Equilibrium In An Endowment Economy

Equilibrium in an Endowment EconomyECON 30020: Intermediate MacroeconomicsProf. Eric SimsUniversity of Notre DameFall 20161 / 17

General EquilibriumIWe previously studied the optimal decision problem of ahousehold. The outcome of this was an optimal decision rule(the consumption function)IThe decision rule takes prices as given. In two periodconsumption model, the only price is rtThree modes of economic analysis:I1. Decision theory: derivation of optimal decision rules, takingprices as given2. Partial equilibrium: determine the price in one market, takingthe prices in all other markets as given3. General equilibrium: simultaneously determine all prices in allmarketsIMacroeconomics is focused on general equilibriumIHow do we go from decision rules to equilibrium? Whatdetermines prices?2 / 17

Competitive EquilibriumIWebster’s online dictionary defines the word equilibrium to be“a state in which opposing forces or actions are balanced sothat one is not stronger or greater than the other.”IIn economics, an equilibrium is a situation in which pricesadjust so that (i) all parties are content supplying/demandinga given quantity of goods or services at those prices and (ii)markets clearIIf parties were not content, they would have an incentive tobehave differently. Things wouldn’t be “balanced” to useWebster’s termsIA competitive equilibrium is a set of prices and allocationswhere (i) all agents are behaving according to their optimaldecision rules, taking prices as given, and (ii) all marketssimultaneously clear3 / 17

Competitive Equilibrium in an Endowment EconomyIAn endowment economy is a fancy term for an economy inwhich there is no endogenous production – the amount ofincome/output is exogenously givenIWith fixed quantities, it becomes particularly clear how priceadjustment results in equilibriumBasically, what we do is take the two period consumptionmodel:IIIIIIIOptimal decision rule: consumption functionMarket: market for saving, StPrice: rt (the real interest rate)Market-clearing: in aggregate, saving is zero (equivalently,Yt Ct )Allocations: Ct and Ct 1This is a particularly simple environment, but the basic ideacarries over more generally4 / 17

SetupIThere are L total agents who have identical preferences, butpotentially different levels of income. Index households by jIEach household can borrow/save at the same real interestrate, rtIEach household solves the following problem:maxCt (j ),Ct 1 (j )U (j ) u (Ct (j )) βu (Ct 1 (j ))s.t.Ct 1 ( j )Yt 1 (j )Ct ( j ) Yt (j ) 1 rt1 rtIOptimal decision rule is the standard consumption function:Ct (j ) C d (Yt (j ), Yt 1 (j ), rt )5 / 17

Market-ClearingIIIn this context, what does it mean for markets to clear?Aggregate saving must be equal to zero:LSt St (j ) 0j 1IIWhy? One agent’s saving must be another’s borrowing andvice-versaBut this implies:L (Yt (j ) Ct (j )) 0 j 1IL Yt (j ) j 1L Ct ( j )j 1In other words, aggregate income must equal aggregateconsumption:Yt Ct6 / 17

Everyone the SameIIISuppose that all agents in the economy have identicalendowment levels in both period t and t 1Convenient to just normalize total number of agents to L 1– representative agent. Can drop j referencesOptimal decision rule:Ct C d (Yt , Yt 1 , rt )IMarket-clearing condition:Yt CtIIYt and Yt 1 are exogenous. Optimal decision rule iseffectively one equation in two unknowns – Ct (the allocation)and rt (the price)Combining the optimal decision rule with the market-clearingcondition allows you to determine both rt and Ct7 / 17

Graphical AnalysisIDefine total desired expenditure as equal to consumption:Ytd C d (Yt , Yt 1 , rt )IIIIITotal desired expenditure is a function of income, YtBut income must equal expenditure in any equilibriumGraph desired expenditure against income. Assume totaldesired expenditure with zero current income is positive – i.e.C d (0, Yt 1 , rt ) 0. This is sometimes called “autonomousexpenditure”Since MPC 1, there will exist one point where incomeequals expenditureIS curve: the set of (rt , Yt ) pairs where income equalsexpenditure assuming optimal behavior by household.Summarizes “demand” side of the economy. Negativerelationship between rt and Yt8 / 17

Derivation of the IS ��� 𝑌𝑌𝑡𝑡𝑌𝑌𝑡𝑡𝑑𝑑 𝐶𝐶 𝑑𝑑 (𝑌𝑌𝑡𝑡 , 𝑌𝑌𝑡𝑡 1 , 𝑟𝑟1,𝑡𝑡 )𝑌𝑌𝑡𝑡𝑑𝑑 𝐶𝐶 𝑑𝑑 (𝑌𝑌𝑡𝑡 , 𝑌𝑌𝑡𝑡 1 , 𝑟𝑟0,𝑡𝑡 )𝑌𝑌𝑡𝑡𝑑𝑑 𝐶𝐶 𝑑𝑑 (𝑌𝑌𝑡𝑡 , 𝑌𝑌𝑡𝑡 1 , 𝑟𝑟2,𝑡𝑡 )𝑟𝑟2,𝑡𝑡 𝑟𝑟0,𝑡𝑡 𝑡𝐼𝐼𝐼𝐼𝑌𝑌𝑡𝑡9 / 17

The Y s CurveIIThe Y s curve summarizes the production side of the economyIn an endowment economy, there is no production! So the Y scurve is just a vertical line at the exogenously given level of Yt𝑟𝑟𝑡𝑡𝑌𝑌 𝑠𝑠𝑌𝑌0,𝑡𝑡𝑌𝑌𝑡𝑡10 / 17

EquilibriumIMust have income expenditure (demand side) production(supply-side). Find the rt where IS and Y s cross𝑌𝑌𝑡𝑡𝑑𝑑 ��𝑡𝑑𝑑 𝐶𝐶𝑡𝑡 𝐶𝐶 𝑑𝑑 (𝑌𝑌𝑡𝑡 , 𝑌𝑌𝑡𝑡 1 , 𝑟𝑟0,𝑡𝑡 �𝑡𝑡𝑌𝑌 0,𝑡𝑡𝑌𝑌𝑡𝑡11 / 17

Supply Shock: Yt𝑌𝑌𝑡𝑡𝑑𝑑 ��𝑡𝑑𝑑 𝐶𝐶𝑡𝑡 𝐶𝐶 𝑑𝑑 (𝑌𝑌𝑡𝑡 , 𝑌𝑌𝑡𝑡 1 , 𝑟𝑟1,𝑡𝑡 )𝑌𝑌𝑡𝑡𝑑𝑑 𝐶𝐶𝑡𝑡 𝐶𝐶 𝑑𝑑 (𝑌𝑌𝑡𝑡 , 𝑌𝑌𝑡𝑡 1 , 𝑟𝑟0,𝑡𝑡 � 𝑠𝑠𝑌𝑌𝑡𝑡𝑌𝑌 𝑠𝑠 ���𝑡12 / 17

Demand Shock: Yt 1𝑌𝑌𝑡𝑡𝑑𝑑 ��𝑡𝑑𝑑 𝐶𝐶𝑡𝑡 𝐶𝐶 𝑑𝑑 (𝑌𝑌𝑡𝑡 , 𝑌𝑌1,𝑡𝑡 1 , 𝑟𝑟0,𝑡𝑡 )𝑌𝑌𝑡𝑡𝑑𝑑 𝐶𝐶𝑡𝑡 𝐶𝐶 𝑑𝑑 𝑌𝑌𝑡𝑡 , 𝑌𝑌0,𝑡𝑡 1 , 𝑟𝑟0,𝑡𝑡 𝐶𝐶 𝑑𝑑 𝑌𝑌𝑡𝑡 , 𝑌𝑌1,𝑡𝑡 1 , 𝑟𝑟1,𝑡𝑡 𝑡𝑡𝑌𝑌 �𝑌𝑡𝑡13 / 17

DiscussionIMarket-clearing requires Ct YtIFor a given rt , household does not want Ct Yt . Wants tosmooth consumption relative to incomeIBut in equilibrium cannotIrt adjusts so that household is content to have Ct YtIrt ends up being a measure of how plentiful the future isexpected to be relative to the present14 / 17

Example with Log UtilityIWith log utility, equilibrium real interest rate comes out to be(just take Euler equation and set Ct Yt and Ct 1 Yt 1 ):1 rt 1 Yt 1β YtIrt proportional to expected income growthIPotential reason why interest rates are so low throughoutworld today: people are pessimistic about the future. Theywould like to save for that pessimistic future, which ends updriving down the return on saving15 / 17

Agents with Different EndowmentsISuppose there are two types of agents, 1 and 2. L1 and L2 ofeach typeIIdentical preferencesIType 1 agents receive Yt (1) 1 and Yt 1 (1) 0, whereastype 2 agents receive Yt (2) 0 and Yt 1 (2) 1IAssume log utility, so consumption functions for each type are:11 β11Ct ( 2 ) 1 β 1 rtCt ( 1 ) IAggregate income in each period is Yt L1 and Yt 1 L216 / 17

EquilibriumIWith this setup, the equilibrium real interest rate is:1 rt 1 L2β L1INoting that L2 Yt 1 and L1 Yt , this is the same as inthe case where everyone is the same!IIn particular, given aggregate endowments, equilibrium rt doesnot depend on distribution across agents, only depends onaggregate endowmentIAmount of income heterogeneity at micro level doesn’t matterfor macro outcomes. Example of “market completeness” andmotivates studying representative agent problems moregenerally17 / 17

Graphical Analysis I De ne total desired expenditure as equal to consumption: Yd t C d(Y t,Y t 1,r t) I Total desired expenditure is a function of income, Y t I But income must equal expenditure in any equilibrium I Graph desired expenditure against income. Assume total desired expenditure with zero current income is positive { i.e. Cd