Econ 712, Corbae Fall 2016

L2. Static Allocations in a Production Economy These notes introduce a labor/leisure choice, e¢ cient allocations, implementation as a competitive equilibrium, income and substitution e¤ects, value functions, envelope conditions, Frisch labor supply elasticity, and the welfare theorems in a static economy.

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Environment Population: Agents (households) are identical. Since we assume there are a large number of them (speci…cally there is a unit measure of agents so no one agent has any in‡uence on allocations, which is sometimes called an “atomless” economy).1 Production: There is a technology available for producing the single good (y) from labor input (n). Let y = f (n) where f ( ) is strictly increasing (i.e. f 0 ( ) > 0) and concave (i.e. f 00 ( ) 0), with f (0) = 0: Households are endowed with a unit of time which they can allocate to leisure and work (the latter is denoted n 2 [0; 1]). Preferences: U (c:n) = u(c)

g(n)

over consumption (c) and labor supplied (n). Assume that u( ) is strictly increasing (i.e. u0 ( ) > 0) and concave (i.e. u00 ( ) 0) and that g( ) is strictly increasing (i.e. g 0 ( ) > 0) and strictly convex (i.e. g 00 ( ) > 0). Assume lim

c!0;n!0

u0 (c)f 0 (n) > g 0 (n) and u0 (f (1))f 0 (1) < g 0 (1):

(1)

These assumptions hold if the marginal utility of consumption is very high at zero consumption and the marginal disutility of work is very high when leisure is zero. 1 Suppose the only di¤erence between agents is their “name”, indexed by i 2 [0; 1]; distributed uniformly. IfR each agent iR takes the same action then xi = X: Then the economywide action is given by 01 xi di = X 01 di = X ij10 = X: Note that even if one agent does something di¤erent from X; the economywide action is still X since that agent is of measure zero.

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Planner’s Problem Given symmetry, the planner chooses fc; ng to maximize the utility of the “representative agent” (named Robinson Crusoe) subject to resource feasibility max

c 0;n2[0;1]

u(c)

g(n)

s.t. c = y = f (n) or max u(f (n))

n2[0;1]

g(n):

(2)

Given the assumptions on u; f; g; the objective function is concave and the constraint set is compact and convex. Given the assumptions in (1), the …rst order condition is given by u0 (f (n )) f 0 (n ) = g 0 (n )

(3)

where n denotes the planner’s solution. Note that the planner sets the marginal utility bene…t from work (the left hand side of (3)) equal to the marginal utility cost of work (the right hand side of (3))

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Competitive Equilibrium We assume competitive markets for goods and labor where agents and …rms interact. Firms are owned by households. Firm pro…ts are returned to the households. Households supply labor to …rms at relative price of labor in terms of consumption goods (i.e. the real wage) w: The competitive assumption implies that household and …rm choices do not in‡uence pro…tability and prices.

3.1

Household Problem Households choose how much labor to supply and consumption to solve max

c 0;n2[0;1]

u(c)

g(n)

(4)

s.t. c = wn + Notice the price normalization (i.e. the price of consumption goods in terms of consumption goods is one).

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Forming a lagrangian for (4) $ = u(c)

g(n) +

[wn +

c] ;

the …rst order conditions are given by c : u0 (c) = n

: g 0 (n) =

(5) w

(6)

Substituting out ; the solution to (5)-(6) is given by ns which satis…es the …rst order condition or u0 (wns + ) w = g 0 (ns )

(7)

– The household sets the marginal utility bene…t from work (the left hand side of (7)) equal to the marginal utility cost of work (the right hand side of (7)). – Note that the solution implicitly de…nes labor supply in terms of wages and other income (pro…ts). We call this a decision rule or policy function ns (w; ): How does labor supply respond to changes in wages? To understand this we can di¤erentiate (7) with respect to w and ns to yield u00 (wns + ) or

w2 dns + w ns dw + u0 (wns + ) dw = g 00 (ns ) dns [u00 (wns + ) w ns + u0 (wns + )] dns = dw [u00 (wns + ) w2 g 00 (ns )]

– The denominator of (8) is positive since u00

(8)

0 and g 00 > 0:

– The …rst term in the numerator is negative; as wages increase, households have more income and thus want to consume more leisure (this is known as the income e¤ect). – The second term is positive; as wages increase, households substitute away from leisure towards work (this is known as the substitution e¤ect). – Which e¤ect dominates depends on the curvature of u( ): These opposing e¤ects are one reason why labor supply does not respond much to wage changes. Once we know the decision rule ns (w; ); then we know the value (or indirect utility) function associated with a solution to (4). That is, V (w; ) = u(w ns (w; ) + )

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g(ns (w; ))

(9)

To understand how wage changes, for example, a¤ect the household’s utility (or value function) di¤erentiate (9) with respect to w to yield dV (w; ) dw

= u0 (w ns (w; ) + )

ns + w

dns dw

g 0 (ns (w; ))

= u0 (w ns (w; ) + ) ns + [u0 (w ns (w; ) + ) w

dns dw g 0 (ns (w; ))]

= u0 (w ns (w; ) + ) ns where the …nal line is due to the “envelope condition”. That is, by the …rst s order condition (7), the term in [ ] multiplied by dn dw is zero. Thus, utility increases from a wage increase whether or not labor supply increases.

3.2

Firm Problem Firms choose labor to maximize pro…ts given by the following problem: = max f (n) n

w n

(11)

The solution to this problem is given by nd which satis…es the …rst order condition f 0 (nd ) = w (12) – This says that the …rm chooses the fraction of hours such that the marginal bene…t (the marginal product of labor) equals the marginal cost (the wage). – This de…nes the …rm decision rule nd (w). By the implicit function theorem 1 dnd = 00 d < 0 f 00 (nd )dnd = dw , dw f (n ) since f 00 ( ) 0: Thus, the labor demand curve is downward sloping. The special case where f (n) = A n; so f 00 ( ) = 0; implies a perfectly elastic (i.e. horizontal) labor demand curve at w = A: That is, the …rm is willing to demand any amount of labor supplied at real wage A:

3.3

Market Clearing Labor and goods market clearing are given by nd

= ns

n

(13)

c

= f (n )

(14)

Again using Walras Law, we can solve for an equilibrium by using only one of the two market clearing conditions. We will use (13). 4

dns dw (10)

3.4

Equilibrium A symmetric competitive equilibrium is an allocation (n ; c ) and prices w such that households optimize (7), …rms optimize (12), and markets clear (13)-(14). Using (7), (11), (12), and (13) we have u0 (w

n

+ [f (n )

0

w

n ]) f (n ) 0

u0 (f (n )) f (n )

= =

g 0 (n ) () g 0 (n )

(15)

Notice that the competitive outcome (15) is identical to the planner’s outcome (3). Further, since n = n we know c = c : Figure L2.1 illustrates equilibrium for both the planner’s problem and the competitive outcome. – An indi¤erence curve is the locus of points of (c; n) for which the household is indi¤erent. Since U = u(c) g(n); then dU = 0 () u0 (c)dc

g 0 (n)dn = 0 ()

dc g 0 (n) jdU =0 = 0 dn u (c)

M RS:

where M RS denotes marginal rate of substitution between consumption and leisure/labor. – The slope of the production function measures the M RT or marginal rate of transformation (of labor into consumption goods) – Equation (15) can be re-written such that at an optimum the M RT 0 f 0 (n) = gu0(n) M RS so that the tangency of the indi¤erence curve (c) and the production function de…nes picks out the optimal allocation. – Since in a decentralized equilibrium, f 0 (n) = w; the tangency de…nes the wage and the intercept de…nes pro…ts. Figure L2.2 illustrates the Hicksian case where substitution e¤ects (from 0 to s) outweigh income e¤ects (from s to 1) so that labor supplied rises as real wages (or marginal product of labor) rise (from f0 (n) to f1 (n)).

3.5

Parametric Example How do we determine the responsiveness of labor supply to wage changes (basically the slope of the labor supply curve)? There are many ways to decompose changes in real wages (Marshallian labor supply holds income constant, Hicksian holds utility constant, Frisch holds the marginal utility of wealth - in (6) - constant).

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Consider the following parametric example which is a special case of Keane and Rogerson (2012, equation 1, with = 1):

where < :

1;

u(c)

=

g(n)

=

f (n)

=

c 1+

1

n1+

1

(16)

A n

0; and to be consistent with Assumption (1) we need

As discussed in Keane and Rogerson (2012, equation 4), to uncover the labor supply elasticity parameter , MaCurdy (1981) ran a panel regression after taking logs of (6) to yield ln

n

1

ln (n)

=

ln (

= b+

(1

) w) ()

ln (w)

(17)

where b = [ln ( ) + ln(1 ) ln ( )] : That is, the coe¢ cient on the regression of log hours on log wages uncovers the Frisch labor supply elasticity :This is an example of how we can estimate parameters of a macro model. Keane and Rogerson make clear that micro estimates of which come from cross-sectional data (somewhere between 0 to 0:5) are much smaller than the parameterization used in macro models which come from aggregated time series data (somewhere between 2 to 4). Intuitively, to get enough variation in aggregate hours in response to labor demand shocks (e.g. technology shocks which alter the real wage), we need an elastic labor supply curve. Since (17) implies that the slope of the labor supply curve is inversely related to ;a large means a moderate slope which in turn means more variation (i.e. a higher implies a more elastic supply curve).2 2 In the data (e.g. Table 1, p. 321 in Hansen (1985)) the standard deviation of aggregate hours is 1.66 and the correlation with output is 0.76. At reasonable parameterizations, the real business cycle model predicts a standard deviation of hours of 0:70 and a correlation of hours with output of 0:98. Hence a model economy with divisible labor is less than half as volatile as in the data. To bring the model closer to the data, Hansen (1985) and Rogerson (1988) introduced “indivisible labor". In their papers, variation in hours comes about by variation in employment (the extensive margin) rather than variation in hours per worker (the intensive margin). That is, aggregate hours Ht = ht Nt where ht is hours per worker and Nt is per capita # workers. In that case, var(log Ht ) = var (log ht ) + var (log Nt ) + 2cov (log ht ; log Nt ) : In the data, var(log ht ) var(log N ) 2cov(log ht ;log Nt ) = 0:2; var(log Ht ) = 0:55; = 0:25: var(log Ht ) var(log Ht ) t So indivisibilities in the workday (i.e. hours per worker) can help explain the observed behavior of variability of aggregate hours over the business cycle. These papers also provided a methodological innovation in real business cycle models since they used lotteries to convexify a nonconvex set.

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Note that pro…ts are given by = f (n)

f 0 (n) n = An

1

A n

n = (1

) f (n)

With constant returns to scale (i.e. = 1) there are no pro…ts while with decreasing returns (i.e. < 1) there are pro…ts. It is also simple to see that under the parameterization in (16), the indifference curves associated with these preferences in Figure L2.1 are given by U

= c

1

1+

c = U+

1+

n1+ ()

1

1

n1+

1

hence the intersection of indi¤erence curves with the vertical axis (where n = 0) is increasing in U .

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The Distortionary E¤ects of Taxation “How much do tax revenues respond to changes in proportional income taxes?”This quantitative question (“How much”) depends on the responsiveness of labor supply to changes in real after-tax wages (which depends on the above Frisch elasticity). Assume that proportional labor taxes 2 [0; 1] are placed on earnings used to …nance government expenditure which is lump sum transferred back to the household (i.e. T = w n ). In this case, (15) can be written g 0 (n ) MC

= u0 ((1

)w

n

+ T + [f (n )

w

n ]) (1

0

g 0 (n ) = u0 (f (n )) (1

) f (n )

MB

0

) f (n ) ()

(18)

Notice that the only di¤erence between the …rst best (15) and (18) is the marginal bene…t with taxes is now lower than the MC (which is the same in both the …rst best and the competitive equilibrium with taxes). Since the bene…t is less than the cost, households will supply less labor and in turn consume less as well. This is the distortionary e¤ect of taxation. Using the parameterization in (16), we can substitute into (18) to yield: (n )

1

n which makes clear that as

=

(1 (1

=

rises, n 7

)

A (n ) )A

falls.

1

()

1=(1+ 1

) (19)

A fundamental result in public …nance is to tax the least sensitive (most inelastic) good if one wants to maximize tax revenues. To see this, The optimal labor tax rate to maximize total revenue is given by: =

arg max

f 0 (n ) n

=

arg max

A

(1

)A

(1+ 1

)

which implies 0

(1

= A +

()

0 @

A 1+

)A

1

= 1+

1+

)

(1+ 1 1

(1

A

1

)A

(1+ 1

)

1

A

1

Note that d d

1+

1

2

1+

= 1+

1

2

1

2

< 0:

Hence, the optimal labor tax decreases in the Frisch labor supply elasticity :

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Welfare Theorems and Equivalence of Planner and Market Outcomes In the last section we saw that the competitive outcome (15) is identical to the planner’s outcome (3) under appropriate assumptions. Thus there is no way to make anyone better o¤ without making someone else worse o¤ in the competitive equilibrium. This is the idea behind the First Welfare Theorem; any competitive equilibrium is pareto optimal. The converse is given by the Second Welfare Theorem; under appropriate assumptions, any pareto optimal allocation can be acheived as a competitive equilibrium with appropriate transfers. The second welfare theorem is actually useful computationally in macro; we can implement the allocation that solves the planner’s problem (which doesn’t involve solving for prices) as a competitive equilibrium (which is harder to solve for).

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