Economics 201B–Second HalfLecture 4, 3/18/10The Robinson Crusoe Model: Simplest Model Incor-porating Production• 1consumer• 1 firm, owned by the consumer• Both the consumer and firm act as price-takers (silly in thismodel, but it shows how equilibrium operates)• 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 strictly concave; first gather low-hanging bananas, then start climbing trees to gather, thentend plants to increase 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 **production**set, taking prices as given– 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 budgetconstraint ispx2≤ w(¯L − x1)+Π(p, w)• Walrasian equilibrium prices are (p∗,w∗) such that marketsclear:x2(p∗,w∗)=q(p∗,w∗) (banana market)z(p∗,w∗)=¯L − x1(p∗,w∗) (labor market)• In the previous diagram showing the firm’s problem, lines per-pendicular to the price vector (w, p) (note this is labelled incor-rectly as (p, w) in MWG) are isoprofit lines. Any two points ona given isoprofit line yield the same profit; this is true whetheror not the point on the isoprofit line is a feasible production. Inparticular, if we consider the isoprofit line through the firm’sprofit-maximizing point on its production set, the x2interceptof this line must beΠ(p,w)p.• The previous diagram superimposes the consumer’s problemon the firm’s problem. If x1=¯L (Crusoe gets no labor in-come), 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-mazimizing pro-duction is the consumer’s budget frontier!2• The previous diagram does not show an equilibrium configu-ration. For market clearing, requirex2(p∗,w∗)=q(p∗,w∗) (banana market)z(p∗,w∗)=¯L − x1(p∗,w∗) (labor market)Markets clear if and only if the firm’s profit-maxizing pointand Crusoe’s demand point coincide. In the diagram, Crusoeis supplying less labor than the firm is demanding, and Crusoeis consuming fewer bananas than the firm is selling• In the following diagram, we dispense temporarily with thefirm and look at the consumer’s overall problem, in which Cru-soe applies the technology directly without going through thestructure of the firm• Notice that Crusoe’s feasible set is just given by the productiontechnology, so the feasible set is nonlinear; it is not a “budgetset;” each point in the feasible set is a feasible consumption forCrusoe.• What consumption would Crusoe choose? The economy hasa unique(!) Pareto optimum, given by the point of tangencybetween the feasible set and Crusoe’s indifference curve.• Second Welfare Theorem:Ifwechoose(p∗,w∗) such that(w∗,p∗) is perpendicular to the common tangent at the ParetoOptimum, then firm’s profit-maximizing production and Cru-soe’s demand point coincide, so the unique Pareto Optimum isa Walrasian Equilibrium (without transfers); that’s the SecondWelfare Theorem.3• First Welfare Theorem:If(p∗,w∗) is a Walrasian Equilib-rium Price, then firm’s profit-maximizing point and Crusoe’sdemand point coincide, (p∗,w∗) supports this common point,so it is Pareto OptimalArrow-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 tran-sitive– 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⎫⎪⎬⎪⎭4Note the budget set depends on prices, the income transfer,and on the firms’ production decisions• 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 ofthe L goods) and that we require equality. The set of feasibleallocations is denoted by A• Walrasian Equilibrium with Transfers: In the Arrow-Debreueconomy, a Walrasian Equilibrium with Transfers is a 4-tuple(p∗,x∗,y∗,T) such that1. T ∈ RIis an income transfer. We don’t put an ∗ onT because T is not determined endogenously by 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)• Pareto 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 intoaccount; only the welfare of the consumers matters. But ofcourse the production technology does play a role in determin-ing whether an allocation and a proposed Pareto improvementare