Equilibrium with Production and Labor Supply

Transcription

1 Equilibrium with Production and Labor Supply ECON 30020: Intermediate Macroeconomics Prof. Eric Sims University of Notre Dame Fall / 20

2 Production and Labor Supply We continue working with a two period, optimizing, equilibrium model of the economy We augment the model with which we have been working along the following two dimensions: 1. We model production and capital accumulation 2. Model endogenous labor supply 2 / 20

3 Firm There exists a representative firm. The firm produces output using capital, K t, and labor, N t, according to the following production function: Y t = A t F (K t, N t ) A t is exogenous productivity variable. Abstract from trend growth F ( ) has the same properties as assumed in the Solow model increasing in both arguments, concave in both arguments, both inputs necessary. For example: Y t = A t K α t N 1 α t, 0 < α < 1 3 / 20

4 Capital Accumulation Differently than the Solow model, we assume that the firm owns its capital and makes the capital accumulation decisions Actually doesn t matter whether household or firm makes capital accumulation decisions Current capital, K t, is predetermined and hence exogenous. Capital accumulates according to: K t+1 = q t I t + (1 δ)k t q t : exogenous variable which measures efficiency of transforming investment into capital Will call investment shock for short. Reduced form way to model health of financial sector. Negative financial shock q t 4 / 20

5 Dividends and Firm Valuation The representative household owns the firm. The firm returns any difference between revenue and cost to the household each period in the form of a dividend Dividend is output less payments to labor less expenditure on new capital, which is investment: D t = Y t w t N t I t D t+1 = Y t+1 w t+1 N t+1 I t+1 Value of the firm is present discounted value of stream of dividends: V t = D t + D t r t 5 / 20

6 Firm Problem Pick N t, N t+1, I t, and I t+1 to maximize V t, subject to capital accumulation equation in both periods: max V t = D t + D t+1 N t,n t+1,i t,i t r t s.t. K t+1 = q t I t + (1 δ)k t K t+2 = q t+1 I t+1 + (1 δ)k t+1 Terminal condition: want K t+2 = 0, implies: I t+1 = (1 δ)k t+1 q t+1 Could call this the liquidation value of the firm 6 / 20

7 First Order Conditions Easiest to (i) impose terminal condition and (ii) re-write problem in terms of choosing K t+1 instead of I t by substituting in the period t capital accumulation equation First order conditions are: w t = A t F N (K t, N t ) w t+1 = A t+1 F N (K t+1, N t+1 ) 1 = 1 [ q t A t+1 F K (K t+1, N t+1 ) + (1 δ)q ] t 1 + r t q t+1 Discussion and intuition 7 / 20

8 Labor Demand First two conditions are static (same in each period) and implicitly characterize a downward-sloping labor demand curve: N t = N d (w t, A t, K t ) + + ww tt AA tt or KK tt AA tt FF NN (KK tt, NN tt ) NN tt 8 / 20

9 Investment Demand Second first order condition implicitly defines a demand for K t+1, which can be used in conjunction with the accumulation equation to get an investment demand curve: I t = I d (r t, A t+1 +, q t, K t ) + rr tt AA tt+1 or qq tt or KK tt II tt = II dd (rr tt, AA tt+1, qq tt, KK tt ) II tt 9 / 20

10 Household There exists a representative household. Households gets utility from consumption and leisure, where leisure is L t = 1 N t, with N t labor and available time normalized to 1 Lifetime utility: U = u(c t, 1 N t ) + βu(c t+1, 1 N t+1 ) Example flow utility functions: u(c t, 1 N t ) = ln C t + θ t ln(1 N t ) u(c t, 1 N t ) = ln [C t + θ t ln(1 N t )] Here, θ t is an exogenous labor supply shock governing utility form leisure (equivalently, disutility from labor) Notation: u C denotes marginal utility of consumption, u L marginal utility of leisure (marginal utility of labor is u L ) 10 / 20

11 Budget Constraints Household faces two flow budget constraints, conceptually the same as before, but now income is partly endogenous: C t + S t w t N t + D t C t+1 + S t+1 S t w t+1 N t+1 + D t+1 + r t S t Household takes D t and D t+1 as given (ownership different than management) Terminal condition: S t+1 = 0. Gives rise to IBC: C t + C t r t = w t N t + D t + w t+1n t+1 + D t r t 11 / 20

12 First Order Conditions Do the optimization in the usual way. The following first order conditions emerge: u C (C t, 1 N t ) = β(1 + r t )u C (C t+1, 1 N t+1 ) This is the usual Euler equation, only looks different to accommodate utility from leisure u L (C t, 1 N t ) = w t u C (C t, 1 N t ) u L (C t+1, 1 N t+1 ) = w t+1 u C (C t+1, 1 N t+1 ) Discussion and intuition 12 / 20

13 Optimal Decision Rules Can go from first order conditions to optimal decision rules Cutting a few corners, we get the same consumption function as before: C t = C d (Y t, Y t+1, r t ) + Or, if there were government spending, with Ricardian Equivalence we d have: C t = C d (Y t G t + +, Y t+1 G t+1, r t ) + 13 / 20

14 Labor Supply First order condition for N t can be characterized by an indifference curve / budget line diagram similar to the two period consumption case: CC tt ww tt + DD tt uu LL uu CC = ww tt CC 0,tt UU 0 DD tt LL 0,tt 1 LL tt 14 / 20

15 Income and Substitution Effects of Higher Wage CC tt ww 1,tt + DD tt CC 1,tt New bundle UU 1 ww 0,tt + DD tt h CC 0,tt Hypothetical bundle with new ww tt but fixed utility CC 0,tt Original bundle UU 0 DD tt h LL 0,tt LL 1,tt LL 0,tt 1 LL tt 15 / 20

16 Labor Supply Function We assume that the substitution effect of a higher wage dominates the income effect This means that labor supply is increasing in w t In principle, labor supply would also be affected by non-wage income and the real interest rate (anything which would impact C t ) We will abstract from this. We assume that labor is an increasing function of w t and a decreasing function of θ t, an exogenous variable which we take to be a measure of preferences for leisure (or more generally anything other than w t which affects labor supply): N t = N s (w t +, θ t ) 16 / 20

17 Labor Supply Graphically ww tt NN tt = NN ss (ww tt, θθ) θθ NN tt 17 / 20

18 Market-Clearing Market-Clearing Requires that S t = 0. Why? No other entity operates in market for bonds Household budget constraint imposing this: C t = w t N t + D t Given definition of D t, this is: Y t = C t + I t Murky definition of saving here. Could write problem where firm finances capital by issuing debt (as opposed to equity, as presented here). Optimal decision rules would be the same see Effectively, firm saves for household by doing investment and not paying a dividend. So even though there is no bond-holding in this economy in equilibrium, one can think about S t = Y t C t being equal to I t. 18 / 20

19 Equilibrium Conditions Four optimal decision rules: C t = C d (Y t, Y t+1, r t ) N t = N s (w t, θ t ) N t = N d (w t, A t, K t ) I t = I d (r t, A t+1, q t, K t ) Market-clearing condition plus production function: Y t = C t + I t Y t = A t F (K t, N t ) Endogenous variables: Y t, N t, C t, I t (quantities), w t and r t (prices) Exogenous variables: A t, A t+1, K t, θ t, q t 19 / 20

20 Adding a Government Doesn t change much. Ricaridian Equivalence still holds: C t = C d (Y t G t, Y t+1 G t+1, r t ) N t = N s (w t, θ t ) N t = N d (w t, A t, K t ) I t = I d (r t, A t+1, q t, K t ) Y t = C t + I t + G t Y t = A t F (K t, N t ) Now G t and G t+1 are exogenous, and T t and T t+1 are irrelevant 20 / 20