econ 100a - fall 2011 - santesteban - midterm 1

7/31/2019 ECON 100A - Fall 2011 - Santesteban - Midterm 1

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Midterm Examination SolutionsIntermediate Microeconomics

Swiss Federal Institute of Technology

Fall Semester, 2008

Prof. Thomas Rutherford

November 10, 2008

Closed book, closed notes, no calculators.

Problem 1 (25 points)

You have been hired by an architect to determine the most energy-efficient specification for a newbox-shaped cabin with a flat roof. The volume of the cabin is to be 1000 m3. The height, width and lengthof the cabin are design parameters (denoted h, w and , respectively). Energy demand of the building isproportional to the exposed surface area, A, excluding the floor.

a Formulate a constrained optimization model and solve for the building shape which maximizes energyefficiency. (15 points)

Solution:

Solve the constrained minimization problem:

min A = 2h + 2hw + w

s.t.hw = 1000

This classical optimization problem has the Lagrangian:

L = 2h + 2hw + w (hw 1000)

The first order conditions are:

w = 2 + 2w hhw = 2h + w

h = 2h + w

The last two conditions imply:

= w =2h

h 1

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Substituting into the first order condition for h we have:

2 = 4 = 4

We then solve:4

=

2h

h 1to conclude that h = 2, or

= w = 2h.

We can then substitute into the volume constraint to find h:

hw = 4h3 = 1000 h = 3

250

andA = 2h + 2hw + w = 4h2 + 4h2 + 4h2 = 12(250)2/3

b Suppose that the architect subsequently requires that on aesthetic grounds, the building length mustbe at least twice its width. How does this constraint affect energy efficiency? (10 points)

Solution:

The aesthetic constraint will be binding, hence we can solve the two-variable problem by substituting = 2w into the original problem:

min A = 6hw + 2w2

s.t.2hw2 = 1000

Substitute for h using the constraint to obtain an unconstrained optimization problem in terms of w:

min3000

w+ 2w2

Hence:w =

3

750,

= 23

750,

h =500

7502/3

,

andA = 6(750)2/3

The increased exposed area is then:

A

A=

12(250)2/3

6(750)2/3=

32/3

2 1.04

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a. Suzy consumes ice cream (x1) and soda (x2) for lunch, and her budget currently permits one icecream and two sodas per week when they both cost 1 CHF.

i) What Cobb-Douglas utility function is consistent with Suzys choices over ice cream and soda?(5 points)

Solution:

A Cobb-Douglas utility function has the form:

U(x1, x2) = x1 x

12

in which is the budget share of good 1. The description of her current choices implies that = 1/3.

ii) The price of ice creamdoubles

. What is the minimum increase in her lunch allowance requiredto compensate for the price increase? (5 points)

Solution: The Cobb-Douglas indirect utility function has the form:

V(p1, p2, M) = U(x1(p, M), x2(p, M)) =

M

p1

(1 )M

p2

1=

M

p1p12

where = 1/((1 )1). We want to find an income level M which compensates, hence wemust solve:

V(p1 = 2, p2 = 1, M) = V(p1 = 1, p2 = 1, M = 3)

or

M

2= 3.

Hence, with = 1/3 we have:M = 3

3

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b. Jim has different preferences than Suzy. Irregardless of relative prices, Jim always has one sodabefore and one soda after eating an ice cream. His budget determines how many times a month hehas ice cream.

i) What utility function is consistent with these choices? (5 points)

Solution:

The description of his choices implies that he has perfect complements preferences. These arecharacterized by the utility function:

U(x1, x2) = min(x1, x2/2)

His benchmark choices, x1 = 1, x2 = 2 are consistent with U(1, 2) = 1.

ii) Write down demand functions which could extrapolate Jims optimal choices to any expenditure(m) and prices (p1 and p2). (5 points)

Solution:

x1 =M

p1 + 2p2

and

x2 = 2M

p1 + 2p2

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c. Toms preferences are yet different again. He currently has three sodas and one ice cream per month

when all cost 1 CHF, yet when his income doubles his consumption of ice cream increases to threeper month and his consumption of sodas increases to five per month. Write down a utility functionwhich is consistent with these choices. (5 points)

Solution:

Income elasticities differ from unity in the case of the LES preferences. In the present example, themarginal budget shares are 0.5 for ice cream and 0.5 for soda. The subsistence demand for good 1can be assigned to zero, and we then calibrate the subsistence demand for good 2 from thebenchmark demands, so:

U(x1, x2) =

x1(x2 2)


Consider the following utility functions:

U(x) =i

i ln(xi) (1)

U(x) =i

(xi i)i (2)

U(x) = mini

ixi (3)

U(x) =i

ixi (4)

U(x) =

i

ixi1/

(5)

Assume that in any case the consumer solves a conventional budget-constrained utility maximizationproblem:

max U(x)

s.t. i

pixi = M

in which the budget is M and pi is the price of the ith commodity.

For each of these functions write down (2.5 points each, 50 points in total):

a. The name commonly given to the utility function.

1. Cobb-Douglas

2. Linear expenditure system

3. Perfect complements

4. Perfect substitutes

5. Constant elasticity of substitution

b. Demand functions which express optimal demand as a function of commodity prices and income,xi(p, M).

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1. Cobb-Douglas

Assuming that

i i = 1 we havexi = i

M

pi

2. Linear expenditure system

Assuming that

i i = 1 we have

xi = i + iMj pjj

pi

3. Perfect complements:

xi =1

i

M

j pj/j4. Perfect substitutes:

xi =

Mpi

if pi/i < pj/jj0 otherwise

Note that when there are pairs of goods i and j for which

pii

=pjj

then there is not a unique choice any combintation of these goods which exhaust the budgetwill be optimal.


xi =

M pij j p1

j

where

=1

1 c. Indirect utility functions are defined as, V(p, M) = U(x(p, M)).

1. Cobb-Douglas:

V(p, M) =i

i ln

i

M

pi

2. Linear expenditure system:

V(p, M) =

i(i + iM

j pjj

pi i)i

=

i

i

M

jpjj

pi

i

=

Mj jpjj jpjj

3. Perfect complements

V(p, M) =Mi pi/i

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4. Perfect substitutes

V(p, M) = maxi

iM

pi


V(p, M) =

i

i p1i

1/(1)

d. Parameter values calibrated to observed consumption choices, ( p, x). (In this response, indicate whichparameters are determined by elasticity assumptions rather than by benchmark budget shares.)

1. Cobb-Douglas:

i =pixi

M

2. Linear expenditure system:

i is determined by the income elasticity of demand:

xiM

=i

pi

so

i xiM

M

xi= M

ipixi

We then insert the benchmark values to solve for the marginal budget share given the incomeelasticity of demand and the average budget share:

i = ipixiM

= isi

where si is defined as the benchmark value share.

Given the marginal budget share, subsistence demand () can be calibrated to replicate thebenchmark expenditure, so they solve:

xi = i + iMj pjj

pi

3. Perfect complements:i =

1

xi

4. Perfect substitutesi = pi


i =pixiM

xi

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econ 100a - fall 2011 - santesteban - midterm 1

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