Computing Produce Scale
22 April 2010
Computing Produce Scale
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pleaseee somebody answer that question?
Suppose the short-run total cost function of a
firm is given by
C(q) = a + bq + cq2 where a; b; c > 0.
a) Compute the marginal and average cost functions. What are
fixed costs equal to?
b) Does the marginal cost curve cross the average cost curve at the minimum of average cost?
c) Compute the region of output over which economies of scale are present. [Economies of scale are present when the average cost of production decreases as quantity increases]
d) Now suppose the same fi
rm starts producing a second product and the cost function becomes
C(q1; q2) = d + bq1 + cq1(square) + eq2 + fq2(square) + gq1q2
For what values of d; b; c; e; f and g are there economies of scope in production? [Economies of scope are
present when it costs less to produce two goods together than separately: C(q1; q2) < C(q1) + C(q2)
a)
C=a+bq+cq²
MC = ∂C/∂q = b+2cq
AC=C/q
FC=a
AC=a/q+b+cq
b)
AC[MIN] if ∂AC/∂q=0
∂AC/∂q=c-a/q²
c-a/q²=0
c=a/q²
q²=a/c
q=√(a/c)
q[ATC→MIN]=√(a/c)
AC=MC
a/q+b+cq = b+2cq
a/q=cq
q²=a/c
q[AC=MC]=√(a/c)
Both parts gives true identity: √(a/c) = √(a/c) or q[ATC→MIN]=q[AC=MC]
Thus for this cost function MC crosses AC at it's minimum point.
c)
AC=C/q = a/q+b+cq
AC[MIN] if ∂AC/∂q=0
∂AVC/∂q = c-a/q²
c-a/q²=0
a/q²=c
a=cq²
q²=a/c
q=√(a/c) [a>0 and c>0]
For this cost function economies of scale exist at range 0≤q<√(a/c) or for q changing to 0→√(a/c)
d)
FC=d
VC(q1) = bq[1]+cq[1]²+gq[1]q[2]= {q[2]=0} = bq[1]+cq[1]²
VC(q2) = eq[2]+fq[2]²+gq[1]q[2]= {q[1]=0} = eq[1]+fq[1]²
C(q1;q2) = FC+VC(q1)+VC(q2)+gq[1]q[2]
C(q1;q2) < FC+VC(q1)+VC(q2)
d+bq[1]+cq[1]²+eq[1]+fq[1]² +gq[1]q[2]< d+bq[1]+cq[1]²+eq[1]+fq[1]²
Simplifies to:
d+gq[1]q[2] < d
Simplifies to:
gq[1]q[2] < 0
Since q[1]≥0 and q[2]≥0 (actually q[1]>0 and q[2]>0 because overvice if q[1] or q[2] =0 then gq[1]q[2] = 0), thus the only solution for this inequality is at values:
g < 0 ; q[1]>0 ; q[2]>0
All other variables does not affect "economies of scope" condition for this cost function.
g < 0
