Emma is a first-year Ph.D. student of economics. She is very lucky because she got a

place to stay in the Aggie Village, the nicest part of Davis. She lives in a beautiful cottage

with a little garden. Her modest wealth from being a TA is w. She spends it on coffee

and gardening. Let x1 denote the amount of coffee and x2 the amount of gardening and

let p1 and p2 denote the corresponding unit prices. Her budget constraint is given by

p1x1 + p2x2 w. (1)

Coffee is really a private good in the sense that she is the sole beneficiary of caffeine

in her coffee (unless she calls up in panic her fellow student in the middle of the night

because she cannot solve her ECN200A homework problem). In contrast, gardening

creates a positive externality on others. But so does the gardening of others come up with a

positive externality on her. There is plenty of gardening in the Aggie Village. Denote

by e the total externality or public good created from gardening in the community.

Her utility function u(x1, x2, e) is concave and continuously differentiable with a strictly

positive gradient on the interior of its domain. The total externality depends in part on

Emma's gardening x2 and on the externality created by the gardening of others, denoted

by ei

. It is assumed to satisfy

e ei + x2 (2)

for some parameter satisfying > 0. Emma does not think that she can affect the

level of externalities provided by others. For instance, Professor Schipper, who also lives

in the Aggie Village, is so busy writing prelim exam questions that talking to him about

keeping up his gardening is no use. Thus, we can safely assume that Emma takes ei as

well as p1, p2, and w as given.

Since Emma diligently studies microeconomic theory for the prelims, she is eager to

maximize her utility function subject to constraints (1) and (2). This yields demand

functions x1(p1, p2, w, ei) and x2(p1, p2, w, ei) as well as her optimal desired amount of

public good e(p1, p2, w, ei).

a.) Write down her Kuhn-Tucker-Lagrangian (ignore non-negativity constraints).

b.) Derive the Kuhn-Tucker first-order conditions (ignore non-negativity constraints).

c.) Use the Kuhn-Tucker conditions and the assumptions that the solution is interior,

that it is unique, and that constraints (1) and (2) are satisfied with equality to derive

a system of three equations and three unknowns that does not involve multipliers

and whose solution defines x1(p1, p2, w, ei), x2(p1, p2, w, ei) and e(p1, p2, w, ei).

(No need to solve it.)

d.) If you think that Professor Schipper acts in a strange way sometimes, this is not

just due to being an economic theorist. The secret is he comes from another

solar system. One feature of these aliens is that they can read immediately the

utility function of others. (Although this sounds quite useful, it is rather a curse.)1

Anyway, as a proof of this claim we print here Emma's utility function:

u(x1, x2, e) = x1 + (a2, ae)

x2

e

1

2

(x2, e)B

x2

e

, (3)

where a2, ae > 0 and B =

b22 b2e

be2 bee

is symmetric positive definite. I know, you

surely must think "Wow" but let's focus again on the prelim exam. Assume that

solutions are interior and that constraints are satisfied with equality. Write out the

system of equations from problem c.) for Emma's utility function.

e.) Provide an interpretation of the partial derivatives x2(p1,p2,w,ei)

ei

and e(p1,p2,w,ei)

ei

and their signs.

f.) Compute x2(p1,p2,w,ei)

ei

and e(p1,p2,w,ei)

ei

.

g.) Assume b2e 0. Derive the signs of x2(p1,p2,w,ei)

ei

and e(p1,p2,w,ei)

ei

.

h.) Assume now b2e < 0. Show that without additional assumptions, the signs of

x2(p1,p2,w,ei)

ei

and e(p1,p2,w,ei)

ei

remain ambiguous in this case. Find additional assumptions on matrix B and that allow you to determine the signs of x2(p1,p2,w,ei)

ei

and e(p1,p2,w,ei)

ei

.

i.) We want to get a better understanding of whether good 2 and the externality are

complements or substitutes. In our particular context, do we need to distinguish

between gross complements/substitutes and (net) complements/substitutes?

j.) To figure out whether good 2 and the externality are complements or substitutes,

we can use the Slutsky substitution matrix. To this end, compute first Walrasian

demand functions as if the externality has a market price pe. That is, compute

Walrasian demand functions x2(p1, p2, pe, w) and e(p1, p2, pe, w).

k.) How does the fact that good 2 and the externality are substitutes or complements

depend on the sign of b2e?

`.) How is the fact that good 2 and the externality are substitutes or complements

related to the sign of the derivatives x2(p1,p2,w,ei)

ei

and e(p1,p2,w,ei)

ei

discussed earlier?