Course regulations Technology Microeconomics 1 Juan Manuel Puerta September 27, 2009 Course regulations Technology First course in micro theory. Roughly consumer and producer theory and rudiments of general equilibrium. Course prerequisites are calculus/algebra to the level of Alpha Chiang or Simon and Blume. 1 Week 1-3: Producer Theory 2 Week 4-6: Consumer Theory 3 Week 7-8: Market Structure: Competition vs. Monopoly 4 Week 9: General Equilibrium 5 Week 10: Revision and Exam This are approximate dates, you are responsible to periodically check for changes (I will try to announce any major change) Evaluation: Midterm and final (80%) and problem sets (20%). “Big” and “Small” Varian Breaks Course regulations Technology The 3-tip method to doing well in Micro 1 Tip 1: Make sure your math level allows you to understand lectures. But don’t panic, there are a lot of good textbooks (see previous slides). Tip 2: Work steadily througout the term. Do the homeworks and exercises at the end of the chapter. Remember: memory vs. solving Tip 3: Talk, ask, participate... Don’t wait to the day of the exam to tell me you haven’t understood anything! We have office hours to help you. Course regulations Technology Slides vs. Blackboard action. Remarks about grad school. Questions? Comments? Short speeches? Course regulations Technology Slides vs. Blackboard action. Remarks about grad school. Questions? Comments? Short speeches? Course regulations Technology Slides vs. Blackboard action. Remarks about grad school. Questions? Comments? Short speeches? Course regulations Technology Chapter 1: Technology In the first part of the course we will be interested in studying the behavior of firms. The amount of output firms produce depends on the characteristics of the available technology. In this first chapter we introduce important concepts regarding technology that will help us in order to study profit maximization (ch. 2-3) and cost minimization (ch. 4-5). We will be interested in the combinations of inputs and outputs that could be produced given the state of technology at a given moment. Think of outputs and inputs as defined over time (flows). The aim of the chapter is to introduce some definitions about technology and a few results that will be useful later. Course regulations Technology Specification Some definitions i o n available goods that can be either inputs or outputs, yj and yj are the quantities of the good j used as an input and output o i respectively. The net ouput of good j is given by yj = yj − yj. A production plan is a list of net outputs for the n goods. A production plan is represented by a vector n y = (y1; y2; :::; yj; :::; yn) in < . If yj > (<)0 then the good is and input(output). The set of all technologically feasible production plans is the production possibility set, Y ⊂ <n In the short run some inputs may be fixed, so some feasible plans may not be “immediately feasible”. Let z denote the vector of fixed inputs, the restricted production possibility set is the set of all feasible net output bundles consistent with z. Course regulations Technology Specification Consider a firm producing one output y using an input vector x. The input requirement set is defined as n V(y)={x in <+ : (y,-x) is in Y} This is simply the set of input requirements that produces at least y. We can also define the concept of isoquant using the input requirement set n 0 0 Q(y)={x in <+ : x is in V(y) and x is not in V(y ) for y > y} In multi-output settings, there is a definition that would come handy, this is the transformation function A transformation function T : <n ! < where T(y) = 0 if and only if y is efficient, i.e. There is no y’ in Y such that y’ ≥ y and y’ 6= y Course regulations Technology Monotonicity Monotonic technologies It may seem natural the possibility to “throw away” inputs. Assume we have 2 units of each of two inputs but we have a technology with V(1)=(2,1),(1,2). It seems reasonable to use (2,1) to produce 1 unit of output and throw away the remaining unit of input 2. If this is possible, we say there is free disposal. The idea of free disposal is related to the monotonicity assumption of the input requirement set. Monotonicity (Input requirement set). If x is in V(y) and x’ ≥ x, then x’ is in V(y). Monotonicity (Production set) If y is in Y and y’ ≤ y, then y’ is in Y. Note the particular sign convention. This is due to the fact that inputs enter as negative values in Y. In our previous example, y=(1,-2,-1) and y’=(1,-2,-2). Course regulations Technology Monotonicity Monotonic technologies It may seem natural the possibility to “throw away” inputs. Assume we have 2 units of each of two inputs but we have a technology with V(1)=(2,1),(1,2). It seems reasonable to use (2,1) to produce 1 unit of output and throw away the remaining unit of input 2. If this is possible, we say there is free disposal. The idea of free disposal is related to the monotonicity assumption of the input requirement set. Monotonicity (Input requirement set). If x is in V(y) and x’ ≥ x, then x’ is in V(y). Monotonicity (Production set) If y is in Y and y’ ≤ y, then y’ is in Y. Note the particular sign convention. This is due to the fact that inputs enter as negative values in Y. In our previous example, y=(1,-2,-1) and y’=(1,-2,-2). Course regulations Technology Monotonicity Monotonicity in action. The left chart represents that original input requirement set. The right one includes all the possible combinations that imply wasting inputs, i.e. the combinations if monotonicity holds. Course regulations Technology Convexity Imagine we want to produce 100 units of output. By a replication argument, (200,100) and (100,200) should be in V(100). Are there any other ways of producing 100 units of output? Imagine we use 50% of the times the first technology (producing 50 units of output) and 50% the second technology (producing the other 50). 0:5(200; 100) + 0:5(100; 200) = (150; 150) In general, t(200; 100) + (1 − t)(100; 200) should also be in V(100) for t = 0:01; 0:02; :::; 1 Convexity. If x and x’ are in V(y), then tx+(1-t)x’ is in V(y) for all 0 ≤ t ≤ 1 Course regulations Technology Convexity Some remarks on convexity 1 Convexity was motivated through a replication argument. Seems appropriate for large output relative to scale. 2 Convexity could be motivated in production plans that are implemented over relatively long times (you can switch production plans in the middle) 3 Convexity for production sets. A production set is convex if y and y’ are both in Y, then ty+(1-t)y’ is also in Y. Note that this is more restrictive assumption. In particular it rules out fixed costs. Why? Course regulations Technology Convexity Some useful results Theorem 1. Convex production sets imply convex input requirement sets Proof: 1 Take y=(y,-x) and y’=(y,-x’) both in Y. This implies x and x’ are in V(y) 2 Convexity of Y implies that ty + (1 − t)y’ is also in Y. But then 3 t(y; −x) + (1 − t)(y; −x’) = (ty + (1 − t)y; −tx − (1 − t)x’) = (y; −tx − (1 − t)x’) 4 But then by definition of V(y), (tx+(1-t)x’) is in V(y). 5 So x and x’ in V(y) imply (tx+(1-t)x’) is in V(y). In other words, V(y) is convex. Theorem 2. Convex input requirement set () quasiconcave production function Course regulations Technology Convexity Does the converse statement hold true, i.e. V(y) convex =) Y convex? It turns out this claim is not true (ex. 1.1) How would you go about proving this is not the case? Counterexample? Contradiction? (You can establish A=)B if you prove that -B=)-A or similarly we the contradiction after assuming A and -B). So imagine a production function that like y = f (x) = x2. This production function does not imply a convex production set (can you see that?) Y={(y,-x) in <2: y ≤ x2} 2 But V(y)={x in <+: x ≥y} which is a convex set. Course regulations Technology Convexity Does the converse statement hold true, i.e. V(y) convex =) Y convex? It turns out this claim is not true (ex. 1.1) How would you go about proving this is not the case? Counterexample? Contradiction? (You can establish A=)B if you prove that -B=)-A or similarly we the contradiction after assuming A and -B). So imagine a production function that like y = f (x) = x2. This production function does not imply a convex production set (can you see that?) Y={(y,-x) in <2: y ≤ x2} 2 But V(y)={x in <+: x ≥y} which is a convex set. Course regulations Technology Convexity Does the converse statement hold true, i.e. V(y) convex =) Y convex? It turns out this claim is not true (ex. 1.1) How would you go about proving this is not the case? Counterexample? Contradiction? (You can establish A=)B if you prove that -B=)-A or similarly we the contradiction after assuming A and -B). So imagine a production function that like y = f (x) = x2.