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Mini Quiz 2 ISHISE, Hirokazu

Graduate Introductory Macroeconomics: Mini Quiz 2

Name: Student ID:

Increasing returns to scale

Suppose that for an operation, a rm needs to employ f > 0 units of labor adding to laborproportional to output. The rm faces a downward sloping demand curve. A rm choosesp(j), y(j) and l(j), for taking z(j), w, p, y and f as given.

maxfp(j);y(j);l(j)g

p(j)y(j) wl(j) wfs.t. y(j) = z(j)l(j)

y(j) =

p

p(j)

y:

Note that the productivity z(j) diers across rms.

Question 1 (2 points)

Set-up a Lagrangian of the rm's optimization problem.

L = p(j)y(j) wl(j) wf + (j)

p

p(j)

y y(j)

+ '(j) (z(j)l(j) y(j)) : (1)


Derive rst order conditions of the rm's optimization problem.

FOCs are

0 = y(j) (j)pp(j)1y; (2)0 = p(j) (j) '(j); (3)0 = w + '(j)z(j): (4)

1 Graduate Introductory Macroeconomics Summer 2014


Question 3 (2 point)

Derive the expression of p(j) (as a function of exogenous variables and parameters).

p(j) =

1w

z(j): (5)


Derive the expression of l(j).

l(j) = ( 1)pywz(j)1 (6)

Question 5 (1 point)

Derive the expression of the prots of the rm.

p(j)y(j) wl(j) wf = 1

w

z(j)z(j)l(j) wl(j) wf

=1

1wl(j) wf

=1

1w( 1)pywz(j)1 wf

=( 1)1pyw+1z(j)1 wf (7)

Question 6 (2 points, bonus, dicult)

In this model, some rms have high z(j), while others have low productivity. If a rm'sproductivity is very low, the rm would earn the negative prots. If a rm's productivity ishigh, the rm earns positive prots. There should be a cut-o productivity, z(j) = z. If a rmhas higher productivity than this cut-o, the rm operates. If a rm has lower productivity



than this cut-o, the rm does not operate. Using the zero-prots condition, determine thecut-o productivity z.

A rm earns zero-prots if the rm's productivity satises the zero-prots condition:

0 = ( 1)1pyw+1z(j)1 wf: (8)

The cut-o productivity z is this z(j). Hence,

0 = ( 1)1pyw+1z1 wf; (9)

or

z =

1

1w

p

1f

y

11

: (10)


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