Title PageInvestment and Goods MarketOctober 19, 20161 / 31Reminder: Where is this Going?We are building different markets:1. Consumers supply labor (i.e., labor markets)2. Consumers save to smooth (i.e., consumption and savings)3. Firms build capital (i.e., investment)4. Connect consumer savings and firm investment5. Short term fluctuations and monetary/fiscal policy2 / 31InvestmentIRecall the capital accounting from Production ModelIThis is the amount spent by private businesses on capitalexpenditures to support future productionIIt translates into the increase in the stock of capital, K, ofthe private sector, and increases the productive capacityIIt includes depreciation expenditures to maintain theexisting capital stock. Let δ be the rate of depreciation:It= Kt+1− Kt+ δKt= Kt+1− (1 − δ) Kt3 / 31Components of InvestmentIFixed Investment :INon residential fixed investmentIExpenditures on new equipment and softwareIPurchases of new office structures and plantsIResidential investmentIConstruction of new homes and apartmentsIChange in inventories (usually small)4 / 31Investment Share of GDPIUsually around 15-25% of GDP for most countries5 / 31Investment and the Busines CycleIFluctuations in investment are often the main source ofshort term cycles and recessions.6 / 31Volatility of Investment vs. GDPIBy far the most volatile component of aggregate spendingI3 times as volatile as GDP, 6 times that of consumption7 / 31Neo-Classical model of InvestmentIRelated to Solow and production model of the firm, butwith optimal choice of capital accumulationIMajor modelling differences between labor and capital :IFirm decision : firms hire labor but usually own the capitalITiming : Labor can usually be quickly adjusted butcapital takes time to installICapital lasts, and it is costly to adjust and install so itsfuture productivity matters8 / 31Firm Profits given CapitalIAssume again that there are only two periodsIUnlike the labor market section, firms will own capitalrather than renting it from the consumers. Won’t matterIThe “cost” per unit of capital is the user cost: opportunitycost to the owners of the firm, replacement cost, etc.IFirm choose its desired stock to maximize profitsπ = AF (K, N) − uc · K − W · NIWhere uc is the adjusted real user cost of capital per unitof capital.9 / 31User Cost of CapitalIn general, the user cost of capital, for a period of time, includestwo components:IA depreciation component : As we use the capital forproduction, it depreciates i.e.,IParts need to be replaced,ILoses economic value.INeeds to be upgraded to match other technological changesIAn interest component:IOpportunity cost to investment in capitalIThe resources that are used for investment could havegenerated some interest revenue r if invested somewhereelse, so owners of capital have to be compensated.10 / 31Desired Stock of CapitalIFocus on capital, so take labor as givenI(Remember in labor markets, we took capital as given)IInvestment today only affects the future capital stock :Kf= (1 − δ)K + IIChoosing I amounts to choosing Kf, so directly choose KfIAssume that firms are able to finance any investment frompublic financial markets without any frictions, so in thepresent they can set Kfand only need to worry aboutpaying the costs for the capital tomorrow.11 / 31Desired Capital StockITo choose Kftoday, needs to know:Ithe user cost of capital tomorrowIthe return of a marginal unit of capital tomorrowIThe firm chooses its desired Kfto maximize profitsmaxKfnAfF (Kf, Nf) − ucf· Kf− wf· NfoIFOCs with respect to Kf:Af∂F (Kf, Nf)∂Kf| {z }Future marginal product− ucf|{z}marginal cost= 0MP Kf− MCf= 0IRemember we are leaving N and Nfcompletely flexible.ICould combine with decisions in the labor markets modelIWe normalized so that all tax terms are all in the user cost12 / 31Optimal Level of CapitalIThe desired Kfis reached when the slope of theproduction function equals that of the cost.IIf MP K > MC, gains to increase K furtherIIf MP K < MC, then capital expensive, so reduceIIf MP K = MC : optimum, no improvements possible13 / 31Optimal Investment and the User Costof CapitalIDepreciation rate δ is known, so once the firm computesthe desired Kf, if capital can be freely adjusted right away,optimal investment today is:I = Kf− (1 − δ)KITwo alternative choices for investment/capital unit:1. Can get MP Kftomorrow and then resell the capital lefttomorrow, (1 − δ)2. Sell the unit of capital today at and earn interest to obtain(1 + r) tomorrow.IIn equilibrium, the two investment possibilities have toyield the same return (no arbitrage):MP Kf+ (1 − δ) = (1 + r)14 / 31User Cost in EquilibriumIRearrange the user cost formulaMP Kf= (1 + r) − (1 − δ)IRecall, the firm’s optimal choice is such thatucf= MP KfIThenucf= r + δISo in equilibrium unit user cost, is the foregone interestrate + depreciation rate15 / 31User Cost EqulibriumIChecking for optimality, from the bookIWhy is MP Kfdecreasing in K?16 / 31Comparative Statics of MPKFor example, a change in future productivity:IEach unit of capital is more productive, so invest more:The optimal amount of Kfgoes up17 / 31Change in Future ProductivityIAlternatively, looking at directly with MP Kfand uc18 / 31Comparative Statics of User CostIChanges in r, δ show up in the user cost :ucf= r + δIIncrease in r :IIt becomes more profitable to put profits in a bank, so theopportunity cost of holding capital increases: ucf↑IIncrease in δ:ICapital depreciates more, so it becomes more costly to holdcapital: ucf↑19 / 31General Effect of User CostIIf the user cost increases from uc1to uc2the cost curveshifts up :IEach unit of capital is more costly, so invest less. Theoptimal amount of Kfgoes down20 / 31Example: Change in Interest rate21 / 31Other Determinants of Desired CapitalStockICurrent productivity goes up ?INo effect on Kf: only future productivity mattersIFuture labor goes up ?IIf production function is Cobb-Douglas, then productioncurve moves up : each unit of capital is more productive, soKfgoes up22 / 31Short Run vs. Long RunTwo basic stages1. Short Run: factors that can adjust immediately and withoutcost or other frictions → adjust immediatelyIexample: labor supply; labor demand2. Long Run: factors that required time to adjust → adjust in thelong run onlyIexample: capitalFrictions/costs/time delays are model dependent.23 / 31IntroductionNow we want to combine the two models together.IThe key assumption is