(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) (could set this up in matrix form, see p.32) - these two equations can then be solved for the comparative static results: dl/dK and dc/dK e.g. - 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. - 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 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> - The Lagrangian for the consumer problem is: (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: ∂/∂c = Uc – P =0 l: ∂/∂l = Ul -  W=0  : ∂/∂ = W∙(1-l)+  -Pc = 0 - combine first two foc's: Ul/Uc = 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)
y*=c* in output market labour supply = labour demand =(L*) - 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 similar decision-makers e.g. many identical consumers, many 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 interdependence between decision- makers; decisions must be consistent (supply=demand) or else prices change. - 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
- 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 - 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) - Take an example: Say U = ln(c1) + ln(c2) so: ∂/∂c1=U'(c1)=1/c1 ∂/∂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: