ECON/MGEC 330
Answers of Study Questions on Cost Functions and Behavior of Competitive Firms
1. TRUE or FALSE
a) The average variable cost curve must always be U shaped. FALSE
b) If the average cost curve is U shaped, then the marginal cost curve must cross the
average cost curve at the bottom of the U. TRUE
c) The cost function c(y) = 10+ 3y has marginal cost less than average cost for all levels
of output. TRUE
d) The area under the marginal cost curve measures total variable costs. TRUE
e) A firm has the cost function c(x) = x2 + 100 if the firm's output x is positive and c(0)
=0. If the price of output is lower than 20, and the firm maximizes profit, then the
firm will produce 0 output. TRUE
f) A firm's production function is f(x1,x2) = x1 + 2x2. This means that x2 is twice as
expensive as x1. FALSE
g) A firm has two variable factors and a production function f(x1, x2) = (2x1 + 4x2)1/2. The
marginal rate of technical substitution between x1 and x2 is constant. TRUE
2.
TC(q)=4q2+ 100q + 100
TVC(q)= 4q2+ 100q
AVC(q)=4q + 100
AVC(25)=4*25+100=200
3.
MC = 6y
Derivative of Total Variable Cost (with respect to output) = Marginal Cost
MC = 6y, so TVC= 3y2 -> TVC(8)=3*82=192
4.
c(y) = 3y2 MC=dc/dy=6y
P=36
In perfect competition: P=MC
So 36=6y -> y=6
5.
w1=1 w2=3 p=16
a) C(y)= w1*x1+w2*x2
Production function=y= (min(x1,3x2))1/2
Observe that efficient production with this production function requires x1=3x2
So by substituting this into the production function;
y2= x1=3x2 -> x1=y2 & x2=y2/3
So C(y)= 1*y2+ 3* y2/3= 2y2
b) Profit max. cond: P=MC
MC(y)= dC(y)/dy= 4y
So 4y=16 -> y=4
6.
q = F(S,L) = S0.5L0.5
Cost function is found as follows:
Min rS + wL
subject to q = F(S,L) = S0.5L0.5
where r is the cost of capital (per unit of capital) and w is the wage rate.
This implies MPS/MPL=r/w -> L/S = r/w
Apply these to country A and Country B to see in which country the cost of production is
going to be cheaper:
Country A: r = 16 w = 9
So in country A, we will have L/S = 16/9 -> q = S0.5((16/9)*S)0.5 = 4/3S -> S = 3/4q and
L = 4/3q, implying the cost of production in Country A as:
CA(q)=16*(3/4q) + 9*(4/3q)= 24q
Country B: s = 27 w = 3
Therefore, in country B we will have L/S = 27/3 -> q = S0.5(9*S)0.5 = 3S -> S = 1/3q and L
= 3q, implying the cost of production in Country B as:
CB(q)=27*(1/3q) + 3*(3q) = 18q
Since for each q it costs less to produce in Country B (18q < 24q), locate in country B
7. The cost function is given as
C(y)=5y3 – 40y2 + 96y + 50
The point where a firm switches from negative profit to positive is the point where average
cost is minimized. Average cost is:
AC(y)= C(y)/y = 5y2 – 40y + 96 + 50/y
AC is minimized at the point where dAC/dy=0
dAC/dy= 10y – 40 – 50/y2 = 0 implies y*=4.274
p=AC(4.274) is the minimun price for non-negative profit and hence production of positive
amount.
8.
Capital and labor are perfect substitutes (in a 1-1 ratio) in this production technology. So
only the input that is cheaper will be used in production. With cost of capital 3 and cost of
labor 1, this means only L will be used.
y = L1/2 --> L(y) = y2, implying the cost function C(y) = 1*L(y) = y2