3 Comparative statics (max. 25 points) Suppose there are only two price-taking firms j, j ∈ (A,B), on a market with a decreasing demand function D(p). The firm’s cost functions Cj(yj , x̄j) are both...

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3 Comparative statics (max. 25 points) Suppose there are only two price-taking firms j, j ∈ (A,B), on a market with a decreasing demand function D(p). The firm’s cost functions Cj(yj , x̄j) are both increasing and strictly convex. y denotes output and x̄j are exogenously fixed input factors for each firm j. Suppose further that marginal costs are decreasing in x̄j , ∂Cj(yj ,x̄j)∂yj∂x̄j < 0.="" a)="" what="" is="" the="" profit="" maximization="" condition="" for="" a="" price="" taking="" firm="" j?="" what="" is="" the="" firm’s="" supply="" function?="" (max.="" 4="" points="" )="" now="" consider="" a="" short-run="" equilibrium="" in="" which="" both="" firms="" are="" active.="" b)="" state="" the="" equations,="" that="" determine="" the="" equilibrium.="" state="" the="" endogenous="" and="" exoge-="" nous="" variables.="" (max.="" 9="" points)="" c)="" explain="" the="" necessary="" steps="" for="" doing="" a="" comparative="" statics="" analysis="" to="" determine="" the="" change="" of="" the="" market="" price="" p="" caused="" by="" an="" exogenous="" increase="" of="" the="" input="" factor="" of="" firm="" b,="" x̄b.="" (max.="" 8="" points)="" d)="" use="" a="" comparative="" statics="" analysis="" to="" show="" that="" ∂p∂x̄b="">< 0. (max. 4 points) 4 general equilibrium (max. 33 points) consider an economy with production and two consumers, a and b, who derive utility from consuming good x and leisure l. utilities of the consumers are described by functions ui(xi, li) with i ∈ (a,b), xi and li denoting quantities of the good and leisure consumed by i. both consumers are endowed with a total amount of time t . there exists two firms, i and ii. both produce good x using only labour n as input. the firm’s technologies can be described by production functions yj = fj(nj), j ∈ (i, ii), yj and nj denoting the firm j’s output and used labour. firm i is owned by consumer a and firm ii by consumer b. suppose all consumers and firms are price takers. prices for the good and labour are denoted by p and w. a) state the definition of a walrasian equilibrium. (max. 5 points) b) state walras’ law. suppose there is an excess supply on the labour market, what does this imply for the market of good x? (max. 5 points) c) set up the system of equations that determines the equilibrium. what are the endoge- nous variables? (max. 18 points) d) suppose the price vector of a walrasian equilibrium is (p, w) = (0.5, 1). consider a different price vector (p′, w′) = (0.25, 0.5). is this a price vector of a walrasian equilibrium as well? explain briefly. (max. 5 points) 3 0.="" (max.="" 4="" points)="" 4="" general="" equilibrium="" (max.="" 33="" points)="" consider="" an="" economy="" with="" production="" and="" two="" consumers,="" a="" and="" b,="" who="" derive="" utility="" from="" consuming="" good="" x="" and="" leisure="" l.="" utilities="" of="" the="" consumers="" are="" described="" by="" functions="" ui(xi,="" li)="" with="" i="" ∈="" (a,b),="" xi="" and="" li="" denoting="" quantities="" of="" the="" good="" and="" leisure="" consumed="" by="" i.="" both="" consumers="" are="" endowed="" with="" a="" total="" amount="" of="" time="" t="" .="" there="" exists="" two="" firms,="" i="" and="" ii.="" both="" produce="" good="" x="" using="" only="" labour="" n="" as="" input.="" the="" firm’s="" technologies="" can="" be="" described="" by="" production="" functions="" yj="fj(nj)," j="" ∈="" (i,="" ii),="" yj="" and="" nj="" denoting="" the="" firm="" j’s="" output="" and="" used="" labour.="" firm="" i="" is="" owned="" by="" consumer="" a="" and="" firm="" ii="" by="" consumer="" b.="" suppose="" all="" consumers="" and="" firms="" are="" price="" takers.="" prices="" for="" the="" good="" and="" labour="" are="" denoted="" by="" p="" and="" w.="" a)="" state="" the="" definition="" of="" a="" walrasian="" equilibrium.="" (max.="" 5="" points)="" b)="" state="" walras’="" law.="" suppose="" there="" is="" an="" excess="" supply="" on="" the="" labour="" market,="" what="" does="" this="" imply="" for="" the="" market="" of="" good="" x?="" (max.="" 5="" points)="" c)="" set="" up="" the="" system="" of="" equations="" that="" determines="" the="" equilibrium.="" what="" are="" the="" endoge-="" nous="" variables?="" (max.="" 18="" points)="" d)="" suppose="" the="" price="" vector="" of="" a="" walrasian="" equilibrium="" is="" (p,="" w)="(0.5," 1).="" consider="" a="" different="" price="" vector="" (p′,="" w′)="(0.25," 0.5).="" is="" this="" a="" price="" vector="" of="" a="" walrasian="" equilibrium="" as="" well?="" explain="" briefly.="" (max.="" 5="" points)="">
Aug 22, 2021
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