In this figure PPF
1
is the original PPF and it
shifts to PPF
2
when z increases from z
1
to z
2
.
The initial equilibrium input is at point A, and
the final equilibrium is at point B after z
increases. The equation for PPF
2
is given by
C
= z
2
F(K, h – l) – G.
Now consider constructing an artificial PPF, called PPF
3
, which is obtained by shifting PPF
2
downwards by a constant amount. The equation is given by C = z
2
F(K, h – l) – G –C
0.
Here,

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C0 is a constant that is large enough so that PPF3 is just tangent to the initial indifference
curve I1. What we are doing here is taking consumption away from the representative
consumer to obtain the pure substitution effect of an increase in z. The substitution effect is
the movement from D to B. The substitution effect is for consumption to increase and leisure
to decrease, so that hours worked increases. The income effect is for both consumption and
leisure to increase, consumption must increase as both goods are normal goods, but leisure
may increase or decrease due to opposing income and substitution effects.
Why must the real wage increase in moving from A to B, even if the quantities of leisure and
employment rise or fall?
Firstly, the substitution effect involves an increase in
MRS
l, c
(the indifference curve becomes
steeper) in moving along the indifference curve from A to D. Secondly, because PPF
2
is just
PPF
3
shifted by a fix amount, the slope of PPF
2
is the same as the slope of PPF
3
for each
quantity of leisure. As the quantity of leisure is higher at point B than at point D, the PPF is
steeper at B than at D, and so
MRS
l, c
also increases in moving from D to B. Thus the real
wage, which is equal to the marginal rate of substitution in equilibrium, must be higher in
equilibrium when
z
is higher.
The increase in total factor production causes an increase in the marginal productivity of
labour, which increases the demand for labour by firms, driving up the real wage. Workers
now have more income given the hours worked, and they spend the increased income on
consumption goods. Because they are offsetting income and substitution effects on the
quantity of labour supplied, hours worked may increase or decrease. An important feature of
the increase in total factor productivity is that the welfare of the representative consumer
must increase. Therefore, the representative consumer must consume on a higher indifference
curve when
z
increases. As a result, increases in total factor productivity unambiguously
increase the aggregate standard of living.

A Simplified One-Period Model with Proportional Income Taxation
We use this model to study the incentive effects of the income tax, and to derive the “Laffer
curve”. In this section we modify the model by including proportional tax on wage income,
instead of a lump sum tax. This tax will then distort the labour supply decision, and the
competitive equilibrium will not be Pareto optimal.

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- Summer '18