Economics 201B–Second HalfLecture 4, 3/18/10The Robinson Crusoe Model: Simplest Model Incorporating Pro duction• 1consumer• 1 firm, owned by the consumer• Both the consumer and firm act as price-takers (silly in this model, but it shows how equilibriumoperates)• 2 goods:– Leisure x1, endowment¯L (24 hours per day)– Consumption goood x2bananas, endowment = 0• p: price of bananas• q: quantity of bananas produced by firm• w: wage rate = price of labor• Production function f(z): z units of labor produces q = f(z) bananas. We assume f is strictlyconcave; first gather low-hanging bananas, then start climbing trees to gather, then tend plants toincrease yield• Firm’s profit: pq − wz. Note that profit is a linear function of (q, z), the vector of inputs and outputs,whether or not (q, z) is feasible. The firm maximizes profit over the feasible set, taking prices asgiven– Labor demand z(p, w) chosen to maximizepq − wz = pf(z) − wz1taking p, w as given. First order conditionpf(z) − w =0– Output q(p, w)=f(z(p, w))– Profit Π(p, w)=pq(p, w) − wz(p, w)• Consumer owns firm, so receives the profit. Crusoe’s budget constraint ispx2≤ w(¯L − x1)+Π(p, w)• Walrasian equilibrium prices are (p∗,w∗) such that markets clear:x2(p∗,w∗)=q(p∗,w∗) (banana market)z(p∗,w∗)=¯L − x1(p∗,w∗)(labormarket)• In the previous diagram showing the firm’s problem, lines perpendicular to the price vector (w, p)(note this is labelled incorrectly as (p, w) in MWG) are isoprofit lines. Any two points on a givenisoprofit line yield the same profit; this is true whether or not the point on the isoprofit line is a feasibleproduction. In particular, if we consider the isoprofit line through the firm’s profit-maximizing pointon its production set, the x2intercept of this line m ust beΠ(p,w )p.• The previous diagram superimposes the consumer’s problem on the firm’s problem. If x1=¯L(Crusoe gets no labor income), Crusoe’s income is Π(p, w), so Crusoe can purchaseΠ(p,w )pbananas,so¯L,Π(p,w )p lies in Crusoe’s budget frontier. The isoprofit line through the firm’s profit-mazimizingproduction is the consumer’s budget frontier!2• The previous diagram does not show an equilibrium configuration. For market clearing, requirex2(p∗,w∗)=q(p∗,w∗) (banana market)z(p∗,w∗)=¯L − x1(p∗,w∗)(labormarket)Markets clear if and only if the firm’s profit-maxizing point and Cr usoe’s demand point coincide.Inthe diagram, Crusoe is supplying less labor than the firm is demanding, and Crusoe is consumingfewer bananas than the firm is selling• In the following diagram, we dispense temporarily with the firm and look at the consumer’s overallproblem, in which Crusoe applies the technology directly without going through the structure of thefirm• Notice that Crusoe’s feasible set is just given by the production technology, so the feasible set isnonlinear; it is not a “budget set;” each point in the feasible set is a feasible consumption for Crusoe.• What consumption would Crusoe choose? The economy has a unique(!) Pareto optimum,givenbythe point of tangency between the feasible set and Crusoe’s indifference curve.• Second Welfare Theorem: Ifwechoose(p∗,w∗) such that (w∗,p∗) is perpendicular to the commontangent at the Pareto Optimum, then firm’s profit-maximizing production and Crusoe’s demandpoint coincide, so the unique Pareto Optimum is a Walrasian Equilibrium (without transfers); that’sthe Second Welfare Theorem.3• First Welfare Theorem:If(p∗,w∗) is a Walrasian Equilibrium Price, then firm’s profit-maximizingpoint and Crusoe’s demand point coincide, (p∗,w∗) supports this common point, so it is ParetoOptimalArrow-Debreu Economy• L commodities, indexed by  =1,...,L– I consumers, indexed by i =1,...,I– Consumption sets Xi⊆ RL+– Endowments ωi∈ RL+– Preference relations ion Xi, assumed complete and transitive– Social endowment¯ω =Ii=1ωi=(¯ω1,...,¯ωL)• J firms, indexed by j =1,...,J– Production Sets Yj⊂ RLassumed closed and nonempty– Shareholdings: Consumer i owns share θijof firm j,Ii=1θij=1(foreachj)• Income Transfer: An income transfer is T ∈ RIsuch thatIi=1Ti= 0 (Budget Balance)• Budget set:Bi(p, y, T )=⎧⎨⎩x ∈ Xi: p · x ≤ p · ωi+Jj=1θijp · yj+ Ti⎫⎬⎭Note the budget set depends on prices, the income transfer, and on the firms’ production decisions4• Demand:Di(p, y, T )=x ∈ Bi(p, y, T ):∀x∈Bi(p,y,T )x ix• An allocation(x, y)=(x1,...,xI,y1,...,yJ)is a specification of xi∈ Xi(i =1,...,I)andyj∈ Yj(j =1,...,J); the allocation is feasible ifIi=1xi=¯ω +Jj=1yjNotice that this is a vector equation (one equation for each of the L goods) and that we requireequality. The set of feasible allocations is denoted by A• Walrasian Equilibrium with Transfers: In the Arrow-Debreu economy, a Walrasian Equilibrium withTransfers is a 4-tuple (p∗,x∗,y∗,T) such that1. T ∈ RIis an income transfer. We don’t put an ∗ on T because T is not determined endogenouslyby market-clearing2. p∗is a price, i.e. p∗∈ RL(don’t require p ∈ RL+)3. for j =1,...,J, y∗j∈ Yjand∀yj∈Yjp∗· y∗j≥ p∗· yj(price-taking profit maximization)4.x∗i∈ Di(p∗,y∗,T) (price-taking preference maximization)55. (x∗,y∗) is a feasible allocation, i.e.Ii=1x∗i=¯ω +Jj=1y∗j(market-clearing)• Par eto Optimality: A feasible allocation (x, y)is– Pareto Optimal if there is no other feasible allocation (x,y) such thatxiixi(i =1,...,I)xiixi(some i)– weakly Pareto Optimal if there is no other feasible allocation (x,y) such thatxiixi(i =1,...,I)Note that the firms’ profits or “preferences” are not taken into account; only the welfare of theconsumers matters. But of course the production technology does play a role in determining whetheran allocation and a proposed Pareto improvement are