Tutors you help is needed here
Exercises Question 1 (Profit Maximization in the Long-Run) In a perfectly competitive industry, all firms face the cost structure and market demand as depicted below. P ($)A MC Market Demand Price ($) Quantity 6 140 5 180 ATC 200 240 2 270 5 1. If the current market price is $5, then each firm will produce units, and there are firms in the industry. What is the long-run equilibrium number of firms in this industry? What is the quantity produced by each firm in this long-run equilibrium? Question 2 (Short-Run vs. Long-Run: Conceptual Question) Each worker (L) costs $10 to hire, each unit of capital (K) costs $10 to rent. The following table denotes the combinations of capital and worker mixes for a representative firm to produce certain quantities of good: L K TC AC 100 20 20 400 200 60 20 800 200 30 30 600 1. What should a firm do in the short-run if it wishes to increase the production from 100 units to 200 units? Why? 2. What should a firm do in the long-run if it wishes to increase the production from 100 units to 200 units? Why? Relate the difference between short-run and long-run average cost curves in light of answers in part 1 and 2. 4. Can you tell if the firm is experiencing increasing, decreasing or constant returns to scale? Business Learning Center - Econ 101 (Hansen) - Session 11 (October 28, 2015) Tutor: Kanit Kuevibulvanich Question 3 (Long-Run Average Cost Curves and Profit Maximization) COST INDUSTRY REPRESENTATIVE FIRM MC1 ACI LRAC MC2 AC2 15 12 400 600 20 40 50 Each firm has the ability to choose Scale 1 or Scale 2, denoted by MC1, AC1 and MC2, AC2, respectively. Consider the industry demand-supply curve and the cost structure of a representative firm and answer the following questions: 1. Suppose the current market price is $15, then each firm would produce units, and there are firms in the industry. Each firm earns an economic profit of . What would happen in the long-run? 3. How many firms are there in the long-run?Question 4 In ECN200A we assumed that consumers have preferences over consumption bundles. In many contexts though it is more natural to think that consumers have preferences over characteristics of consumption bundles like "lots of vitamins", "gluten free", "lots of horse power", "no tail pipe emissions", "many mega pixels" etc. In the following I outline an alternative model of consumer theory, in which preferences are defined over characteristics rather than consumption bundles. You are right, we have never discussed it in class. But there is no need to freak out since we know all the tools that are required to think about such a model. Goods are indexed by / = 1, ..., L and characteristics are indexed by i = 1, ..., I. We denote by a; > 0 the quantity of characteristics i possessed by one unit of good . If denotes the quantity of good 6. z, is the quantity of characteristic i. We let = = (21, ..., 27) and assume that z e Z C R. Moreover, as usual we let r = (21, .... II) EX CR+. We assume 2 = 2 diet for i = 1, ..., I. This is the amount of characteristic z; derived from a bundle of goods I = (T1, ..., IL). We arrange A = (ait)izl,.,1,/=1...,L into a matrix, in which rows refer to characteristics and columns to goods. a.) Consider a binary relation > on the space of characteristics, Z. State conditions on _ that are sufficient for the existence of a utility function over characteristics u : Z - R that represents _. b.) Given prices of goods p = (P1, ...; PL) > > 0 and wealth w 2 0, define the budget set on the characteristics space by Kow,A := {= c Z : there exist r ( X s.t. = = Ar,p. ISw). Show that for any p > > 0 and w 2 0, the budget set Kow,A is convex. c.) Assume that u is monotone and continuously differentiable. Consider the consumer problem max u(z) s.t. z E Kpu,A. Show that necessary conditions for a utility maximum are pe 2 du (2) for { = 1, .... L. where A is the Lagrange multiplier w.r.t. to the budget constraint p. r S w. d.) Provide an economic interpretation of the term 1 du(z) A dziQuestion 1 Consider an economy with / consumers and L goods. For each consumer i e {1, .... / }, the consumption set is R4. Her utility function is given by u'(x') = -) are-Biz, K=1 where ox, ; > 0 for k e {1. ..., L}. As usual we denote r' = (r;, ...,;). Consider price vectors (p1, .... pz) and wealth levels (w], ...,w') for which the solution to the utility maximization problem is interior for every consumer i e {1, .... /}. a.) Derive the Walrasian demand function for good j by consumer i. (Be careful with your calculations. Double-check! It is easy to make mistakes.) b.) What is the slope of consumer i's Engel curve for good j at (p, wi)? c.) Find a condition as general as possible on parameters ox, By, i e {1,..., / }, k E {1. ..., L} guaranteeing the existence of a positive representative consumer. Do we need restrictions on parameters ox, ke {1, .... L}? d.) Consider now the special case with just a single consumer and two goods. The consumer's utility function is given by Derive the wealth-expansion path for a given price vector (P1, p2)- e.) In problem d.), when does the wealth-expansion path intersect the r,-axis and when does it intersect the ry-axis
