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Econ 370
Fall 2002
Homework #7
Hendrix
1.
Suppose you are interested in opening a retail store in the mall. You have a choice
of three different floor plans: 200 square feet, 500 square feet, and 1000 square
feet. The monthly rent will be $1 per square foot. You have estimated that if you
have F square feet of floor space and sell y units of merchandise each month, then
2
variable costs will be cv ( y )  y
each month.
F
a.
b.
c.
2.
Suppose a producer is faced with the production function
f ( x1 , x2 )  (min{ x1 ,2 x2 })1 / 2 . Let w1 and w2 be the prices of inputs.
a.
b.
c.
d.
3.
For each square footage option, find the cost function, marginal cost
function and average cost function. Also find at what amount of output
average cost is minimized. Then find the level of average cost at that
output.
Graphically show the average and marginal cost curves for each floor
plan.
On the same graph, show the long-run average and marginal cost curves.
What is the cost function for this firm?
If the factor prices are equal to 1, what is the marginal cost of producing y
units of output? How many units of output would be supplied at price p?
What would be the cost per unit of output?
Suppose the producer is in a competitive market where p=48 and factor
prices are each 1. How many units of output will be produced? How much
profit will the firm make?
In a more general sense (allowing w1 and w2 be the prices of inputs),
what is the marginal cost function? What is the supply function for this
firm?
Consider an industry with three firms. Suppose the firms have the following
supply functions: S1 ( p)  p, S 2 ( p)  p  5, S 3 ( p)  2 p .
a.
b.
Graph each supply function and the resulting industry supply curve.
If the market demand curve has the form D ( p )  15 , what is the resulting
market price? Market output? What is the output level for each firm at this
price?
4.
Consider a competitive industry with a large number of firms, all of which have
identical cost functions c( y)  y 2  1 for y  0 and cost of zero for output of
zero. Suppose that initially the demand curve for this industry is given by
D( p)  52  p . (For this problem, the output of the firm does not have to be an
integer, but the number of firms does.)
a.
b.
c.
d.
e.
f.
g.
h.
i.
5.
What is the supply curve of an individual firm? If there are n firms in the
industry, what is the industry supply curve?
What is the smallest price at which the product can be sold?
What will be the equilibrium number of firms in the industry?
What will be the equilibrium price? What will be the equilibrium output of
each firm?
What will be the equilibrium output of the industry?
Now suppose that the demand curve shifts to D( p)  52.5  p . What will
be the new equilibrium number of firms?
What will be the new equilibrium price? What will be the equilibrium
output of each firm? What will be the equilibrium output of each firm?
What will be the equilibrium profits of each firm?
Now suppose that demand shifts to D( p)  53  p . What will be the
equilibrium number of firms? What will be the equilibrium price?
What will be the equilibrium output of each firm? What will be the
equilibrium profits of each firm?
Suppose that Morris and Philip consume wine and books. Morris has an initial
endowment of 60 books and 10 bottles of wine. Philip has an initial endowment
of 20 books and 30 bottles of wine. Morris and Philip only trade with each other
For Morris, a book and a bottle of wine are perfect substitutes. His utility function
is U (b, w)  b  w , where b is the number of books he consumes and w is the
number of bottles of wine he consumes. Philip has more convex preferences
which are represented by the Cobb-Douglas function, U (b, w)  bw .
a.
b.
c.
Draw an Edgeworth box diagram in which Morris’s consumption is
measured from the lower left, and Philip’s is measured from the upper
right corner of the box. Mark the initial endowment and label it E. Draw
the indifference curve for each consumer that passes through E.
At any Pareto optimum, where both people consume some of each good, it
must be that their marginal rates of substitution are equal. No matter what
he consumes, what is Morris’s MRS? What is Philip’s? Write an equation
that every Pareto optimal allocation will satisfy. Draw the locus of Pareto
optimal allocations on the graph.
At a competitive equilibrium, Morris will consume both books and wine.
What will the ratio of price of wine to price of books be? If the price of
books is assumed to be 1, what is the price of wine in competitive
equilibrium?
d.
e.
f.
At the equilibrium prices found in part (c), what is the value of Philip’s
endowment? How many books and bottles of wine will Philip chose to
consume? If Morris consumes all that Philip does not, what will he
consume?
What is Morris’s income at competitive equilibrium? At these prices, is
the cost to Morris of consuming all of the books and all of the wine that
Philip does not consume the same as, more than, or less than his income?
At these prices can Morris afford a bundle that he likes better than the
bundle (55,15)?
If there are 1000 Philips and 1000 Morrises, would the equilibrium prices
that you found still hold? Would supply equal demand for both wine and
books?