EconS 501 - Micro Theory IAssignment #5 - Due date: October 23rd, in class.

1. Exercises from JR (3rd edition):

(a) Chapter 3: Exercise 3.31, and 3.42.

2. Exercises from NS:

(a) Chapter 12: Exercise 12.9.

3. Exercises from Varian (1992):

(a) Chapter 5: Exercise 5.16.

4. [Cost minimization]

(a) Prove that the cost function c(w; q) is a concave function of the input price vectorw.

(b) Assuming that the input demand correspondence z(w; q) is continuously di¤eren-tiable, show that

@zi(w; q)

@wj=@zj(w; q)

@wi

for any two inputs i 6= j.

5. [Marginal cost being independent of an input price] Consider the productionfunction f(h(z1) + z2), where f(�) is increasing, h(�) is an increasing concave functionwhich satis�es h0(0) =1 and h0(1) = 0.

(a) Given the input price vector w, show that for large enough output levels, theinput demand correspondence of input 2, z2(w; q), must be strictly positive.

(b) Assuming that z2(w; q) > 0 (as shown in the previous part), write down the �rm�scost function as a function of f�1(q) and z1 alone. Hence, show that the inputdemand correspondence of input 1, z1(w; q), is independent of q.

(c) Show that hte marginal cost is independent of w1.

6. [Returns to scale] Consider a production function f(z) that exhibits increasingreturns to scale everywhere (i.e., for all input levels).

(a) Show that the scale elastity of f(z) satis�es "(f(�z); �) > 1 for all input vector zand for any common increase in all inputs by a factor � > 0.

(b) Show that@ ln f(�z)

@�>1

�

1

(c) Then, show that for any � > 1,

lnf(�z)

f(z)=

Z �

1

@ ln f(�z)

@�d�

7. [Scale of �rms in a perfectly competitive market] Consider an industry in whichthere is free entry, whereby all �rms have the same U-shaped AC curve.

(a) Show that for each �rm the marginal cost curve is

MC =nXj=1

wjzjq"(zj; q)

where "(zj; q) represents the output elasticity (i.e., the percent increase in q as aresult of a 1% increase in input zj).

(b) Now show that output level along the average cost function must satisfy

1 =nXj=1

kj"(zj; q),

where kj � wjzjw�z represents the expenditure share of input zj in the cost-minimizing

input vector of the �rm.

(c) From the above results, show that in the case that the �rm only uses two inputs,1 and 2, if "(z2; q) > "(z1; q), then

"(z2; q) > 1 > "(z1; q).

(d) Let us now consider a parametric example. A �rm has production function f(z) =(z1 � a)�z�2 , where z1 � a > 0, and z2 � 0. The input price vector is w. Showthat along the output expansion path the following equality must hold

w2z2 =�

�(w1z1 � aw1) :

Then, show that along the output expansion path "(z2; q) > 1 > "(z1; q) holds.

(e) Finally, show that the scale of the active �rms will rise if one of the input pricesrises, while the scale will fall if the other input price rises.

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