subject to (3/2)e + 2f * 4/3. The solution to this problem is in fact e* - 4/9 and f* « 1/3.
(c) As calculated in the answer to (b), the lump sum tax is 2/3.
(d) Let e e [0,1] be a planned level of education and suppose that it is
produced by the input combination L^ = T^ = e . (It can be shc^n that this equality always holds at all Pareto optimal allocations.) Let p^ = 1, then, as we saw in the answer to (h)t the zero profit cohoition and the cost minimization condition of the food production implies that p = 1 and pf ~ 2.
Hence, if the government follows the marginal cost pricing rule, then it sets
. 1/2 . . . 0 1/2 ,1/2 1/2 d = l/e and the deficit is 2e - e/e ® e This is tne trans: er
- ex from the landowner when the planned level of education is e.
(e) As we saw in the answer to (d), at any marginal cost pricing equilibrium with a planned level e of education, the equilibrium price vector must be (a
scalar multiple cfj (p. ,p ,p ,pf) = (1,1,l/e ,2). Faced with this price
vector, the demand of the labor owner is (e /2,l/4) and the demand of the
1/2, T _ 1/2} _ fil/2 landowner is ( —-e-, ---- )* Hence the aggregate demand for
food is ———-. The inputs used for education are (L ,TJ = (e ,e )
and thus the food supply is 1 - e . Hence we must have e = 4/9. This is the planned level of education compatible with a marginal cost pricing equilibrium.
We shall now prove that the marginal cost pricing equilibrium is Pareto optimal by showing that the production possibility frontier is below the Scitovsky contour (page 120), as illustrated in the figure below.
Note first that the production possibility frontier is given by the
1/2 2 equation f « 1 - e , or equivalently, e « (1 - f) . On the other hand, since the aggregate demand at the marginal cost pricing equilibrium is (f*,e*)
= (1/3,4/9) and both the laborowner and the landowner have the identical,
homothetic utility function u(e,f) = e f , the Scitovsky contour is given
by e i ~ = (4/9) (1/3) , or equivalently, e = 4/27f. It is sufficient to show that the Scitovsky contour touches on the production possibility frontier only at (f*,e*) = (1/3,4/9) and the former is above the latter
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