The Ramsey Model Lectures 11 to 14 Topics in Macroeconomics November 10, 11, 24 & 25, 2008 Lecture 11, 12, 13 & 14 1/50 Topics in Macroeconomics
The Ramsey Model: Introduction 2 Main Ingredients Neoclassical model of the firm (Topics 1 & 2) Consumption-savings choice for consumers (Topic 3, Certainty) Solow model + incentives to save (recall example with taxes) Lecture 11, 12, 13 & 14 2/50 Topics in Macroeconomics
Model Setup Markets and Ownership Representative Firm Representative Household Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Government Consumption & Lump-Sum Taxes Gvmt. Cons, Lump-Sum Taxes: Permanent Increase Gvmt. Cons, Lump-Sum Taxes: Temporary Increase Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Lecture 11, 12, 13 & 14 3/50 Topics in Macroeconomics
Markets and Ownership Representative Firm Representative Household Model: Markets and ownership 4 Agents Firms produce goods, hire labor and rent capital Households own labor and assets (capital), receive wages and rental payments, consume and save There are N t households and many firms Markets Inputs: competitive wage rates, w, and rental rate, R Assets: free borrowing and lending at interest rate, r Output: competitive market for consumption good Lecture 11, 12, 13 & 14 4/50 Topics in Macroeconomics
Markets and Ownership Representative Firm Representative Household Model: Firms / Representative Firm* 5 (Recall equivalence!) Seeks to maximize profits Profit t = F(K t, L t ) R t K t w t L t The FOCs for this problem deliver F(t) F(t) = R t = w t K t L t In per unit of labor terms, let f(k t ) F(k t, 1) f (k t ) = R t f(k t ) k t f (k t ) = w t Recall Euler s Theorem: factor payments exhaust output Lecture 11, 12, 13 & 14 5/50 Topics in Macroeconomics
Markets and Ownership Representative Firm Representative Household Model: Households / Representative household 6 Preferences U 0 = β t u(c t ) t=0 Budget constraint c t + a t+1 = w t l t + (1 + r t )a t, for all t = 0, 1, 2,... a 0 given Note: labor supplied inelastically, l t = 1, i.e. L t = N t Lecture 11, 12, 13 & 14 6/50 Topics in Macroeconomics
Markets and Ownership Representative Firm Representative Household Model: Households / Representative household 7 Intertemporal version of budget constraint t=0 s=0 t ( 1 1 + r s ) c t = a 0 + t=0 s=0 t ( 1 1 + r s We rule out that debt explodes (no Ponzi games) a t+1 B for some B big, but finite ) w t More compactly, PDV(c) = a(0) + PDV(w) Lecture 11, 12, 13 & 14 7/50 Topics in Macroeconomics
Markets and Ownership Representative Firm Representative Household Model: Household s problem 8 max (a t+1,c t ) t=0 s.t. β t u(c t ) t=0 c t + a t+1 = w t + (1 + r t )a t, for all t = 0, 1, 2,... a t+1 B for some B big, but finite a 0 given Lecture 11, 12, 13 & 14 8/50 Topics in Macroeconomics
Markets and Ownership Representative Firm Representative Household Model: Household s problem 9 Euler equation In general, u (c t ) = β(1 + r t+1 )u (c t+1 ) From here on, CES utility, u(c) = c1 σ 1 σ, Euler eqn. becomes, ( ) σ ct+1 = β(1 + r t+1) c t Lecture 11, 12, 13 & 14 9/50 Topics in Macroeconomics
Markets and Ownership Representative Firm Representative Household Model: Household s problem 10 Budget constraint c t + a t+1 = w t + (1 + r t )a t, for all t = 0, 1, 2,... Lecture 11, 12, 13 & 14 10/50 Topics in Macroeconomics
Markets and Ownership Representative Firm Representative Household Model: Household s problem 11 Transversality condition HH do not want to end up with positive values of assets lim t βt u (c t )a t 0 HH cannot think they can borrow at the end of their life lim t βt u (c t )a t 0 Hence, lim t βt u (c t )a t = 0 Lecture 11, 12, 13 & 14 11/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Definition of Equilibrium* 12 A competitive equilibrium is defined by sequences of quantities of consumption, {c t }, capital, {k t }, and output, {y t }, and sequences of prices, {w t } and {r t }, such that Firms maximize profits Households maximize U 0 subject to their constraints Goods, labour and asset markets clear Choices are consistent with the aggregate law of motion for capital K t+1 = (1 δ)k t + I t Lecture 11, 12, 13 & 14 12/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Characterizing Equilibrium Quantities* 13 From the equilibrium conditions derived before, we find: There cannot be arbitrage opportunities in equilibrium R t δ = r t In equilibrium it does not pay to invest in capital directly. The riskless asset and capital have the same payoff. Lecture 11, 12, 13 & 14 13/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Characterizing Equilibrium Quantities* 14 From the equilibrium conditions derived before, we find: Substituting out all the prices leads to the following set of necessary and sufficient conditions for an equilibrium in terms of quantities only. k t+1 + c t = f(k t ) + (1 δ)k t c t+1 c t = [β(1 + f (k t+1 ) δ)] 1/σ lim t βt u (c t )k t = 0 k 0 > 0 Prices can be determined from the firm s problems FOCs. Lecture 11, 12, 13 & 14 14/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Benevolent planner s problem* 15 What is the allocation of resources that an economy should feature in order to attain the highest feasible level of utility? Central Planner s optimal choice problem max (k t+1,c t ) t=0 s.t. β t u(c t ) t=0 c t + k t+1 = f(k t ) + (1 δ)k t, for all t = 0, 1, 2,... k 0 > 0 given Lecture 11, 12, 13 & 14 15/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Benevolent planner s problem 16 Welfare Socially optimal allocation coincides with the equilibrium allocation. The competitive equilibrium leads to the social optimum. Not surprising: no distortions or externalities Welfare Theorems hold Lecture 11, 12, 13 & 14 16/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Notes: simplifying features* 17 We are considering an economy without population growth. There is no exogenous technological change, either. Lecture 11, 12, 13 & 14 17/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Steady state* 18 Definition A balanced growth path (BGP) is a situation in which output, capital and consumption grow at a constant rate. If this constant rate is zero, it is called a steady state. We can usually redefine the state variable so that the latter is constant (i.e. the growth rate is zero) Recall from the Solow model: aggregate capital stock for (n = 0, g = 0) capital per unit of labor for (n > 0, g = 0) capital per unit of effective labor for (n > 0, g > 0) Lecture 11, 12, 13 & 14 18/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Steady state 19 From the Euler equation, c t+1 c t = [β(1 + f (k t+1 ) δ)] 1/σ, for all t If consumption grows at a constant rate (BGP), say γ 1 + γ = [β(1 + f (k t+1 ) δ)] 1/σ, for all t Thus RHS must be constant k t+1 = k t = k must be constant along the BGP Lecture 11, 12, 13 & 14 19/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Steady state 20 But then, from the resource constraint with k t = k t+1 = k : c t + k t+1 = f(k t ) + (1 δ)k t, for all t i.e., c t = f(k ) δk c t+1 = f(k ) δk We find that consumption must be constant along the BGP, c t+1 = c t = c or γ = 0 Hence we have a steady state in per capita variables. Lecture 11, 12, 13 & 14 20/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Steady state* 21 Hence from the Euler equation 1 + γ = 1 = [β(1 + f (k ) δ)] 1/σ or, simplified f (k ) = 1 β (1 δ) = ρ + δ we can solve for k and from the (simplified) resource constraint we can solve for c c = f(k ) δk Lecture 11, 12, 13 & 14 21/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Modified golden rule* 22 The capital stock that maximizes utility in steady state is called the modified golden rule level of capital Using f(k) = k α, we get f (k ) = ρ + δ k = k MGR = [ ] 1 α 1 α ρ + δ Compare to golden rule level of capital(max cons o in st. st.) [ α ] 1 k GR 1 α = δ (see Problem set 2, Q 2.2, assume A = 1 and set s = α) Lecture 11, 12, 13 & 14 22/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Modified golden rule 23 Since ρ > 0 and α (0, 1), k MGR = [ ] 1 α 1 α ρ + δ < [ α δ ] 1 1 α = k GR This result reflects the impatience of agents. As long as ρ > 0, they d always prefer to consume earlier rather than later, thereby reducing investments for next period and hence the steady state level of capital (and consumption)! One of Ramsey s points was that this is the steady state that we should aim at because it makes people the happiest - not the one that maximizes consumption per se. Lecture 11, 12, 13 & 14 23/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Off steady state dynamics* 24 Off the steady state, consumption and capital adjust to reach the steady state eventually. To analyze these dynamics, consider the movements of c and k separately. Let c = c t+1 c t and k = k t+1 k t. See graphical analysis. Lecture 11, 12, 13 & 14 24/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Off steady state dynamics* 25 We use 2 equilibrium conditions: Euler equation (EE) c t+1 c t = [β(1 + f (k t+1 ) δ)] 1/σ Resource constraint (RC) c t + k t+1 = f(k t ) + (1 δ)k t Lecture 11, 12, 13 & 14 25/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Off steady state dynamics* 26 Let c = c t+1 c t and k = k t+1 k t Use EE to determine points where c = 0 Use RC to determine points where k = 0 Look at dynamics left and right of c = 0 Look at dynamics above and below k = 0 Steady state is where c = 0 and k = 0 Lecture 11, 12, 13 & 14 26/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Off steady state dynamics* 27 Consider the set of points such that c = 0, then from the Euler eqn, the optimal k satisfies f (k) = ρ + δ draw vertical line at k (< k GR ) To the left: k t < k f (k t ) > f (k ) c > 0 c To the right: k t > k f (k t ) < f (k ) c < 0 c Consider the set of points such that k = 0, then from the Resource cstrt, the optimal c satisfies c = f(k) δk draw hump-shaped line from origin, maximized at k GR cross 0 again for k such that f(k) = δk Above: c t > f(k t ) δk t k = f(k t ) δk t c t < 0 k Below: c t < f(k t ) δk t k = f(k t ) δk t c t > 0 k Lecture 11, 12, 13 & 14 27/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Equilibrium path toward steady state 28 Suppose k 0 < k Then, what consumption level should the household pick? above k = 0 curve? This has c rising but would eventually lead to k = 0 and from RC jump of c to c = 0 violates EE cannot be an equilibrium decision below k = 0 curve? Yes, for some c 0 all equilibrium conditions will be satisfied Intuition: K-stock too low, marginal product high invest a lot if too low HH oversaving leads to c = 0 and k = 0 violates TC Lecture 11, 12, 13 & 14 28/50 Topics in Macroeconomics
Definition and Characterization of Equilibrium Benevolent Planner s Problem Steady State Off Steady State Dynamics Equilibrium path toward steady state 29 Suppose k t > k Then, what consumption level should the household pick? below k = 0 curve? This would lead to k such that f(k) = δk and c = 0 (u (0) =, >< transversality) cannot be an equilibrium decision above k = 0 curve? Yes, for some c t all equilibrium conditions will be satisfied Intuition: K-stock too high, marginal product low consume a lot If too high, get to k = 0 and jump to c = 0 again. Lecture 11, 12, 13 & 14 29/50 Topics in Macroeconomics
Government Consumption & Lump-Sum Taxes Gvmt. Cons, Lump-Sum Taxes: Permanent Increase Gvmt. Cons, Lump-Sum Taxes: Temporary Increase Fiscal policy (Romer 1996 p.59 72) 30 Government cons. spending (per capita), g t Financed by lump-sum taxes τ t borne by households The main modifications to the model are: Public sector τ t = g t, for all t = 0, 1, 2,... Household pays taxes c t + a t+1 = w t + (1 + r t )a t τ t, for all t = 0, 1, 2,... Lecture 11, 12, 13 & 14 30/50 Topics in Macroeconomics
Government Consumption & Lump-Sum Taxes Gvmt. Cons, Lump-Sum Taxes: Permanent Increase Gvmt. Cons, Lump-Sum Taxes: Temporary Increase Fiscal policy (Romer 1996 p.59 72) 31 In equilibrium, EE : RC : c t+1 c t = [β(1 + f (k t+1 ) δ)] 1/σ c t + g t + k t+1 = f(k t ) + (1 δ)k t At the steady state (assume g t = g, constant), EE : f (k ) = ρ + δ RC : c = f(k ) δk g k and output per capita unaffected by g g reduces c on 1 to 1 basis: full crowding out Lecture 11, 12, 13 & 14 31/50 Topics in Macroeconomics
Government Consumption & Lump-Sum Taxes Gvmt. Cons, Lump-Sum Taxes: Permanent Increase Gvmt. Cons, Lump-Sum Taxes: Temporary Increase A permanent increase in g * graph 32 If there is a permanent increase in g and HH perceive it as such Graphically, k = 0 shifts down by the magnitude of g The economy adjusts instaneously through a downward jump of c wealth effect No dynamic effect on capital accumulation Hence no effect on output Lecture 11, 12, 13 & 14 32/50 Topics in Macroeconomics
Government Consumption & Lump-Sum Taxes Gvmt. Cons, Lump-Sum Taxes: Permanent Increase Gvmt. Cons, Lump-Sum Taxes: Temporary Increase A temporary increase in g * graph 33 If there is a temporary increase in g and HH perceive it as such Consumption falls but by less than the increase in g: wealth effect but consumption smoothing Therefore, capital falls initially Hence output declines initially g back k returns to initial path toward steady state Lecture 11, 12, 13 & 14 33/50 Topics in Macroeconomics
Government Consumption & Lump-Sum Taxes Gvmt. Cons, Lump-Sum Taxes: Permanent Increase Gvmt. Cons, Lump-Sum Taxes: Temporary Increase First conclusions 34 Government spending cannot increase the steady state level of output per capita. It can even decrease it in the short run, e.g. when changes are temporary. Lecture 11, 12, 13 & 14 34/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Ricardian equivalence: lump sum taxes vs. debt 35 Does it matter whether the government chooses to finance expenditures through debt or non distortionary taxes? Two approaches: Lecture 11, 12, 13 & 14 35/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Ricardian equivalence: allowing for govmt. debt 36 (1.) The main modifications to the model are The government may borrow or lend d t+1 + τ t = g t + (1 + r t )d t The HH s budget constraint includes public debt In equilibrium, c t + a t+1 + d t+1 = w t + (1 + r t )(a t + d t ) τ t EE : RC : c t+1 c t = [β(1 + f (k t+1 ) δ)] 1/θ c t + g t + k t+1 = f(k t ) + (1 δ)k t Lecture 11, 12, 13 & 14 36/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Ricardian equivalence: allowing for govmt. debt 37 In equilibrium, EE : RC : c t+1 c t = [β(1 + f (k t+1 ) δ)] 1/θ c t + g t + k t+1 = f(k t ) + (1 δ)k t Method of financing irrelevant for allocation of resources. Public debt changes the distribution of taxes over time, but not its total value. Lecture 11, 12, 13 & 14 37/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Ricardian equivalence: allowing for govmt. debt 38 (2.) The intertemporal budget constraint of the household is not affected by the sequence of public debt and taxes. The HH s IBC PDV(c) = a 0 + d 0 + PDV(w) PDV(τ) The Government s IBC = k 0 + d 0 + PDV(w) PDV(τ) d 0 + PDV(g) = PDV(τ) Combining PDV(c) = k 0 + PDV(w) PDV(g) Lecture 11, 12, 13 & 14 38/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Ricardian equivalence: allowing for govmt. debt 39 PDV(c) = k 0 + PDV(w) PDV(g) All that matters for household s behaviour is the present value of government expenditures, irrespective of how the government decides to pay for it. Lecture 11, 12, 13 & 14 39/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Key assumptions for Ricardian equivalence to hold 40 The debate on Ricardian equivalence: Infinite horizon OR OLG framework Perfect capital markets liquidity constraints Lump-sum taxation distortionary taxes: next Full rationality rule of thumb? are key assumptions underlying Ricardian Lecture 11, 12, 13 & 14 40/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Distortionary taxes on capital income 41 Tax rate on capital income, τ k, with receipts being used to finance government consumption, g. The main modifications to the basic model are: Government s budget constraint τ kt r t a t = g t Household s budget constraint c t + a t+1 = w t + (1 + (1 τ kt )r t )a t The tax rate influences the after-tax rate of return on savings. Lecture 11, 12, 13 & 14 41/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Distortionary taxes on capital income 42 The Euler equation becomes: EE : c t+1 c t = [β(1 + (1 τ kt+1 )r t+1 )] 1/θ In equilibrium, EE : RC : c t+1 c t = [β(1 + (1 τ kt+1 )(f (k t+1 ) δ)] 1/θ c t + g t + k t+1 = f(k t ) + (1 δ)k t in addition to k 0 > 0 and the transversality condition. Notice that now the EE changes, hence the intertemporal consumption-savings decision is distorted. Lecture 11, 12, 13 & 14 42/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Switching from LS taxes to DT on K to finance g (*) 43 Recall: Permanent increase in g financed with LS taxes If there is a permanent increase in g and HH did not expect it before but perceive it as permanent now Graphically, k = 0 shifts down by the magnitude of g The economy adjusts instaneously through a downward jump of c wealth effect No dynamic effect on capital accumulation Hence no effect on output Lecture 11, 12, 13 & 14 43/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Switching from LS taxes to DT on K to finance g (*) 44 Switching to distortionary taxes on K income to finance g A higher tax rate reduces steady-state capital per capita.* k τ k = < = ( ) 1 (1 τk )α 1 α ρ + (1 τ k )δ ( (1 τ k )α (1 τ k )ρ + (1 τ k )δ ( α ρ + δ ) 1 1 α ) 1 1 α = k τ LS = k k τ k < k τ LS = k Lecture 11, 12, 13 & 14 44/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Switching from LS taxes to DT on K to finance g (*) 45 Switching to distortionary taxes on K income to finance g Graphically, the c = 0 curve shifts to the left.*(k τ k < k ) Adjustment features an immediate upward jump of consumption to reach the path to the new steady state. This is because saving is less profitable. In the long-run, consumption will also be lower bec. output falls. Lecture 11, 12, 13 & 14 45/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Taxes on labour income 46 Tax rate on labour income, τ n, with receipts being used to finance government consumption, g. The main modifications to the basic model are: Government s budget constraint τ nt w t = g t Household s budget constraint c t + a t+1 = (1 τ nt )w t + (1 + r t )a t The tax rate influences the after-tax wage. Lecture 11, 12, 13 & 14 46/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Distortionary taxes on labour income 47 The Euler equation becomes: c t+1 EE : = [β(1 + r t+1 )] 1/θ c t In equilibrium, c EE : t+1 = [β(1 + f (k t+1 ) δ)] 1/θ c t RC : c t + g t + k t+1 = f(k t ) + (1 δ)k t in addition to k 0 > 0 and the transversality condition. Notice that now the EE and RC are the same as for LS taxes. This is because labour is supplied inelastically in this model. That means a change in the after-tax return to work (price) does not affect any decision. Next: Example with elastic labour supply and labour income taxes Lecture 11, distortion 12, 13 & 14 47/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Distortionary taxes on labour income 48 Static (one period) model Household solves max c,l s.t. u(c) + v(l) c = (1 τ n )(1 l)w where c is consumption, l is leisure, w wage, τ n is labour income tax, 1 is time endowment, (1 l) is hours worked Lecture 11, 12, 13 & 14 48/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Distortionary taxes on labour income 49 Static (one period) model: rewritten After substituting for c in the utility function, household solves The first order condition is: max u((1 τ n )(1 l)w) + v(l) l u (c)((1 τ n )w) = v (l) Lecture 11, 12, 13 & 14 49/50 Topics in Macroeconomics
Ricardian Equivalence: Lump-Sum Taxes vs. Debt Distortionary Taxes on Capital Income Distortionary Taxes on Labour Income Distortionary taxes on labour income 50 Static (one period) model: solution In terms of Marginal rate of substitution: Thus if the labor tax goes up, MRS c,l = v (l) u (c) = (1 τ n)w Substitution effect: leisure becomes cheaper relative to consumption work less Income effect: less after-tax income even if work the same consume less, work more Lecture 11, 12, 13 & 14 50/50 Topics in Macroeconomics