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Intermediate Macroeconomics Lecture 12 - A dynamic micro-founded macro model Zsófia L. Bárány Sciences Po 2014 April

Overview A closed economy two-period general equilibrium macroeconomic model: households supply labor, consume and save firms produce output, demand labor and invest government consumes resources, levies taxes and issues debt

Three markets 1. Goods market supply: output produced by firms demand: consumption by households, investment (accumulation of capital) by firms, government spending Y = C + I + G 2. Labor market supply: work done by households demand: hiring by firms 3. Financial market supply: debt issued by government + equity issued by firms demand: household savings S = I where S = S p + S g = S p B g Note when the labor and the goods market clear, also the credit market clears (Walras law)

Dynamics two time periods for simplicity the future period captures everything that will happen beyond the present the length of time in each period may depend on the context: for business cycles: the current period will be the current quarter or year for longer term issues (e.g. changes in productivity trends, demographics): the current period may be the current decade or even longer

Representative Household

Representative household preferences over consumption and leisure, now and in the future assumptions: diminishing marginal utility in each good both consumption and leisure are normal goods (in both periods) all indifference curves are convex sources of income: labor income dividends paid out by firms payout from bonds (savings and interest) must pay taxes to government

Representative household current budget constraint: future budget constraint: C + S p = w(h l) + π T C = w (h l ) + π T + (1 + r)s p combining gives the inter-temporal budget constraint: C + C 1 + r = w(h l) + π T + w (h l ) + π T 1 + r

Consumption/leisure decision intra-temporal choice, within each period: giving up an hour of leisure to work increases income by w, so budget constraint has slope w. household requires MRS l,c units of consumption to compensate for loss of utility from one hour of leisure at an interior solution optimality requires MRS l,c = w in the future period MRS l,c = w Note: here the current period intra-temporal choice is shown in the future period the kink in the constraint is at: l = h, C = π T + (1 + r)s p

Consumption/saving decision inter-temporal choice giving up a unit of consumption today allows an extra 1 + r units of consumption to be purchased in the future household requires MRS C,C units of future consumption to compensate for the loss of utility from one unit of current consumption optimality requires MRS C,C = 1 + r

Timing of labor/leisure decision inter-temporal choice working an extra hour today provides extra income w, which has value (1 + r)w in the future this extra income allows future hours of work to be reduced by (1+r)w w, with the consumption plan remaining affordable household requires MRS l,l hours of future leisure to compensate for loss of utility from one hour of current leisure optimality requires MRS l,l = (1+r)w w

Summary of household optimality conditions 1. intra-temporal consumption/leisure choice MRS l,c = w and MRS l,c = w 2. inter-temporal consumption choice MRS C,C = 1 + r 3. inter-temporal leisure choice MRS l,l = (1+r)w w This third condition is redundant, as it is implied by 1. for both periods and 2. : to see this recall the definition of MRS and expand MRS l,l = MU l MU l = MU l MU C MU l MU C = MU l MU C MU l MU C = MRS l,c MRS C,C = MU C MRS l MU,C C w(1 + r) w a reduction in current leisure can be compensated for by MRS l,c units of current consumption, each of which is equivalent to MRS C,C units of future consumption, which is in turn equivalent to 1/MRS l,c hours of future leisure optimality sufficiently characterized by optimal intra-temporal consumption/leisure choice in each period and optimal inter-temporal consumption

Wage and interest changes analyze how the representative household responds to changes in market prices: w and r household response can, of course, be decomposed into the substitution effect: response to price change with hypothetical income transfer to ensure household remains on original indifference curve the income effect: response to the hypothetical income transfer used to find the substitution effect with the price held constant we ignore these two types of income effects in the closed-economy macro model (but certainly not all types of income effects) representative household owns the representative firm: higher wage rate increases labor income, but lowers profits by same amount, leaving total income unchanged aggregate economy neither borrowing or lending in bonds income effect of a change in interest rate is zero (provided that the Ricardian Equivalence holds)

Substitution effect of a higher wage higher w increases the slope of the budget constraint substitution effect: find new point on original indifference curve satisfying the optimality condition MRS l,c = w leisure l decreases, labor supply N s = h l increases with w intra-temporal substitution

Substitution effect of a higher interest rate 1. higher r increases the slope of the budget constraint substitution effect: find new point on original indifference curve satisfying the optimality condition MRS C,C = 1 + r current consumption C decreases inter-temporal substitution of consumption)

Substitution effect: higher interest rate 2. higher r increases the slope of the budget constraint substitution effect: find new point on original indifference curve satisfying the optimality condition MRS l,l = (1+r)w w current leisure l decreases, current labor supply N s = h l increases for any w inter-temporal substitution of leisure in the (N, w) space N s shifts to the right so as r C, l both tend to increase savings

Income/wealth effects income effects also occur directly when the household budget constraint shifts higher productivity, z, boosts income increased present value of taxes reducing present value of disposable life-time income in macroeconomics, it is conventional to refer to these income effects as wealth effects (these terms are interchangeable) a positive shift of the budget constraint increases the demand for all normal goods C, l, C, l given diminishing marginal utility, there is a desire for consumption smoothing (at constant prices)

Wealth effect: consumption rightward shift of the household budget constraint demand for consumption increases (normal good) at any r in the (C, r) space C d shifts to the right when there is a positive wealth effect

Wealth effect: labor supply rightward shift of the household budget constraint greater demand for leisure (normal good) at any (w, w, r) in the (N, w) space N s shifts to the left when there is a positive wealth effect

Representative Firm

Representative firm Exactly as in the investment topic: representative firm maximizes the present value of profits V = π + π 1+r where π = Y wn I current profits π = Y w N + (1 d)k future profits Y = zf (K, N) current output Y = z F (K, N ) future output K = (1 d)k + I future capital (evolution of capital) firm chooses current and future employment, and future capital (current investment), given initial level of capital

Firm s optimality conditions mathematically: choosing I is equivalent to choosing K as I = K (1 d)k from the capital accumulation equation given K, choose N d, N d, K to max V = π + π 1+r V = Y wn (K (1 d)k) + Y w N + (1 d)k First order conditions: (N d ): Y N w = 0 MPN = w (N d ): Y N w 1+r = 0 MPN = w (K ): 1 + Y K +1 d 1+r = 0 MPK d = r 1 + r

Optimal firm behavior Note: MPK d = r pins down the optimal K, which implies I = K (1 d)k. It is possible that I < 0.

Shifters of labor demand Example: * increase in current TFP higher z higher MPN * increase in the current capital stock higher z higher MPN

Shifters of investment demand Example: * increase in expected future TFP higher z higher MPK * decrease in the current capital stock lower K no change in MPK, requires higher I to reach the same K

Government

Government Exactly as in the Ricardian equivalence topic: given government spending G, it must satisfy its inter-temporal budget constraint: G + G 1 + r = T + T 1 + r government sells bonds B = G T to finance its deficit the present discounted value of spending equals the present discounted value of taxes Note: in case of an endogenous change in r, the present value of taxes has to adjust to satisfy the budget constraint

General Equilibrium

General equilibrium now we put together the demand and supply functions of households, firms, and the government to find the equilibrium of the economy do this in two steps: 1. find the equilibrium in the labor market for a given real interest rate r that is the real wage w adjusts to clear the labor market w s.t. N s (r, w) = N d (w) = N(r) this yields the supply of output: Y s (r) = zf (K, N(r)) 2. find the equilibrium in the goods market that is the real interest rate r adjusts: r s.t. Y s (r) = Y d (r) = Y using the demand of output Y d (r) = C d (r) + I d (r) + G and the supply of output obtained at the first stage Y s (r) = zf (K, N(r))

Labor market equilibrium taking r as given for now, there is the upward sloping N s w adjusts to ensure labor supply is equal to labor demand, N s (r, w) = N d (w) = N(r) K is predetermined, the production function reveals how much output firms supply given the level of employment: Y = zf (K, N)

The output supply curve describes how much output is supplied by firms for each possible level of the interest rate can trace out the output supply curve by considering the output supplied at two different levels of interest rate consider an increase in the real interest rate: r 1 r 2 labor supply shifts to the right: inter-temporal subst. of leisure employment rises (and real wage falls) higher employment implies a larger supply of output (for a given capital stock), Y s (r) = zf (K, N(r)) output supply curve is upward sloping when drawn against r

Shifts in the output supply curve an increase in current or future government spending a decrease in the lifetime wealth of the consumer reduced demand for current leisure increased labor supply for any real wage higher equilibrium employment for any r the output supply curve shifts to the right an increase in TFP or the capital stock more output produced for any labor input the marginal product of labor is higher the labor demand curve shifts to the right higher equilibrium employment for any r the output supply curve shifts to the right

The output demand curve describes how much output is demanded in the aggregate economy for each possible level of the interest rate arises from: consumption by households C d (r) falling in r: inter-temporal substitution of consumption also increasing in lifetime wealth investment by firms I d (r) falling in r: higher user cost of capital spending by the government G - exogenous for each interest rate r: summing up the components of demand yields the output demand curve aggregate demand Y d (r) = C d (r) + I d (r) + G depends negatively on the real interest rate r

Shifts in the output demand curve shifts in C d (r), I d (r) or G shifts the output demand curve an increase in G a decrease in the present value of taxes by increasing lifetime wealth an increase in future income Y by increasing lifetime wealth an increase in future TFP, z by increasing investment demand a decrease in current capital stock, K by increasing investment demand the output demand curve is shifted right

General equilibrium putting the output supply and output demand curves together allows us to determine goods market equilibrium real interest rate adjusts to clear the goods market, r s.t. Y s (r) = Y d (r) = Y this market clearing interest rate determines the location of the N s (r)-curve and the labor market equilibrium

Credit market: Walras law there are only three markets here the goods and labor markets are in equilibrium the bond market is automatically in equilibrium this is a consequence of Walras law: intuitively, since income derives from selling in markets, after summing up over all market participants, total spending (demand) in the markets that clear is exactly equal to the income generated there if all markets except one are known to clear then there is no surplus income available to generate an excess demand in the remaining market, and an incentive to spend more in the case of excess supply so the final market clears

Credit market: Walras law Formally: private demand for bonds (= household saving, net of share purchases): S p = wn s + π T C d income from dividends (= profits of representative firm): π = Y s wn d I d by substituting the expression for dividends into the first equation: S p = Y s + w(n s N d ) I d T C d supply of bonds (= the government s budget deficit): B = G T bond market equilibrium requires S p = B, i.e. Y s + w(n s N d ) I d T C d = G T Y s (C d + I d + G) + w(n s wn d ) = 0 (Y s Y d ) + w(n s N d ) = 0 where Y d = C d + I d + G hence Y s = Y d and N s = N d imply bond market clearing

Shifters of output demand and supply Examples: increase in labor productivity (higher TFP or higher capital stock): Y s shifts right (and N d shifts right) increase in expected MPK (higher future TFP or lower current capital stock): I d and Y d shift right increase in government spending (ignoring increase in tax burden): Y d shifts right wealth (income) effects also shift Y d and Y s a positive wealth effect on hh increases demand for all normal goods: shifts C d and hence Y d to the right shifts N s and hence Y s to the left change in current or expected future TFP change in G or G implies a higher T but no wealth effect from a change on T without a change in G, by Ricardian equivalence (exogenous) change in initial capital stock

Applications of the Model

Example 1: temporary increase in government spending increase in G by G, leaving future G unchanged higher G directly shifts Y d to the right higher G increases the present value of taxes by G negative wealth effect BUT the average per-period increase in taxes < G C d falls, but by less than G, hence Y d shifts right overall N s rises, which causes Y s to shift right future consumption and leisure are normal goods: both fall inter-temporal smoothing the sum of rightward shift in Y s and leftward shift in C d from wealth effect is less than change in Y d due to G (direct effect) at the initial interest rate there is excess demand for goods/credit the interest rate needs to rise to clear the goods market/credit market in equilibrium both the real interest rate and output increase

Higher G: effect on Y s First look at the effect on Y s in detail: negative wealth effect reduces demand for leisure, shifting labor supply curve right at any real interest rate, employment is higher output supplied is higher output supply curve shifts right

Temporary increase in G: equilibrium effects first rightward shifts of N s due to wealth effect at a given r second due to the increase in r N rises; w falls both Y s and Y d shift right: Y rises Y d shift is larger than Y s shift shock is temporary & hhs smooth consumption and leisure over time r rises

Temporary increase in G: crowding out output Y increases by less than G the government spending multiplier, the ratio of the equilibrium increase in Y to the increase in G, is less than 1 this is a reflection of Y = C + I + G and that consumption and investment fall C : negative wealth effect and substitution effect of higher interest rate I : higher interest rate implies an increase in the user cost of capital this displacement effect of higher government spending is known as crowding out

Example 2: temporary increase in TFP increase in z, leaving future z unchanged increases production for any N and K directly shifts Y s to the right increases MP N shifts N d to the right tends to increase equilibrium employment N Y s shifts further to the right increases income (temporarily), triggering a positive wealth effect: C d increases (but by less than the increase in Y since shock is temporary), shifting Y d right N s falls, shifting Y s to the left (by less than the original shift) future consumption and leisure are normal goods: both increase inter-temporal smoothing sum of Y d and Y s shifts due to wealth effect is smaller than the direct shifts in Y s at initial interest rate there is excess supply for goods/credit in equilibrium the real interest falls and output increases

Higher z: effect on Y s We first look at the effect on Y s in detail: TFP increase shifts N d to the right employment is higher at each real interest rate (we do not include the wealth effect here) higher employment and TFP by itself both increase output, so Y s shifts right

Temporary higher z: equilibrium effects Explicitly taking into account the wealth effect (which is small for a temporary shock in the current period) Note: the textbook neglects/omits it in figure 11.24 rightward shift of N d leftward shift of N s w rises; N ambiguous diagram shows case where N Y d shifts right; Y s (on net) shifts right too: Y rises Y s shift is larger (since wealth effect is small)

Example 3: expected increase in future TFP good news arrives about future TFP (z ), leaving current z unchanged higher z the expected future marginal product of capital MP K increases, increasing I d at a given interest rate and therefore shifting Y d to the right increase in future output implies a positive wealth effect: C d increases, leading to a further shift of Y d to the right N s decreases, implying Y s shifts left without further assumptions, cannot draw conclusions about relative sizes of these shifts

Temporary higher z: equilibrium effects Explicitly taking into account the wealth effect Note: the textbook neglects/omits it in figure 11.26 net effect on N s is ambiguous: w ambiguous; N ambiguous Y d shifts right; Y s shifts left: r rises relative size ambiguous, so diagram shows case where N effect on current Y rises and w falls ambiguous Note: If wealth effect sufficiently small: Y and N increase; w falls