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Consider the following economy. There is continuum of workers with mass 1, each endowed with L units of labor, and a continuum of goods of mass N. They have the same utility given by U = [ 1 - e-be b -di, where N is the number of goods, which is endogenous. Each differentiated good is produced by a monopoly. There is a fixed overhead cost equal to / units of labor. There is no variable cost (an arbitary large quantity of the good can be produced: these goods are like software, music, etc). 1. Show that if the price of good i is p;, then the demand for good i by a consumer with income R is where R + & So pi In padi So pidi 2. Show that each firm will charge a price p; = p = eba-1, where & is defined as above and common to all workers. N is endogenously determined by the free entry condition. We normalize the common price level to p = 1. 3. Compute the wage level w (as defined by the wage of 1 unit of labor, so that a worker's income is wL). How does it depend on the overhead labor cost I? Explain why. 4. Compute the utility of a worker. How is it affected by total productivity (as measured by L) and overhead costs? We now modify the model and assume that each worker is also endowed with q units of managerial quality. A firm employing a manager of quality q has a total overhead cost now equal to 1/q (instead of just ?). q is uniformly distributed in the population over [9min: 9max]: i.e. with c.d.f. F(q) = 9-1min and density f(q) = F'(q) = 4max -9mm Each worker has to work either as a worker or a manager, and can't do both. There is free entry of firms which compete to hire managers. Let w(q) be the wage paid to a manager with quality 4 in equilibrium. 5. Show that (with the same price normalization as before), one must have w(q) = 1/b - wl/q 6. Show that all workers with managerial quality q > q" become managers in equilibrium, where LF(q') = ;f(9)dq 7. Show that this condition defines a unique q* such that both q* and 1/q* go up when I rises. 8. Show that the equilibrium wage is - b(L + 1/q* ) 9. How does an increase in overhead costs I affect (i) The absolute income level for production workers wL? (ii) Their utility iii) The number of managers? (iii) Inequality (as measured by income ratios) between production workers and low-quality managers? (iv) Inequality between production workers and high-quality managers? v) Inequality between two managers who remain in that activity after the increase in / 10. Same questions for a change in productivity L.1. Let X be a finite set of choices. A a non-empty set of non-empty subsets of X. and c : A > 2~"\\{ii}. where for each A s All y; .:{A} c A. a choice rule. We say that c satisfies expansion {E} if for each A, B s A. I E cfAli'ich} =:- I E :[AUH]. 1. {5 marks} State the Weak Axiom of Revealed Preference {WARP}. 2. {14 marks} Prove or disprove that WARP implies E. 3. {14 marks} Prove or disprove that E implies WARP. 2. The supply side of a perfectly convective market is represented by 1oo identical producers each one endowed with a technology that uses two inputs and produces one output described by the following cost function: C(w1,IU2-.yl= (aucwi''f) s where 3; denotes the output. 191 and us the input prices and we assume aeucaaaL {i} {-l- marks] What are the returns to scale of each producer's technol- ogy? Explain your answer. {ii} {4 marks} Identify the production function associated with each pro- ducer's technology. {iii} {4 marks] Identify and plot in a graph me aggregate suppty function of commodity y in this market The demand side of this market is represented by 100 identical con- sumers. These consumers consume only the output commodity y and their preferences are represented by the strictly monotonic utility function uEy}. Each oonsum er's exogenous income is normalized to a size of 1 . in other words his income is p, where p is the price of 3;. {iv} {4 marks] Identify the Marshallian demand of each individual con- sumer. {v} {4 marks} Identify and plot in a graph the consumers' aggregate demand in this market {vi} {5 marks] What is the equilibrium price and quantity in this market? Explain your answer. {vii} {4 marks} Can you identify the quantity supplied by each individual producer in this market? Explain your answer. {viii} {4 marks} What profit each individual producer obtains at the man-set equilhrium price? Explain your answer. 3. In a two-consumer this. A and Mr. B}. two-commodity, :1 and :2, pure ex- change economy, A's preferences are represented by the utility function: Win". a?) = 13"- Gonsumer B's preferences are represented by the following utility func- tion: Utxf'e I?) = If\" where a} denotes the quantity of commodity j E {1,2} consumed by consumer i: E {A, B}. The total quantities available in the economy of commodity I1, denoted by it. and of commodity n. denoted by is, are identical, a1 = 3-7:- Further. 2 assume that the endowment allocation is such that A owns half of the entire quantity available of commodity 1, 51,32, and half of the quantity available of commodity 2. 52,42. {i} {9 marks] Find the offer curves of both his. A and Mr. Ii. {ii} {16 marks] Find the set of Walrasian equilibrium prices and alloca- tions in this economy. {iii} {9 marks] Find the set of Pareto efficient allocations. {iv} {4 marks] Are the Walrasian equilbrium allocationis} you found in {ii} above Pareto efficient?I Explain your answer. 4. Consider two players playing the following normal form game. {i} {4 marks] Does either player have a strictly dominated strategy? {ii} {1 Ci marks} identify the mired strategy Nash equilibnia of this game. Assume now that the the normal form game above is played in two con- secutive periods. Assume also that each player observe the outcome of play at the end of the first period and that both players discount the future with a common discount factor 5 such that 0
