HOUSEHOLD INCOME OF FARMER time BGSE/UPF, Macroeconomics, SLIDE SET 3

FIGURE 2: INCOME AND "KEYNESIAN CONSUMPTION" HOUSEHOLD INCOME OF FARMER HOUSEHOLD CONSUMPTION OF FARMER (“KEYNESIAN” theory) time BGSE/UPF, Macroeconomics, SLIDE SET 3

FIGURE 3: CONSUMPTION SMOOTHING HOUSEHOLD INCOME OF FARMER HOUSEHOLD CONSUMPTION OF FARMER (EMPIRICAL OBSERVATION) time BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

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

GRAPHICALLY: INCOME LEVELS AND CONSUMTION C[1] Lw[1] Lw[0] C[0] BGSE/UPF, Macroeconomics, SLIDE SET 3

THE INTERTEMPORAL BUDGET CONSTRAINT C[1] Lw[1] 1+r Lw[0] C[0] BGSE/UPF, Macroeconomics, SLIDE SET 3

INTERTEMPORAL UTILITY MAXIMIZATION C[1] Lw[1] 1+r Lw[0] C[0] BGSE/UPF, Macroeconomics, SLIDE SET 3

Lw[1] C[1] 1+r C[0] Lw[0] C[0] BGSE/UPF, Macroeconomics, SLIDE SET 3

BORROWING FOR CURRENT CONSUMPTION C[1] Lw[1] REPAY C[1] BORROW 1+r C[0] Lw[0] C[0] BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

"CONSUMPTION FUNCTION" C[0] 0.5*Lw[0]+0.5*Lw[1] 0.5*Lw[1] Lw[0] BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

DISCOUNTING OF FUTURE UTILITY, AND INTEREST MAXIMIZATION WITH DISCOUNTING&INTEREST with respect to C subject to INTERTEMPORAL BUDGET CONSTRAINT BGSE/UPF, Macroeconomics, SLIDE SET 3

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

INCREASE IN INTEREST RATE C[1] HIGH INTEREST RATE LOW INTEREST RATE Lw[1] C[1] 1+r C[0] C[0] Lw[0] BGSE/UPF, Macroeconomics, SLIDE SET 3

AN EXAMPLE Take the following utility function: with FIRST-ORDER CONDITION BECOMES or BGSE/UPF, Macroeconomics, SLIDE SET 3

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

TIMING YOU ARE HERE C[0] C[1] C[2] t=0 t=1 t=2 w[0]L w[1]L w[2]L Q[0] 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 - interest r[0] - utility discount - interest r[1] - utility discount INITIAL WEALTH BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

INTERTEMPORAL BUDGET CONSTRAINT BGSE/UPF, Macroeconomics, SLIDE SET 3

BUDGET CONTRAINT AND TIME EVOLUTION OF WEALTH t=0 t=1 t=2 C[3] C[0] C[1] C[2] Q[0] w[0]L w[1]L w[2]L BGSE/UPF, Macroeconomics, SLIDE SET 3

INTERTEMPORAL BUDGET CONSTRAINT BGSE/UPF, Macroeconomics, SLIDE SET 3

THE “PRESENT-VALUE BUDGET SURPLUS” = PERMANENT INCOME minus PRESENT VALUE CONSUMPTION = PRESENT VALUE OF END-OF-LIFE WEALTH BGSE/UPF, Macroeconomics, SLIDE SET 3

OPTIMAL SOLUTION OF CONSUMPTION PROBLEM MAXIMIZE BETWEEN ADJACENT PERIODS plus BUDGET CONSTRAINT WITH EQUALITY BGSE/UPF, Macroeconomics, SLIDE SET 3

INFINITE HORIZON =TIME ZERO (PRESENT) VALUE OF 1 EURO PAID AT (end of) PERIOD t BGSE/UPF, Macroeconomics, SLIDE SET 3

INTERTEMPORAL BUDGET CONSTRAINT NO-PONZI-GAME condition BGSE/UPF, Macroeconomics, SLIDE SET 3

WHAT IF: e TIME T BGSE/UPF, Macroeconomics, SLIDE SET 3

CAN INCREASE TIME-0 CONSUMPTION  CONSUMPTION PLAN NOT OPTIMAL!
NECESSARY FOR OPTIMALITY: BGSE/UPF, Macroeconomics, SLIDE SET 3

TIME CONSISTENCY of HOUSOLD CONSUMPTION PLANS BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

2. Intertemporal budget constraint Wealth in discrete time Wealth in continuous time BGSE/UPF, Macroeconomics, SLIDE SET 3

Intertemporal budget constraint in continuous time satisfied with equality if BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

CONSTANT CONSUMPTION IN TIME OPTIMAL CONSUMPTION PATH r = r C(t) C(0) TIME BGSE/UPF, Macroeconomics, SLIDE SET 3

INCREASING CONSUMPTION IN TIME OPTIMAL CONSUMPTION PATH r > r C(t) C(0) TIME BGSE/UPF, Macroeconomics, SLIDE SET 3

DEACREASING CONSUMPTION IN TIME OPTIMAL CONSUMPTION PATH r < r C(0) C(t) TIME BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

THE INTERTEMPORAL BUDGET CONSTRAINT without initial wealth
HENCE BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

Take the following utility function: with BGSE/UPF, Macroeconomics, SLIDE SET 3

FIRST ORDER CONDITIONS FOR THE TWO PERIODS IN TIME making use of the utility function BGSE/UPF, Macroeconomics, SLIDE SET 3

REWRITING THIS CONDITIONS YIELDS subtracting 1 from both sides BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

Apply Hopital’s rule BGSE/UPF, Macroeconomics, SLIDE SET 3

HENCE as two periods become VERY CLOSE WHICH IS WHAT WE WANTED TO SHOW BGSE/UPF, Macroeconomics, SLIDE SET 3

SUMMARIZING QUESTION: What characterizes the optimal consumption PATH that solves subject to BGSE/UPF, Macroeconomics, SLIDE SET 3

ANSWER: and or BGSE/UPF, Macroeconomics, SLIDE SET 3

2. The Ramsey-Cass-Koopmans model
BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

CAPITAL MARKET EQUILIBRIUM E(4) E(5) BGSE/UPF, Macroeconomics, SLIDE SET 3

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

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

SO WE HAVE OUR TWO EQUATIONS: and BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

k-ISOCLINE c k-ISOCLINE: CAPITAL DOES NOT GROW k BGSE/UPF, Macroeconomics, SLIDE SET 3

CHANGES IN k in PHASE DIAGRAM c k-ISOCLINE: CAPITAL DOES NOT GROW k BGSE/UPF, Macroeconomics, SLIDE SET 3

Continue with the optimal consumption equation FIRST: Find ISOCLINE, which are the (c, k) combinations such that or BGSE/UPF, Macroeconomics, SLIDE SET 3

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

c-ISOCLINE: NO CONSUMPTION GROWTH k-ISOCLINE: NO CAPITAL GROWTH k k* BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

(1) Wealth=Capital Q(t)=K(t) or q(t)=k(t) (2) Intertemporal budget constraint with equality BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

NEGATIVE INTEREST RATE time t BGSE/UPF, Macroeconomics, 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/UPF, Macroeconomics, SLIDE SET 3

EQUILIBRIUM (“SADDLE”) PATH c c-ISOCLINE: NO CONSUMPTION GROWTH k-ISOCLINE: NO CAPITAL GROWTH k(0) k k* BGSE/UPF, Macroeconomics, SLIDE SET 3

SADDLE PATH SATISFIES INTERTEMPORAL BUDGET CONSTRAINT Capital market equilibrium: Income per worker=Labor income + Capital income: Hence: BGSE/UPF, Macroeconomics, SLIDE SET 3

Moreover: As: given that interest rates>0 for k<=k* BGSE/UPF, Macroeconomics, SLIDE SET 3

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

OPTIMAL AND EQUILIBRIUM ALLOCATION c c-ISOCLINE: NO CONSUMPTION GROWTH k-ISOCLINE: NO CAPITAL GROWTH k(0) k k* BGSE/UPF, Macroeconomics, SLIDE SET 3

MACROECONOMICS BGSE/UPF

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