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Advanced Microeconomics Simple Economy with Production

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Advanced Microeconomics Simple Economy with Production Advanced Microeconomics Simple economy with production Jan Hagemejer November 27, 2011 Jan Hagemejer Advanced Microeconomics Introducing production We start with a simplest possible setting: single consumer and a single rm. Two goods: labor/leisure and a consumption good produced by the rm This is often referred to as Robinson Crusoe economy. Jan Hagemejer Advanced Microeconomics The rm The rm uses labor to produce the consumption goods. Increasing and strictly concave production function f (z), where z is the labour input. To produce output the rm purchases some labour from the consumer. The rm takes market prices as given. Jan Hagemejer Advanced Microeconomics The rm The rm solves the standard prot maximization problem: Max pf (z) − wz: z≥0 Given prices (p; w) the rms optimal: labor demand is z(p; w) [labor demand function] output is q(p; w) [supply function] prots are π(p; w): [prot function] Jan Hagemejer Advanced Microeconomics The rm Jan Hagemejer Advanced Microeconomics Consumer Consumer is the sole owner of the rm and receives all the prots. Consumer (labour owner) is hired by the rm at the same time. He has endowment L¯ of labour. Consumer preferences are represented by a utility function u(x1; x2): x1 is the consumption of leisure, therefore labour supply is L¯ − x1 x2 is the consumption good The consumer problem given prices is: Max u(x1; x2) s.t. px2 ≤ w(L¯ − x1) + π(p; w): x x 2 ( 1; 2)2R+ Budget constraint reects the two sources of income for a consumer. Jan Hagemejer Advanced Microeconomics Consumer Jan Hagemejer Advanced Microeconomics Demand and equilibrium Consumer optimal demands at prices (p; w) are denoted by: (x1(p; w); x2(p; w)) A Walrasian equilibrium in this economy involves a price vector (p∗; w ∗) at which the consumption and labour markets clear: ∗ ∗ ∗ ∗ x2(p ; w ) = q(p ; w ) ∗ ∗ ∗ ∗ z(p ; w ) = L¯ − x1(p ; w ): Jan Hagemejer Advanced Microeconomics The rst order conditions for an interior optimum From the consumer maximization problem: w MU1 = p MU2 From the rms prot maximization: w pMP w MP z = ! p = z Therefore: MU1 MPz = MU2 (remember that consumption of good 1 is L¯ − z). Jan Hagemejer Advanced Microeconomics Equilibrium Jan Hagemejer Advanced Microeconomics Welfare properties A particular consumption-leisure combination can arise in a competitive equilibrium if and only if it maximizes the consumers' utility subject to the economy's technological and endowment constraints. The same allocation as the social planner solution (under certain conditions, such as existence of prot maximizers). 1st and 2nd welfare theorem hold. For 2nd we need convexity of preferences and strict convexity of the aggregate production set. Jan Hagemejer Advanced Microeconomics The social planner solution We can nd the optimal use of resources in our economy by maximizing the utility function of the consumer: Max u(x1; x2) x1;x2 subject to the resource constraint L¯ = x1 + z ! x1 = L¯ − z subject to technology constraint:x2 = q = f (z) Subsitute to the utility function to get a function of z solely: Max u(L¯ − z; f (z)): z So the social planner will make the agent work the optimal amount of hours. Jan Hagemejer Advanced Microeconomics The social planner solution First order condition @u(L¯ − z; f (z)) @u(x1; x2) @f (z) u(x1; x2) = − + = 0 @z @x1 @z @x2 −MU1 + MPz MU2 = 0 MU1 MPz = MU2 So as long as the interior solution to both problems (comp. eq. and soc. plan.) exist, they are the same. Jan Hagemejer Advanced Microeconomics Three examples with dierent technology Example 1 (constant returns to scale): Suppose we have one consumer with a Cobb-Douglas utility function for consumption x2 and leisure x1: u(x1; x2) = a ln x1 + (1 − a) ln x2: Endowment of leisure is one L¯ = 1. The rm has with a constant-returns-to-scale technology q = f (z) = z. Find the competitive equilibrium. Find the Pareto optimal allocation through a social planner problem. Do the two coincide? Jan Hagemejer Advanced Microeconomics DRS Example 2 (decreasing returns to scale): Suppose we have one consumer with a Cobb-Douglas utility function for consumption x2 and leisure x1: u(x1; x2) = a ln x1 + (1 − a) ln x2: Endowment of leisure is L¯ = 1. Consumer can supply its leisure to the rm with a p decreasing-returns-to-scale technology q = z. Find the competitive equilibrium. Find the Pareto optimal allocation through a social planner problem. Do the two coincide? Jan Hagemejer Advanced Microeconomics IRS Example 3 (increasing returns to scale): Suppose we have one consumer with a Cobb-Douglas utility function for consumption x2 and leisure x1: u(x1; x2) = a ln x1 + (1 − a) ln x2: Endowment of leisure is L¯ = 1. Consumer can supply its leisure to the rm with an increasing-returns-to-scale technology q = L2: Does the competitive equilibrium exist? Find the Pareto optimal allocation through a social planner problem. Do the two coincide? Jan Hagemejer Advanced Microeconomics Welfare properties Jan Hagemejer Advanced Microeconomics.
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