7/30/2019 Notes Problem Set2

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Problem Set 2. Microeconomic foundationsa.grodecka@lse.ac.uk

1 Question 1

How do we know that MPn = w (marginal product of labour equals wage) andthat MRSl,c (marginal rate of substitution) equals (1 t)w?

The first result comes from the assumption about the form of our production function(Y = zNd). When we differentate this function w.r.t. Nd, we will get z.

dY

dNd= z (1)

How do we know that w = z in equilibrium? This result comes from the optimizationproblem of a firm, which wants to maximize its profit .

= YwNd = zNd

wNd (2)

When we differentiate this function with respect to Nd we see that z= w, which givesus also the 0-profit condition in the equilibrium. Because z is a constant here, we will geta horizontal labor demand curve in this simple case.

MRSl,c = (1t)w is derived from the optimization problem of a consumer. Consumerhas some utility U that depends on his consumption c and leisure l (remember thatl+Ns = h). He also faces the budget constraint C= +(1t)wNs = +(1t)w(hl).How do we calculate MRSl,c?

MRSl,c =MUlMUc

. We get the marginal utilities from the consumers optimization prob-lem which can be solved by writing down a Lagrangian function:

L(l, C) = U(l, C) + (C (1 t)w(h l)) (3)

, where denotes the Lagrangian multiplier.Now lets maximize this function w.r.t. c and l to get marginal utilities.w.r.t. c

dU(l, C)

dC+ = 0 (4)

w.r.t. ldU(l, C)

dl+ (1 t)w = 0 (5)

So MRSl,c =MUlMUc

=dU(l,C)

dldU(l,C)

dC

= (1t)w

= (1 t)w.

(6)

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7/30/2019 Notes Problem Set2

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2 Question 2

Does the hump-shaped form of the Laffer curve imply that on the good side ofthe Laffer curve the income effect dominates (and not the substitution effect,as we usually assume)?

No, throughout the book and exercises we assume that the substitution effect is larger

than the income effect, i.e. with a rise in real wage our labor supply will increase, so weface an upward-sloping labor supply curve. The income effect says that after an increasein income we will increase both our consumption and leisure which are assumed to benormal goods. Since leisure increases, the income effect would suggest a falling labor sup-ply after a rise in wage. However, we assume that the substitution effect always prevails.That is, when the real wage increases, the leisure becomes more costly in relation to con-sumption, so we substitute from leisure to consumption. The substitution effect causesthus the leisure to fall and the labor supply increases. The two effects suggest oppositedirections of the movement of labor supply. But because of our assumption that thesubstitution effects is larger than the income one, the labbor supply will always increase

after an increase in real wage.

Now if we are on the left, good side of the Laffer curve, when the tax rate goes up, wewill get more revenue from taxes. But it does not mean that our assumption about theupward-sloping labor supply curve does not hold any more! When the taxes are higher,we will supply less labor, because our real wage decreases. The reason for the rise ingovernment revenue on the good side of the Laffer curve is not the increase in the taxbase w(h l) = wN, because the tax base will be lower exactly because of our assumptionthat the substitution effect is higher than the income effect. The reason for the rise is themechanical effect of a rise in the tax rate t which constitutes another part of governmentrevenue given by REV = tw(h l). When the positive effect of a rise in t outweights thedecrease in tax base, we are on the good side of the Laffer curve.

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