Chapter 23CapitalRate of ReturnSlide 4Slide 5Slide 6Slide 7Rate of Return and Price of Future GoodsSlide 9Demand for Future GoodsUtility MaximizationSlide 12Intertemporal ImpatienceSlide 14Slide 15Slide 16Effects of Changes in rSlide 18Supply of Future GoodsEquilibrium Price of Future GoodsSlide 21The Equilibrium Rate of ReturnRate of Return, Real Interest Rates, and Nominal Interest RatesSlide 24The Firm’s Demand for CapitalDeterminants of Market Rental RatesSlide 27Nondepreciating MachinesOwnership of MachinesSlide 30Theory of InvestmentPresent Discounted ValueSlide 33Slide 34Simple CaseSlide 36Slide 37General CaseSlide 39Slide 40Slide 41Slide 42Slide 43Cutting Down a TreeSlide 45Slide 46Slide 47Optimal Resource Allocation Over TimeSlide 49Slide 50Slide 51A Mathematical DevelopmentSlide 53Slide 54Slide 55Slide 56Slide 57Exhaustible ResourcesSlide 59Slide 60Slide 61Slide 62Important Points to Note:Slide 64Slide 65Slide 66Slide 67Chapter 23CAPITALCopyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONSEIGHTH EDITIONWALTER NICHOLSONCapital•The capital stock of an economy is the sum total of machines, buildings, and other reproducible resources in existence at a point in time–these assets represent some part of the economy’s past output that was not consumed, but was instead set aside for future productionRate of ReturnTimeConsumptiont1t2t3C0sAt period t1, a decision is made to hold s from current consumption for one periods is used to produce additional output only in period t2 xOutput in period t2 rises by xConsumption returns to its long-run level (C0) in period t3Rate of Return•The single period rate of return (r1) on an investment is the extra consumption provided in period 2 as a fraction of the consumption forgone in period 111sxssxrRate of ReturnTimeConsumptiont1t2t3C0sAt period t1, a decision is made to hold s from current consumption for one periods is used to produce additional output in all future periods yConsumption rises to C0 + y in all future periodsRate of Return•The perpetual rate of return (r) is the permanent increment to future consumption expressed as a fraction of the initial consumption foregonesyr Rate of Return•When economists speak of the rate of return to capital accumulation, they have in mind something between these two extremes–a measure of the terms at which consumption today may be turned into consumption tomorrowRate of Return andPrice of Future Goods•Assume that there are only two periods•The rate of return between these two periods (r) is defined to be101CCr•Rewriting, we getrCC1110Rate of Return andPrice of Future Goods•The relative price of future goods (P1) is the quantity of present goods that must be foregone to increase future consumption by one unitrCCP11101Demand for Future Goods•An individual’s utility depends on present and future consumptionU = U(C0,C1) and the individual must decide how much current wealth (W) to devote to these two goods•The budget constraint isW = C0 + P1C1Utility MaximizationC1C0WW/P1W = C0 + P1C1U0U1The individual will maximize utility by choosing to consume C0* currently and C1* in the next periodC0*C1*-Utility Maximization•The individual consumes C0* in the present period and chooses to save W - C0* to consume next period•This future consumption can be found from the budget constraintP1C1* = W - C0*C1* = (W - C0*)/P1C1* = (W - C0*)(1 + r)Intertemporal Impatience•Individuals’ utility-maximizing choices over time will depend on how they feel about waiting for future consumption•Assume that an individual’s utility function for consumption [U(C)] is the same for both periods but period 1’s utility is discounted by a “rate of time preference” of 1/(1+) (where >0)Intertemporal Impatience•This means that)(11)(),(1010CUCUCCU•Maximization of this function subject to the intertemporal budget constraint yields the Lagrangian expressionrCCWCCU1),(1010LIntertemporal Impatience•The first-order conditions for a maximum are0)('00CUCL01)('1111rCUCL0110rCCWLIntertemporal Impatience•Dividing the first and second conditions and rearranging, we find)('11)('10CUrCU•Therefore,–if r = , C0 = C1–if r < , C0 > C1–if r > , C0 < C1Effects of Changes in r•If r rises (and P1 falls), both income and substitution effects will cause more C1 to be demanded–unless C1 is inferior (unlikely)•This implies that the demand curve for C1 will be downward slopingEffects of Changes in r•The sign of C0/P1 is ambiguous–the substitution and income effects work in opposite directions•Thus, we cannot make an accurate prediction about how a change in the rate of return affects current consumptionSupply of Future Goods•An increase in the relative price of future goods (P1) will likely induce firms to produce more of them because the yield from doing so is now greater–this means that the supply curve will be upward slopingEquilibrium Price ofFuture GoodsC1P1SDC1*P1*Equilibrium occurs at P1* and C1*The required amount of current goods will be put into capital accumulation to produce C1* in the futureEquilibrium Price ofFuture Goods•We expect that P1 < 1–individuals require some reward for waiting–capital accumulation is “productive”•sacrificing one good today will yield more than one good in the futureThe Equilibrium Rate of Return•The price of future goods isP1* = 1/(1+r)•Because P1* is assumed to be < 1, the rate of return (r) will be positive•P1* and r are equivalent ways of measuring the terms on which present goods can be turned into future goodsRate of Return, Real Interest Rates, and Nominal Interest Rates•Both the rate of return and the real interest rate refer to the real return that is available from capital accumulation•The nominal interest rate (R) is given byrate) inflation expected 1)(1(1  rR)1)(1(1-ePrRRate of Return, Real Interest Rates, and Nominal Interest Rates•Expansion of this equation yields--eePrPrR 11small, is that Assuming-ePr• -ePrRThe Firm’s Demand for Capital•In a perfectly competitive market, a firm will choose to hire that number of machines for which the MRP is equal to the market rental