The question is mix of Micro and Macro. Please answer part a, b and c of the question. Please solve (or reject) it as soon as possible. Thank you.
![7. Consider an economy where identical agents (of mass 1 ) live for two periods: youth<br>(period 1) and old age (period 2). The utility function of a representative agent born at<br>time t is given by<br>u (C1t, C2,t+1) = log (C1,4) + B log (C2++1)<br>where ci denotes consumption in youth, c2 denotes consumption in old age, and 0 <<br>B < 1 is the discount factor reflecting her time preference. In her youth the represen-<br>tative agent supplies her endowment of 1 unit of labour inelastically and receives the<br>market-determined wage rate w1. So in her youth the agent faces the budget constraint<br>C1t + $t = wo where s; denotes her savings. When old, she just consumes her savings<br>from youth plus the interest earning on her savings, sri+1, where ri+1 is the market-<br>determined interest rate in period t + 1. That is, when old, her budget constraint is<br>C2,1+1 = (1+ r+1) s1<br>(a) Set up the agent's utility maximization problem by showing her choice variables<br>clearly.<br>(b) Write down the first order conditions for this maximization problem and derive the<br>savings function. Explain how savings. 8t. if it does, depends on the interest rate<br>Te+1<br>The production function of the economy is given by Y, = AK;L;¯ª 0 < a < 1,<br>where K and L denote the amounts of capital and labour in the economy, respec-<br>tively. Capital depreciates fully after use, that is. the rate of depreciation of capital<br>is one. Factor markets being competitive, the equilibrium factor prices are given by<br>their respective marginal products.<br>(c) Derive the equilibrium wage rate (w.) of the economy in terms of K,.[ Keep in mind<br>that the mass of agents is 1 and each agent supplies her endowment of 1 unit of<br>labour inelastically.]<br>](https://s3.us-east-1.amazonaws.com/storage.unifolks.com/qimg-008/008_7jhxyp0-go240c1q.png)
Extracted text: 7. Consider an economy where identical agents (of mass 1 ) live for two periods: youth (period 1) and old age (period 2). The utility function of a representative agent born at time t is given by u (C1t, C2,t+1) = log (C1,4) + B log (C2++1) where ci denotes consumption in youth, c2 denotes consumption in old age, and 0 < b="">< 1="" is="" the="" discount="" factor="" reflecting="" her="" time="" preference.="" in="" her="" youth="" the="" represen-="" tative="" agent="" supplies="" her="" endowment="" of="" 1="" unit="" of="" labour="" inelastically="" and="" receives="" the="" market-determined="" wage="" rate="" w1.="" so="" in="" her="" youth="" the="" agent="" faces="" the="" budget="" constraint="" c1t="" +="" $t="wo" where="" s;="" denotes="" her="" savings.="" when="" old,="" she="" just="" consumes="" her="" savings="" from="" youth="" plus="" the="" interest="" earning="" on="" her="" savings,="" sri+1,="" where="" ri+1="" is="" the="" market-="" determined="" interest="" rate="" in="" period="" t="" +="" 1.="" that="" is,="" when="" old,="" her="" budget="" constraint="" is="" c2,1+1="(1+" r+1)="" s1="" (a)="" set="" up="" the="" agent's="" utility="" maximization="" problem="" by="" showing="" her="" choice="" variables="" clearly.="" (b)="" write="" down="" the="" first="" order="" conditions="" for="" this="" maximization="" problem="" and="" derive="" the="" savings="" function.="" explain="" how="" savings.="" 8t.="" if="" it="" does,="" depends="" on="" the="" interest="" rate="" te+1="" the="" production="" function="" of="" the="" economy="" is="" given="" by="" y,="AK;L;¯ª" 0="">< a="">< 1,="" where="" k="" and="" l="" denote="" the="" amounts="" of="" capital="" and="" labour="" in="" the="" economy,="" respec-="" tively.="" capital="" depreciates="" fully="" after="" use,="" that="" is.="" the="" rate="" of="" depreciation="" of="" capital="" is="" one.="" factor="" markets="" being="" competitive,="" the="" equilibrium="" factor="" prices="" are="" given="" by="" their="" respective="" marginal="" products.="" (c)="" derive="" the="" equilibrium="" wage="" rate="" (w.)="" of="" the="" economy="" in="" terms="" of="" k,.[="" keep="" in="" mind="" that="" the="" mass="" of="" agents="" is="" 1="" and="" each="" agent="" supplies="" her="" endowment="" of="" 1="" unit="" of="" labour="">
Extracted text: The role of the financial sector (banks, stock market, and so on) is to mobilize the savings of households to bring it for effective use by the production sector. But the financial sector does not work well and a fraction 0 < 0="">< 1="" of="" aggregate="" savings="" gets="" lost="" (vanishes="" in="" thin="" air)="" in="" the="" process="" of="" intermediation.="" (d)="" derive="" the="" law="" of="" motion="" of="" capital="" (that="" is.="" express="" capital="" in="" period="" t="" +="" 1,="" kt+1;="" in="" terms="" of="" capital="" in="" period="" t,="" k;)="" (e)="" derive="" the="" steady="" state="" anount="" of="" capital="" of="" the="" economy.="" (f)="" how="" does="" the="" steady="" state="" amount="" of="" capital="" depend="" on="" the="" inefficiency="" of="" the="" financial="" sector="">