MACROECONOMIC THEORY 1
MACROECONOMIC THEORY 1 ECON 7020
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ECON7020 MACROECONOMIC THEORY I Martin Boileau II TWOPERIOD ECONOMIES A REVIEW 1 Introduction These notes brie y review two period economies This includes the consumer s problem the producer s problem and general equilibrium In what follows I borrow freely from Chiang I984 Farmer I993 Feeney I999 Obstfeld and Rogoff 1996 and Smith 1997 2 Constrained Optimization Throughout we make extensive use of constrained optimization That is we aim to solve problems of the following form maxfltx1xngt subjectto gltx1xngt y This problem is solved using the Lagrangian Lfx177xn ly79x177xnl7 where A is a Lagrange multiplier Assuming an interior solution the necessary rst order necessary conditions for a maximum are a a a 8 98 iAa i0 for j1n The second order su cient conditions for a maximum are that the bordered hessian the matrix of second derivatives of the function bordered by the rst derivatives of the function g is negative de nite ie the bordered principal minors alternate in sign To simplify our life most of our applications will deal with a simple case for which there exists an interior absolute maximum That is we will assume that the objective function is explicitely quasiconcave and that the constraint set is convex o A function is explicitely quasiconcave if for any pair of distinct points u and U in the domain of f and for 0 lt0 lt I gt f0u1 70M gt I f A twice continuously differentiable function is quasiconcave if its Hessian bordered by its rst derivatives is negative de nite o A function is convex only if for any pair of distinct points u and U in the domain of g and for 0 lt 0 lt 1 09u 1 0g39u 2 g0u 1 0 u f A twice continuously differentiable function is convex if its hessian is every where positive semide nite 3 Consumption 31 The Standard Consumer s Problem The standard consumer s problem is max Ucl 02 subject to p101 p202 W where 01 an 02 denote consumption of goods 1 and 2 p1 and p2 are the prices of these goods and W is the consumer s income The above optimization problem is composed of an objective function and a constraint The objective function is the utility function It summarizes the consumer s preferences for goods 01 and 02 Your basic graduate microeconomics course should layout the conditions for this function to exist and be quasiconcave In general it requires that preferences adhere to the assumptions of completeness re exivity transitivity continuity strong monotonicity non satiation and convexity The constraint is simply a convex budget set Finally we assume throughout that the consumer is a price taker the consumer takes prices as given As stated above the assumptions of a quasiconcave utility function and a convex budget constraint are suf cient to ensure the existence of an absolute maximum To analyze this simple problem then we will only consider the rst order conditions Proceeding with the Lagrangian we can solve this problem as follows max U01 02 7 p101 7 p202 The rst order conditions are 3U 7 A U 7 A 0 801 P1 1017 02 P1 7 3U 8 02 A192 U2C17 02 A192 0 The ratio of these conditions is U2C17 C2 7 P2 U101 02 i The above equation has two parts On the left is the marginal rate of substitution It describes a consumer s willingness to substitute one good for another On the right is the price ratio which describes the market s willingness for the substitution This equilibrium condition can be shown on a diagram It is the point at which the budget set is tangent to the indifference curve The equation of the budget line is K P1 7 017 02 P2 P2 such that its slope is iplpg lt 0 The indifference curve is derived from Ucl 02 constant 1ts slope is dC2 U1C1702 7 lt 0 dC1 U2C1702 32 The Two Period Consumer s Problem The main difference between the standard problem and the two period one is that we must allow for time In a dynamic framework the consumer receives an income in periods 1 and 2 Denote these income 31 and 32 Then in the rst period the consumer chooses to consume or save his income C1S1 91 where 01 is the amount consumed and 51 is the amount saved 1n the second period the consumer receives both his period income and principal plus interest on its savings Thus the consumer chooses to consume 02 92 1 T517 where 02 is the amount consumed and 7 is the real rate of interest Note that the consumer will not save in the second period because of its nite lifetime The two period budgets can be consolidated to form the intertemporal budget con straint This constraint is 01 1 1 c 1 T 2 91 1 T92 It states that the present value or discounted sum of consumption equals the present value of income There are obvious similarities between the intertemporal budget constraint 3 and the budget constraint of the standard consumer s problem We can write this last constraint as 01 p 202 P1 P1 A comparison suggests that 11 r pgpl That is the market discount factor is the relative price of second period consumption Now that we have dealt with the budget we wish to give a dynamic interpretation to the utility function As stated before goods 01 and 02 now simply correspond to consumption in periods 1 and 2 In most application we will assume that the utility function is time additive Uc1 c2 uc1 uc2 where uc is the period or instantaneous utility function with u c gt 0 and u c S 0 0 lt 3 11 p lt 1 is the subjective discount factor and p is the rate of time preference Large values of p show a preference for consumption today as the implied low value for 3 indicate a large discount on future utility The two period consumer s problem is 1 1 b t t maxuc1 uc2 su JeC 0 c1 1Tc2 31 1Ty2 The Lagrangian is L B A 1 1 u c u c i c i c 1 2 91 1 T92 1 1 T 2 The rst order conditions can be summarized by u c2 1 7 c1 1 r where u c 3ucBc This condition the so called Euler equation simply states that the intertemporal marginal rate of substitution equals the market discount factor As before this can simply be represented by a diagram The budget line is 02 1Ty1y2 1T017 such that its slope is 71 7 lt 0 The indifference curve is derived from ucl u02 constant 1ts slope is d 7 u 01 lt 0 dc1 u c2 4 This slope is an increasing function of rst period consumption ie as 01 increases the slope becomes atter 1202 i 7 u cl u c u cl d cg suQ luau02de mm 7 Mon W022 3 luauems 2039 Two interesting cases come out of our analysis The rst consumption smoothing occurs when p r such that B 11 T It implies that u cl 1 lm02 u 02 and that 01 02 The second consumption tilting occurs when p a T For example assume that p lt i such that 3 gt 11 7 the consumer discounts the future less than the market or the consumer is less impatient than the market In that case iiicl 1 ow02 gt iii02 and 01 lt 02 because marginal utility is decreasing Finally the consumer s problem can be used to yield demand functions In general demand functions are found by i solving the consumer s problem and substituting the rst order conditions in the budget constraint We show how this can be accomplished using a simple example 1 1 max ucl 31402 subject to 01 02 31 myg where 1 1 c 7 7 1 uci 171U The rst order conditions imply This condition can be substituted in the budget to yield the demand functions 0 i 1 1 i 91 my 1 was 1 91 my The above framework can be used to discuss the relation between interest rate and 02 the price of a bond Assume that the consumer has two vehicles for his savings First he can purchase an amount 1 of a bond that has the same price as rst period consumption and remits 1 7 unit of second period consumption Second he can purchase an amount 5 a of a bond at price 1 that remits a xed d units of second period consumption In that case is period 1 budget is C1b1qa1 91 The period 2 budget is 02 32 1 T b1 dal We can write the optimization problem for this case simply as maxuy1 7171 7 qa1 u 32 1 rb1 dal The rst order conditions in that case are summarized by 1022 1 i q u cl 1r T d 3 The above implies an inverse relation between bonds prices and interest rates d 1 r 1 Also it is clear that 51 131 qal 33 Savings and the Elasticity of Intertemporal Substitution So far we have discussed consumption It is however interesting to gure out how savings is related to the interest rate In general we think that an increase in the interest rate increases the return to savings and thus leads to more savings draw the savings schedule In what follows we verify this conjecture To do so however we must de ne the elasticity of intertemporal substitution We de ne the elasticity of intertemporal substitution as 7 7 7W0 00 7 cu c39 This elasticity describes the curvature of the utility function Macroeconomists use a variety of utility functions The most popular is the so called constant relative risk aversion CRRA utility In that case the period utility is Clilail uc 110 lnc if 039 1 For this utility function the elasticity of intertemporal substitution is constant 00 039 This elasticity of substitution can inform us of the response of relative consumption to an interest rate change Using the Euler equation for our CRRA case we can show that 02 Cl 301 r Taking logs and a total differential yields 02 dlog Udlog1 r 1 Thus for large values of 039 the response would be large We can now verify our initial intuition Using the budget to eliminate both 01 and 02 in the Euler equation we write 501 Tay1 51 92 1 TS1 Then 151 0W1 rlt01gtlty1 7 51 i 51 neg1 m i 51 W 1 r 01 no 1 r C201 39 Clearly a rise in the interest rate has an ambiguous effect on savings The previous result should not shock anyone There are several channels through which an increase in the rate of interest affects savings These are the substitution effect the income effect and the wealth effect 239 The substitution effect implies that a rise in the interest rate makes savings more attractive That is an increase in 7 is a reduction in the price of second period consumption pgp1 11 This should stimulate second period consumption For a given present value of income this can only be achieved by a rise in savings amp 39 The income effect implies that a rise in the interest rate increases future consumption for a given present value of income and savings In general the consumer would spread this increase to consumption in both periods The rise in rst period consumption reduces savings 51 31 7 cl The wealth effect implies that a rise in the interest rate increases the market discount factor that reduces the present value of income 31 This lowers rst period consumption and increases savings All three effects are summarized in the demand function 1 1 Cl 7 1 1r0 1 7 y1 1ry239 The substitution and income effects are related to the rst term on the right side and the wealth effect to the second term Clearly there are cases where the substitution effect dominates the income effect 0 gt 1 and cases where it does not 0 S 1 4 A Pure Exchange Economy 41 The Equilibrium A pure exchange economy is one in which there is no production and consumers only trade their endowments In what follows we assume that aggregate demand the sum of all the consumers individual demand functions can be represented by the demand function of only one consumer We call this consumer the representative agent Finally we will proceed informally and let your graduate microeconomics course deal with questions of existence and uniqueness of this equilibrium The essential elements of our economic environment are 1 Preferences Ucl 02 ucl 31402 2 Endowments 31 and 32 3 Market Clearing Conditions 01 31 and 02 32 The rst element is simply the preferences of our representative agent The second element endowments represents the income of each individual Here we assume that the representative agent has a tree in its backyard that yields fruits He gets an amount of 31 fruits in the rst period and 32 fruits in the second period The nal element shows the market clearing conditions This condition simply imposes that the demand for fruits 0 equals the supply of fruits yi Our goal is to nd the equilibrium for this economy The pure exchange equilibrium is an allocation 01 02 and prices 101102 such that 239 the consumer maximizes its utility subject to his budget constraint and that markets clear Note that I have used prices p1 and p2 instead of 7 You will recall from your intermediate microeconomic theory courses however that we can only identify 1 of those two prices or a ratio of them That is one of the two goods must be the numeraire In what follows we use rst period consumption to be the numeraire The above de nition itself suggests how to proceed to solve for the equilibrium The rst step is to solve the consumer s problem and nd the demand functions The second step is to impose the market clearing conditions This approach should allow us to solve for equilibrium values of cl 02 and 7 We show this approach using our example with CRRA preferences The consumer s problem is max ucl 31402 subject to 01 og 31 y The rst order conditions are summarized by B 01 107 I 02 71r39 9 The demand functions are 1 i 1 01 HT i 1 ab 1 C2 7 91 The market clearing conditions are At this point we have 2 demand functions and 2 market clearing conditions but only three unknowns cl 02 and 7 This is simply an illustration of Walras s law one of these equations is redundant Substituting 01 31 in the rst demand function yields 1 lty 1gt1a 1r 32 39 Then our pure exchange equilibrium is de ned by an allocation C1 91 02 92 p 2 1 lty 1gt1a P1 1T 32 The above example suggests a simple shortcut That is in the representative agent and a price ratio pure exchange economy there is no need to nd the demand functions The equilibrium is found as follows 1 1 max ucl 31102 subject to 01 1 T02 31 my The rst order conditions can be summarized by lmposing market clearing the equilibrium is 01 91 10 C2 92 m gm 1 T 91 This equilibrium can be shown using a diagram Note that the budget line must go through the pair 31 32 Also note that in equilibrium nobody saves 4 2 Macroeconomics The main question at this point is How does this relate to macroeconomics The simple answer is that we have derived aggregate consumption and the real interest rate for an economy with no production no government and no trading partner To see this recall the national account identity Y G I G X 7 M where Y denotes gross national product GNP G is aggregate consumption I is aggregate investment G is government expenditures X is exports and M imports If we abstract from investment government and trading partners the identity reduces to Y G Thus in our simple pure exchange economy GNP is 31 in the rst period and 32 in the second Aggregate consumption is 01 in period 1 and 02 in period 2 Also note that in general we would have that S Y 7 G 7 T I G 7 T X 7 M In our case there is no investment no government and no foreign trade which explains why there is no savings in equilibrium 5 A Simple Production Economy 51 The Economic Environment We wish to extend our macroeconomic analysis to the case of a production economy As a rst step we will consider the labor leisure choice only In subsequent sections we will consider capital and investment We retain our assumption of a representative consumer and add a representative rm to the economy This rm will produce consumption goods using labor only In this section we abstract from population growth Our economic environment is described by 1 Preferences ucll1 Buczl2 We assume that the period utility function has the following characteristics u1cl gt 0 u2cl gt 0 u11cl lt 0 u22cl lt 0 and u12cl 0 sim plifying assumption 2 Technology 31 rm and 32 rm We assume that the production function exhibits gt 0 and f n lt 0 de creasing returns 3 Goods Market Clearing Conditions 01 31 and 02 32 4 Factors Market Clearing Conditions 1 711 1 and Z2 712 1 Our main addition is the production function y this function can be shown using a diagram 1n the context of our environment I is leisure n is hours worked or labor supplied Also we have xed the endowment of time per period to unity We wish to impose a number of restrictions on this production function These are 1 No labor no output 0 2 It is increasing and concave gt 0 and f n lt 0 3 lnada conditions apply 111117190 00 and limnnm 0 A good example of a function that satis es these conditions is Ana for 0 lt a lt 1 This function can be shown on a diagram There are basically two ways to nd the equilibrium of this production economy The rst is via a competitive equilibrium The second is via a planner s problem 52 The Competitive Equilibrium The idea behind the competitive equilibrium is to let the price system allocate resources As before we de ne the equilibrium as an allocation 01 02 31 32 n1 n2 1 l2 and prices T 1111 1112 such that the consumer maximizes its utility subject to budget the rm maxi mizes pro ts and markets clear We again use rst period consumption as our numeraire and de ne 1111 and 1112 as real wages The consumer s problem is max 1101 1 7 711 31102 1 7 n2 subject to C181 w1n17r1 C2 7112712 1T81 7F2 The two period budget constraint can be consolidated to yield an intertemporal budget 01 C2w1n17r1 w2n27r2 1r 1r The Lagrangian is 1 1 L 1 i 1 i A i i W21 711 B11022 712 mm 7n 1 T wznz W2 01 1 T02 12 These rst order conditions are summarized by u2lt01 1 7 H1 7 i 7111 u1011 7771 1 7 2027 n2 w2 u1021 7772 u1021 7772 1 u1cl1 7m 1r39 The rst two of these conditions state that the marginal rate of substitution between leisure and consumption equals the wage rate These can be shown on a diagram These are static or intratemporal conditions The last condition states that the intertemporal marginal rate of substitution equals the market discount factor The rm s problem is to maximize the present value of its pro ts max m 7T2 1 1 r where TF1 fn1 7111711 TF2 f 2 7112712 As for the consumer the rm is a price taker That is the rm takes interest rates and wages rates as given The rm s problem is 1 maxfn1 7111711 06012 7112712 The rst order conditions are f 711 1111 fM 7112 These intratemporal conditions simply state that the marginal product of labor equals the wage rate To solve for our equilibrium we require values for 01 02y1y2n1n2l1l2 and 7311114112 There are thus 11 unknowns that must be solved using the following 11 equations u2lt0171 7771 7 1111 f n1 U1C171n1 u2021 7772 1112 f 2 7 u1021 7772 u1021 7772 1 u1cl17n1 1r39 C1 31 HM and C2 3J2 KM 1TL11 and l2 21 53 The Planner s Problem The planner s problem is to nd the optimal allocation of resources lts problem is max Mob 1 7 m 31102 1 7 n2 subject to C1 f 1 C2 f 2 The Lagrangian is L M0171 n1 3110271 n2 A1fn1 C1 A2 712 i 02 These rst order conditions can be rearranged to yield 1120171 7 H1 1110171 711 fYh u202 1 7 n2 u1c27 1 7 n2 f 2 These conditions state that the marginal rate of substitution between leisure and consump tion equals the marginal product of labor Thus the planner must nd values for 8 unknowns 01 02 31 32 n1 n2 1 2 using the 8 equations 1120171 77711 i u10171 7771 7 f n17 1120271 77712 i u1c271 7772 7 f n27 5 4 Discussion A comparison of the equations required to solve for the equilibrium in both the compet itive equilibrium and the planner s problem demonstrate that the allocation is the same under both solution method Given that the planner s problem yields the Pareto optimal allocation it follows that the competitive equilibrium is also Pareto optimal This fact is just an example of the fundamental theorems of welfare economics There are two such theorems I will leave the proofs of these for your graduate microeconomic instructor The rst theorem is First Welfare Theorem Every competitive equilibrium is Pareto optimal It states that under certain conditions the allocation under the competitive equilib rium is Pareto optimal That is it is not possible to nd another allocation that would make one person better off while making nobody worse off The conditions ensure that there are no imperfections such as externalities public goods and altruism The second theorem is Second Welfare Theorem Every Pareto optimum can be decentralized as a competitive equilibrium This one states that the planner s allocation can be supported as a competitive equi librium This last theorem might require the existence of a tax and transfer system to ensure that the distribution of endowments is compatible with the desired allocation of resources 6 A Production Economy with Capital Accumulation 61 Production and Investment The economy that we are about to discuss has a more complex production technology The rm produces output using both labor and capital as inputs The rst addition to our economy will be the production function In general the production function is of the form Y FK N where Y is output K is the capital stock and N is hours worked We assume that this function has constant returns to scale That is it is linear homogenous A function is linear homogenous or homogenous of degree 1 when JFKN FJKJN 15 Linear homogenous functions have four useful properties Property I If the function Y FK N is linearly homogenous then it can be written in terms of the capital labor ratio k KN This is easy to show K YNF 1 N k N f or y where y YN Property II If the function Y FKN is linearly homogenous then the average products of labor and capital can be written in terms of the capital labor ratio 14 The average products are Y APN N Y Y N fk AP K K N K k Property III If the function Y FK N is linearly homogenous then the marginal products of labor and capital can be written in terms of the capital labor ratio 14 To show this rst note that iaKNiii d iaKNil aN 8N N2 an 8K 8K N39 Then the marginal products are MPN F2K7N BAg k M mmj J Ni 7 MW MPK F1K N ag kl WU Property IV Euler s Theorem If the function Y FK N is linearly homogenous then Y KF1KNNF2KN This is shown as follows Y Kf Uf N W ff1 Kf kNfk Kf Uf Nfk 16 In what follows we impose a number of restrictions on the function These are 1 No capital no output 0 2 It is increasing and concave gt 0 and f k lt 0 3 lnada conditions apply limkno 00 and limknm 0 An example of a production function that satis es these properties is the Cobb Douglas function 0 lt a lt l i FagNAKNPa In its intensive form it is y mAW It can be shown on a diagram The second addition is capital accumulation For this addition the effect of time is important We will assume that capital accumulation involves a time to build component That is investment 1 today only increases the capital stock tomorrow K2 1 1 7 5K1 where 5 is the rate of depreciation This accumulation equation can also be transformed in per capita as follows or ynkig 1 1 7 In what follows we abstract from population growth and assume that labor is supplied inelastically Accordingly we set N1 N2 l and 7 1 Our accumulation equation then becomes k2 1 1 7 In this case our economic environment is described by Preferences ucl 31402 Technology Y1 FK1 N1 and Y2 FK2 N2 Capital Accumulation K2 1 l 7 5K1 0 2 l 7 5K2 and K1 given Goods Market Clearing Conditions 01 1 Y1 and 02 2 Y2 Factor Market Clearing Conditions n1 l and n2 1 Note that xing labor supply to unity implies that per capita variables are the variables Hgt9390N14 themselves As before the equilibrium of this production economy can be found either via a competitive equilibrium or via a planner s problem 17 62 The Competitive Equilibrium The equilibrium is an allocation 0102n1n2 901311312 kg and prices n 1111 1112 1 such that the consumer maximizes its utility subject to budget the rm maximizes pro ts and markets clear We again use rst period consumption as our numeraire The consumer s problem is max 1101 31102 subject to 01 171 1amp1 1111711 11 Duo 02 7112712 1 Tb1 d2a17 where 10 is given It is worthwhile to discuss the two budget constraint First the consumer supplies labor inelastically such that n1 n2 1 Second there are two different assets that the consumer purchases In the rst period the consumer can purchase a bond that pays 1 7 units of second period consumption and shares of the representative rm at price 11 that remits dividends in the form of d2 units of second period goods The rst period income stems from his labor income and his previous holdings of shares An alternative speci cation of the rst period budget might remove confusions 01 171 1011 i 10 7111711 11110 This states that the consumer changes his holdings of shares and receives income from his current shares This market for shares will add a market clearing condition later on The optimization problem is maxuw1 all qa0 7171 7 qa1 3110112 1 rb1 algal The rst order conditions are summarized by 1022 1 q u cl T 1r The rm s problem is to maximize its value which is the sum of its current dividends and share values V d1 qAo where A0 is the number of shares initially issued We assume that the rm issues a constant number of shares that we normalize to unity A0 A1 1 Clearly the consumer s rst order conditions imply that q lg1 7 Then the rm s problem is 1 1 r d27 maxV maxd1 18 where d1 FK17n1 7 I1 7 w1711 FK17n1 7 K2 1 7 5K1 7 w1711 d2 FK27 L2 7 I2 7 w2TL2 FK27 L2 1 7 7 1112712 The optimization is 1 1r maxFK1n1 7K2 1 7 7 1111774 FK27 L2 1 7 7 1112772 The rst order conditions are 1 T 1171097712 1 7 5 WW2 1 7 5 w1 F2K17n1 fk17 Infk1 1112 F2K27n2 fk2 7 k2f k2 The rst condition states that the rm will invest to the point where the marginal product of capital equals the gross rate of interest This condition can be used to argue that an increase in the rate of interest reduces investment dki2 1 0 dr f k2 lt where k2 K27 L2 K2 n2 1 The last two conditions state that the wage rate equals the marginal product of labor To solve for our equilibrium we require values for 01 02 n1 Hg 11 1 2311312 kg and 73 1111 1112 1 There are thus 14 unknowns that must be solved using the following 14 equations gm m 1 u cl 1r d27 f k2 1 7 5 1 r 1111 fkr1 7 kidkn w2 fkr2 7 kid1 C1901y1fk31 and C2902y2fk32 k2 1 1 7 and 0 2 1 7 5 31 n1n2a11 63 The Planner s Problem As before the planner s problem is to nd the optimal allocation of resources Its problem is max ucl 31402 subject to C1 k2 fk11 5W1 02 fk2 1 5W2 The optimization problem is maxufki1 1 7 5W1 k2 3 k 2 1 5W2 The rst order condition is Zia02 1 W01 f k32 1 i 5 To solve for our equilibrium the planner solves for 01 02 n1 mg x1 x2 31 32 k2 There are thus 9 unknowns that must be solved using the following 9 equations u 02 1 W01 f k32 1 i 5 C1961y1fk31 and C22y2fk32 3 k2 1 1 7 and 0 2 1 7 5 31 7L1TL21 64 Discussion Once again a comparison of the equations required to solve for the equilibrium in both the competitive equilibrium and the planner s problem demonstrate that the allocation is Pareto optimal The allocation can be displayed on a diagram The rst element is the production possibility frontier This frontier is 02 ffk1 1 5W1 C1 1 5fk711 5W1 01 Its slope is d02 d 7f k2 1 75 lt 0 C1 20 This slope is a decreasing function of rst period consumption ie as 01 increases the slope becomes steeper d2 7 f k lt 0 dc T 2 39 The second element is the indifference curve It is derived from ucl u02 constant lts slope is d 2 u w lt 0 dcl u cz This slope is an increasing function of rst period consumption ie as 01 increases the slope becomes atter 1202 i 7 u cl u c u cl d cg u 02 luau02de mm 7 Mon W022 3 luauems 20 We can relate this case to our national account identity In this last economy we have consumption and investment Y C I It follows that savings must equal investment S Y 7 C I In our economy aggregate savings in the rst period is Y1 7 01 1 We can also use our competitive equilibrium to gure out the price of a rm s share and the rm s value The price of a rm s share is d2 1 qi 1T 7 1TFK2TL2175K271U27L2 l 1 T K2F1K27 2 H2F2K27n2 1 5K2 7112712 l 1 F1K27n21 5lK2 1 k 175 K 1Tlf2 gt1 2 Kg Thus the price of the share is the value of the capital stock Therefore purchasing the share of a rm is like purchasing a claim on its capital stock The rm s value is Vqd1 Y1175K1 711117711 Finally using the consumer s rst period budget 01 171 1amp1 1111711 11 Duo 21 we nd C1b1K2Y115K1 given that 11 a0 1 Clearly the national income identity 01 1 Y1 suggests that 131 0 In this economy there is only one vehicle for aggregate savings and it is via the rm s shares and thus its capital stock 7 What Have We Learned About Macroeconomics 1 E0 0quot 4 OT 9 The aggregate consumption function depends on the discounted sum of future income 1 1 01 7 1 1 r0 1601y1 1 ry239 The market discount factor and thus the real interest rate is related to the relative price of future consumption 1 12 1 r P1 I Both preferences and technology affect it u 02 1 1 7 f k2 1 7 5 u cl 1r T There is an inverse relation between bonds prices and interest rates d 1r q An increase in the wage rate increases labor supply and reduces labor demand Thus aggregate employment and the real wages respond to both preferences and technology u2lt01 1 7 H1 110171 7 n1 w1 Hm lnvestment responds negatively to increases in the interest rate and positively to increases in the marginal product of capital partial equilibrium In an economy with no government and no trading partners saving equals investment 51 1 Moreover it must be the case that the only vehicle for aggregate savings is the capital stock That is purchasing the share of a rm is like purchasing a claim on its capital stock References Chiang Alpha 0 1984 Fundamental Methods of Mathematical Economics New York McGraw Hill Farmer Roger E 1993 The Macroeconomics of Self Ful lling Prophecies Cambridge MIT Press Feeney JoAnne 1999 Supplemental Notes Obstfeld Maurice and Kenneth Rogofl 1996 Foundations of International Macroeco nomics Cambridge MIT Press Smith Gregor W 1997 MAcroeconomics Lecture Notes mimeo Queen s University