Preliminary Material: Some Basic Models

- This section looks at some basic models:

One-person, One-period economy (Robinson Crusoe Model)

Two-period, inter-temporal model;

Two-period Robinson Crusoe (with capital).

- Why? Methods and results are similar to more complex models.

- Some Sources:

Doepke, Lehnert and Sellgren Macroeconomics Ch. 2, 3 (by Sellgren)

Williamson Macroeconomics 4th ed. Math. Appendix p. 664-72, 677-80

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Robinson Crusoe Model: A One-Person Economy

- Crusoe lives alone on an island and is the only producer and only consumer.

Production

- Crusoe’s production function for output (y) is:

y = F(K, L)

K = quantity of capital input (machines, buildings, tools etc. could include natural resources)

L = quantity of labour (how much Crusoe works)

- Assume both inputs are always productive (marginal products of labour and capital are positive):

∂ y∂ K

≡ FK> 0∧∂ y∂ L

≡ F L>0

- Common to assume diminishing returns (second derivatives negative):

∂2 y∂ K 2 ≡ FKK < 0∧∂2 y

∂L2 ≡ F ¿<0 (vs. corner solutions)

- The Cobb-Douglas function is an example:

y = A K L withand A>0e.g for labour:

FL = A K L A (K/L) >0

FLL = ( A K L(since )

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- Assume K is fixed in this model e.g. Crusoe has only so much land on the island and tools from the shipwreck.

- could define A’ =A Kthen the Cobb-Douglas production function can be written as:

y = A’ Lwith FL = A’ L >0 FLL = ( A’ L

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Preferences:

- Crusoe’s well being (utility) is determined by amount of consumption (c) and amount of leisure time (time not worked, l).

Total time =1 , L is time spent working, l=1-L is leisure time.

- Utility function:U = U(c,l)

c and l are goods so:

∂U∂ c

≡U c>0∧∂U

∂ l≡U l>0

(Doepke et al: use U(c,L) – note that UL<0 -- a bad not a good!)

- typically we assume that the second derivatives of c and l are < 0.(diminishing marginal utility)

- indifference curves (combinations of c and l with same U):

i.e. these are the level curves for the function U(c,l).

along an indifference curve c, l change so as to keep U constant, so:

dU = 0 = Uc dc + Ul dl

slope: dcdl =

−U l

U c <0 ( -(marginal rate of substitution) )

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- Marginal rate of substitution (MRS) = U l

U c

MRS measures the value of a unit of l in terms of c.

e.g. if U l

U c =2 c per l, then the person is willing to trade up to

2 c for one l.

- Example in Doepke et al:

U = ln(c) + ln(l) (with l=1-L)

( MRS = c/l )

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Crusoe’s Problem:

- Constraints:

- Say that all production is consumed: y =c (budget constraint for this economy)

- Crusoe’s possible choices are also constrained by:- limited time (1 unit of total time to allocate between L and l)- production function: translates time into production (=consumption)

- So constraints faced are:

1 = L + l y=F(L) (or if you prefer F(K,L) with K fixed)y=c (all production is consumed)

- Crusoe faces a constrained maximization problem:

max U(c, l) s.t. y=F(L) , y=c and 1=L+l<c, y,l,L>

(c, y, L, l are variables to be chosen)

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- Solving:

(1) Could substitute the constraints into the objective function

i.e. combine constraints to get: c= F(1-l ) then substitute for c in the utility fn.

then:

max U( F(1-l), l) <l>

- this is now an unconstrained problem (substitution imposes constraints) with one choice variable (l).

- first-order condition for a maximum:

dUdl = -Uc FL + Ul = 0

- So optimal choice requires:

U l

U c = FL

(with constraints substituted into U this implies a value of l (call it: l*). Given l* you can use the constraints to find optimal L (L*), y (y*) and c (c*) )

- 2nd order condition for maximum requires: d2Udl2 <0

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(2) Use Lagrange’s method: set up a Lagrangian

- For example you could combine the constraints into a single-constraint:

i.e.rewrite the constraints ( y=c, 1=L+l, y=F(L) ):

c = F(1-l) as: F(1-l)-c =0

- then define a Lagrangian function (L):

L(c,l,)= U(c,l) + [ F(1-l)-c) ]

= Lagrange multiplier

- maximizing this WRT c, l and gives the f.o.c’s:

c: ∂ L∂ c = Uc – =0

l: ∂ L∂ l = Ul – FL=0

: ∂ L∂ λ = F(1-l) -c = 0

- combining f.o.c.’s for c and l gives:

U l

U c = FL (same as first method: so along

with constraints gives the same solution c*, l*)

- Interpretation of the Lagrange multiplier here?

- value (in terms of the objective) of relaxing the constraint by a marginal amount.

- value in utility of an extra unit of c (marginal utility of c)

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- notice f.o.c. for c says this too: Uc = - Interpretation of the condition WRT l:

Ul = FL

value of time as leisure (LHS) = value of time in terms of utility of extra output (RHS).

(units? measured in terms of utility)

- Interpretation of: U l

U c = FL

- dcdl in terms of utility = - dc

dl in terms of production

i.e. tradeoff between the two goods in terms of utility must equal tradeoff between the two goods in production.

- LHS is the value of extra l to Robinson Crusoe (amount of c could trade for one l and still be just as well off)

e.g. say U l

U c =2 c per l then Crusoe is willing to give up as

much as 2 c to get an extra l.

- RHS is the amount of output Crusoe must give up to have an extra unit of l.

- So if the condition is not true Crusoe can alter his choice and raise U

e.g. say U l

U c =2 c per l > FL =1 c per l

can raise l by 1, this means producing and consuming 1

less c but U l

U c =2 so the extra l is worth 2 c to Crusoe –

Crusoe must be better off!

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- So the condition says the same thing as Ul = FL but measured in different units (in terms of c rather than U).

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- Diagram: solution is tangency of Production fn. and indifference curve

(notice: L is measured from right-to-left so the production functionhas been ‘flipped over’ to reflect this)

- The outcome implies optimal c (c*) and optimal l (l*) as functions of the "exogenous" variables.

- Exogenous variables here? parameters of the production and utility functions.

- In the Lagrangian above the constraints were combined into a single constraint.

- alternative? allow for each constraint individually each with its own Lagrange multiplier.

L(c,y,l,L,)= U(c,l) + 1[ F(L)-y) ] +2[ 1-L-l) ] +3[ y-c ]

- solve by maximizing this WRT c, y, l, L and the multipliers.

- this gives the same result as above (Assignment!)

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- Example with specific functional forms:

Production function: y = A’ L =A’(1-l) with ’

so: FL=A’LA’(1-l)

Utility function : U = ln(c) + ln(l) so Ul/Uc =c/( l)

Combined constraint: c = A’(1-l)

Condition for optimum:U l

U c = FL

Is: c/ l) A’(1-l)

Combine with the constraint (substitute constraint for c) then solve for l:

l = 1/(1+) (=l*)

Substitute l* in constraint:

c* = A’ (1-l*)

c* ’/(1+)]

(Notice : solution for c*, l* involves solving the f.o.c.’s from the Lagrangian – including the constraint)

(Note that we have built ‘K’ into A’ here – so the solution really depends on K as well, i.e. the A’ is really A K1-I mention this since the comparative static exercise will be using K)

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Comparative statics: how the outcome changes if an “exogenous” variable changes

- Exogenous? determined outside the model. The exogenous variables here are technology (parameters, form of F), tastes (parameters, form of U)

and K.

- Comparative statics methods:

- intermediate/undergraduate level: change exogenous parameter, shift the relevant curve, graphically determine the new c, l outcome.

- we want to do this more formally.

- Crusoe model: outcome is characterized by the optimality condition and the constraint

i.e.

Ul(c*,l*) - Uc(c*,l*) FL(K, 1-l*)= 0 (U l

U c =FL)

F(K,1-l*)=c* (solving gives the outcome c*,l*)

- Say the exogenous variable ‘K’ now changes (by a tiny bit: dK).

- outcomes c*, l* will change so that the two conditions above still hold, i.e. dc and dl balance the effect on the conditions of the change in

K).

- so the change in the two condition resulting from dK is:

(Ucl - Ucc FL)dc +(Ull - Ucl FL + Uc FLL)dl – (Uc FKL)dK = 0

FK dK - FL dl = dc (2 eqns in 2 unknowns: dc,dl)

- these equations are solved for the comparative static results: dl/dK, dc/dK e.g.

dldK

=U c F KL−(U cl−U cc FL) FK

(U ¿−U cl FL+U c F ¿)−(U cl−U cc F L)FL

- With specific functional forms? Just take derivative of the solution of l WRT to

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the relevant variable or parameter.

- What does this type of model have to do with macroeconomics?

- Microfoundations approaches: decision-makers make decisions in an optimal way.

- Single decision-maker models common in the “new macroeconomics”.

- Underlying this? “Representative agent” assumption – one representative decision-maker can represent actions of

many similar decision-makers.

- Resource constraint: maximizes well-being s.t. limited resources and technology.

(an economy must satisfy such constraints)

- A general equilibrium problem:

- consumption, labour supply, output supply all jointly determined.

- goods produced are consumed (goods market equilibrium)

- labour supplied is employed to produce output (labour market equilibrium)

- The Robinson Crusoe model is an example of a "Centralized Economy".

- one decision-maker: natural here -- only 1 person!

- New macro: often assumes many people (all identical). Then the Crusoe problem is like a “social planner’s” problem showing ` the `best outcome.

- as noted below the “planner’s” solution and the competitive market outcome are the same in certain circumstances.

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- sometimes it is easier to solve the planners problem. A Decentralized – Market Economy Version of Cruose’s Model:

- Separate Crusoe the consumer and supplier of labour from Crusoe the producer and seller of goods.

- Assume Crusoe is a “price taker” (as if the markets are competitive)

- Coordination of Crusoe`s decisions occurs via markets (market prices key).

- This is unnatural in a one-person econony (useful if assume a “Representative Agent” – an economy with many identical people

whose actions can be modelled as if one decision-maker)

- Crusoe as a Consumer:

- Buys goods at the price 'P' per unit (P given: price taker)

- Sells labour time at a wage rate 'W' per unit (W given too).

- Crusoe is the only person in the economy so he owns the firm and receives its profits () as income.

- Budget constraint: income = spending

WL + = P c (is assumed fixed by Crusoe the consumer)

using 1=L+l:W∙(1-l) + = P c

(this is in 'nominal' terms: divide by P to put it in real terms)

- Crusoe's consumer problem:

max U(c,l) s.t. W∙ (1-l) + = P c <c,l>

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- The Lagrangian for the consumer problem is:

L(c,l,)= U(c,l) + [ W(1-l) + – P c ]

= Lagrange multiplier

- maximizing this WRT c, l and gives the f.o.c’s:

c: ∂ L∂ c = Uc – P =0

l: ∂ L∂ l = Ul - W = 0

: ∂ L∂ λ = W∙(1-l)+ -Pc = 0

- combine first two foc's:

U l

U c = W/P

using this and the budget constraint (f.o.c. for ) implies solutions for:

Output demand: c*=c(w/P, parameters)

Leisure demand: l*=l(w/P, parameters)

Since: 1 = L + l

Labour supply: L*=L(w/P, parameters) = 1 - l(w/P, parameters)

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- Crusoe's business problem (produces and sells y, employs L and takes P and W as fixed):

max Profit (= P y - WL s.t. y=F(L)<y,L>

substitute y=F(L) into the profit function (F(K,L) but K constant)

The problem is then:

max P F(L) - W L<L>

f.o.c: P FL - W = 0

P FL = value of marginal product ($ value of output produced by extra L)

Solution in real terms implies: FL = W/P

Labour demand: L*=L(W/P, parameters)

Output supply: y = F(L*) = y(W/P, parameters)

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- General Equilibrium in a competitive model:

- Prices in competitive markets:

W, P adjust until supply=demand

y*=c* in output market

labour supply = labour demand =(L*) in labour market

- price mechanism here? - artificial here since the buyer is the seller in a one-

person economy!

- more natural if the decision-maker is a representative agent (one of many identical consumers and

identical firms)

- Then pricing reflects interaction of buyers and sellers.

- Characterizing the general equilibrium:

- equilibrium: must satisfy constraints on behavior ;decision-makers must be acting optimally;allow for decision-maker interdependence; decisions must be consistent (supply=demand) or

else prices change.

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- Constraints:

- Consumer budget constraint: W∙(1-l) + = P c

- note that consumer receives profit income from the business which depends on business’ decisions

i.e. = P y - W L

- substitute this into the budget constraint:

W∙(1-l) + P y - W L = P c then use L=1-l

y = c

or : F(1-l)=c

(as in the Centralized version with no markets)

- Optimal behavior implied :U l

U c = FL (both =W/P)

- So can solve: F(1-l)=c and U l

U c = FL

for equilibrium: c*, l*

(as a function of production function and utility function parameters)

can use this solution to get equilibrium (W/P)*, L*=1-l*, y*(=c*).

(also as functions of taste and technology parameters)

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- Notice that together the consumer and business solutions implied:

U l

U c = W/P = FL

where y*=c*, labour demand = labour supply

- result is the same as in the “centralized” version above.

- The “Centralized” (Planner) and “Decentralized” outcomes are often the same in later, more complicated models.

- Solving the Centralized problem is sometimes easier.

- So the Centralized (or “social planner’s) problem is sometimes solved even if we think decision-making is decentralized.

- This reflects a major result in general equilibrium welfare economics.

- Fundamental Theorems of Welfare Economics:

(1) perfectly competitive outcomes are Pareto efficient;

(2) any Pareto efficient outcome can be achieved as a competitive equilibrium.

Pareto Efficient: can’t change the outcome and make someone better off without making someone else worse off.

Crusoe model: outcome is Pareto efficient if Crusoe’s utility is

maximized – just like centralized model!)

More generally: social planner choosing the outcome to maximize the representative agent’s utility

gives a Pareto efficient result.

i.e. like a decentralized competitive market outcome.

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A Two-Period Model of Consumer Behavior:

- Course is mainly concerned with dynamic models with many time periods.

- Two period models are a simple introduction to this type of model.

- Inter-temporal consumer problem:

- Consumer maximizes well-being (utility):

U = U(c1, c2) ∂U /∂c1>0 , ∂U /∂c2>0 (and second derivatives<0)

c1 = consumption in period 1 c2 = consumption in period 2

- In inter-temporal models U is often assumed to be:

U = U(c1) + U(c2)

- additively separable (separate terms for c1 and c2): marginal utility in one time period doesn’t depend on c in other periods.

- discount factor: 0<<1 less weight on future utility. (positive rate of time preference)

- Notice utility here is not a function of “leisure” (assuming labour supply is fixed here): adding leisure is a common extension.

- Consumer is constrained by available financial resources:

Income earned in periods 1 and 2: y1 and y2 (exogenous)

Initial assets: b0 (think of as bonds that pay a return "r" so r= interest rate) (b0 and r exogenous)

Will assume price level is constant so everything is in real terms.

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- Budget constraints for each period:

Period 1: c1 + b1 = b0+y1 (can spend period 1 income and initial bonds on period 1 consumption

or bonds)

b1>0 consumer saves some of their initial financial resources to finance period 2 consumption.

b1<0 consumes more than period 1 financial resources, i.e. is borrowing and will pay back the loan in period 2.

- Assume the consumer pays the same interest rate (r) on a loan as they receive when they save by buying a bond.

(letting rate at which you borrow > rate at which you lend does not alter the results much but makes things far messier!)

Period 2: c2 = y2 + (1+r)b1 (spend income, interest on bonds and bonds on period 2 consumption)

- period 2 constraint has no “b2” term (b2=0)

- won’t keep any resources beyond period 2 since there are only two periods (so won’t want b2>0);

- assume can’t “die in debt” (b2<0): no one will lend to you if they won’t be paid back.

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- You can combine the two budget constraints to get the “intertemporal budget constraint”.

- Period 1 constraint implies: b1 = b0+y1 - c1

- Substitute this for b1 in Period 2 constraint to get:

c2 = y2 + (1+r)( b0+y1 - c1)

divide by (1+r) and collect terms to get:

c1+c2

(1+r )=b0+ y1+

y2

(1+r )

Present value of consumption = Present value of income + initial bonds

(Reminder: Present value is the value of a future payment right now in the present.Say x2 is a payment to be received one period in the future then “present value” answers the question: “how much would you have to invest now for one period at the interest rate r to receive exactly x2 in the future? or equivalently, how much could you borrow in the present at the interest rate r and just be able to pay off your loan with the x2 received one period in the future. The present value of x2 received one period in the future is x2/(1+r) )

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- Lagrangian for the consumer problem is:

L(c1,c2,b1,)= U(c1) +U(c2) + [ b0+y1 - c1 -b1] + [ b1(1+r)+y2 - c2 ]

Two constraints (budget constraint from each period). t is marginal utility of extra financial resources in period t.

f.o.c's: (those for the 's are just the two budget constraints)

c1: U'(c1) -

c2: U'(c2) -

b1: + (1+r) = 0

: b0+y1 - c1 -b1 = 0

: b1(1+r)+y2 - c2 = 0

(note: U'(ci) ≡ ∂U/∂ci )

- Use conditions for c1 and c2 to substitute for the s in the condition for b1 to get:

-U'(c1) + U'(c2)(1+r) = 0

or: U'(c1) = U'(c2)(1+r)

U'(c1) = marginal utility (MU) of $1 of consumption in period 1

U'(c2)= marginal utility of $1 of consumption in period 2

An extra $1 of period 1 consumption means giving up $1∙(1+r) of period 2 consumption.

So the condition says that must be indifferent between using $1 as consumption now and transferring that $1 to the future.

(otherwise can change c1 and c2 and be better off)

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- In much of the macro literature conditions like this are stated in the form (often referred to as the Euler equation):

U'(c 2)(1+r) =1 U'(c1)

- says the same thing as the condition above but in different units (now is in period 1 dollars not in utility).

i.e. giving up $1 of c1 (RHS) gives (1+r) of c2 multiplying this by U'(c2)/U'(c1) converts this into period 1 dollars.

(i.e. U'(c2)/U'(c1) is the marginal rate of substitution (dc1/dc2 that keeps intertemporal utility the

same)

- if not true can change c1 and c2 and raise utility.

- note: whether c1 or c2 is larger depends only on vs. (1+r)

- The five first order conditions imply implicit solutions for the endogenous variables: c1, c2, b1 , and in terms of the exogenous variables (r, b0, y1, y2, and the form of the utility function).

- Comparative statics:

Outcome is described by the optimality condition and the constraints (here they are combined into one intertemporal constraint):

-U'(c1) + U'(c2)(1+r) = 0

c1+c2

(1+r )=b0+ y1+

y2

(1+r )

- take derivatives of these conditions WRT c1, c2 and the desired exogenous variable e.g. b0 then solve for dc1/db0 and dc2/db0

(see Williamson p.678 – a version of this model with taxes)

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- Take an example:

Say U = ln(c1) + ln(c2)

so: ∂U∂ c1

=U'(c1)=1/c1∂U∂ c2

= U'(c2)= /c2

Euler equation becomes:

c1(1+r) =1 c2

- notice: rise in r or means c1/c2 falls ;

if: (1+r) then c1=c2

Exact solution:

c1(1+r) =1 (combines foc's for c1,c2,b1) c2

c1+ c2/(1+r) = b0+y1+ y2 /(1+r) (inter-temporal budget) (combines foc's for 1 and1 )

Two equations in 2 unknowns (c1 and c2).

Solving gives:

c1* = (b0+y1)/(1+) + y2/[(1+)(1+r)]

c2* = (1+r)(b0+y1)/(1+) + y2/(1+)

Using budget constraint and solution for c1 gives bonds:

b1* = b0 + y1 - c1* (insert value for c1 from above)

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Notice comparative statics are easy when you have an explicit solution:

∂ c2

∂r = (b0+y1)/(1+) >0

∂ c1

∂r = -y2/[(1+)(1+r)2] <0

∂ c j

∂ y i >0 for i=1,2 and j=1,2

∂ c j

∂ b0 >0 for j=1,2

∂ c1

∂ β <0 , ∂ c2

∂ β >0

- An alternative Lagrangian for this problem:

L(c1,c2,)= U(c1) +U(c2) + [ b0+y1+ y2 /(1+r) - c1- c2/(1+r) ]

- use the combined budget constraint (this eliminates one choice variable b1)

- this will give the same results (Check for yourself!)

- Link between period 1 and period 2: via saving (lending) and borrowing.

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- See Figures below for the possible outcomes in the 2 period model.

Indifference curves: show combinations of c1 and c2 with the same utility.

Utility= U(c1) +U(c2)

so on an indifference curve:

d(Utility)=0= U'(c1)dc1 + U'(c2)dc2

d c2

d c1 = - U'(c1)/U'(c2) is its slope.

Intertemporal budget: c1+ c2/(1+r) = b0+y1+ y2 /(1+r)

slope : d c2

d c1 = -(1+r)

Case A: Saves in period 1:

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Case B: Borrows in period 1:

(note: if the consumer neither lends nor borrows it will consume y1+b0 in period 1 and y2 in period 2)

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A Robinson Crusoe-style two period model: Introducing Capital Accumulation

- Crusoe's utility: U = U(c1) + U(c2)

- Assume this time that he works a fixed amount of time in each period (ignore leisure).

- Production: yt=F(Kt) t=1 or 2 (builds in fixed L)

- Crusoe's decision? current output can be consumed or invested to create new capital (K) next period.

- Let K1 be given (initial stock of capital is exogenous).

- Define: i1=investment in period 1 (none in period 2: World ends!)

K2-K1= i1-K1 (growth in K between period 1 and 2: equals investment minus depreciation -- is the

share of K1 that wears out between period 1 and 2) = depreciation rate

so : i1 = K2-K1+K1

- investment and it's effect on current consumption and future K is what links time periods in this model.

- Budget constraints:

Period 1 (t=1): y1= c1 + i1 (with y1=F(K1) predetermined since K1 is

exogenous)

substitute for i: y1 = c1 + K2-K1+K1

Period 2 (t=2): y2= c2 (=F(K2)) (assumes you can't consume your capital – if you can consume undepreciated K2:

F(K2)+(1-)K2=c2 )

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- The Lagrangian is:

L(c1,c2,K2,)= U(c1) +U(c2) + [ y1- c1- (K2-K1+K1)] + [F(K2)-c2]

f.o.c's (note those for the ’s are not stated below and that F'=∂F/∂K):

c1: U'(c1) -

c2: U'(c2) -

2: + F'(K2) = 0 ( + [F'(K2)+1-] = 0 if can consume K)

Combine the f.o.c's to get the Euler equation:

U'(c 2)F'(K2) =1 ( U'(c 2)[F'(K2)+1- ] =1 U'(c1) U'(c1)

if can consume K )

or: U'(c2)F'(K2) = U'(c1)

(gain in utility (cost of extra ‘i’ from extra ‘i’) in terms of utility)

- this condition along with the budget constraints imply a solution for c1, c2, K2.

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- A specific case:

U = ln(c1) + ln(c2)F(K) = AK (with A>0 and 0<<1)

so: U'(c 2) = c1/c2 and F'(K)= AK

U'(c1)

Euler equation becomes:

(c1/c2)AK2

and budget constraints are:AK1

- c1- (K2-K1+K1)] =0 AK2

-c2=0

3 equations, 3 unknowns (c1,c2, K2) but non-linear!

Solving gives:c1*= [AK1+K1(1-)]/(1+)c2*= AK1-c1*-K1(1-K2*= [c2*/A]1/

i.e. in terms of K1 (a given) and parameters ,A.

- Comparative statics for c1, c2 can be done the usual way.

- exogenous variables: K1, , utility function parameters and production function parameters.

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( Blue line: F(K2) with K2=F(K1)-c1+(1-)K1 slope is –F')

(How would the diagram differ if the person could consume undepreciated K in period 2?)

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