國立高雄大學九十六學年第一學期
應用經濟學系
經濟數學期中考(II)
1. A firm uses one input, L, to generate output, q, according to the production
function q  16L2 . The input price is w and fixed costs are C0  0 . Show that the
firm’s cost function is given by C (q)  C0 
while
w q
dq
, and that
increases with L
4
dL
dC
decreases with q.
dq
「Solution」
Inverting the production function we have L  q / 4 . This is the level of labour
required to produce q units of output. Since there is a fixed cost of C0  0 , and
each unit of labour cost w, the total cost to the firm of producing q units of output is
w q
. The first and second-order derivatives of the production
4
function are 32L and 32, respectively. Since the second-order derivative is positive,
C (q)  C0 
dq
increases with L. Similarly, the first and second-order
dL
w
w
derivatives of the cost function are
and
3 . Since the input price and the
8 q
16q 2
this shows that
quantity are both positive, the second-order derivative is negative. This shows that
dC
decreases with q.
dq
2. Consider a consumer that faces the following optimization problem
max
U  U ( x, y )
s.t.
px x  p y y  B
x, y
,
(a) Use the Lagrangian method and solve for the demand for x and y.
(b) Use the total-differential approach to (i) obtain the results of optimal x and y, and
(ii) perform the comparative static analyses.
「Solution」
The Lagrangian function of this optimization problem is
Z  U ( x, y )   ( B  p x x  p y y ) .
The first-order-condition yields the following set of simultaneous equations:
Z x  U x  p x  0 , Z y  U y  p y  0 , Z   B  p x x  p y y  0 .
The first two equations implies that
Ux Uy

  , which, in turn, suggests that
px
py
consumers must allocate their budgets so as to equalize the ratio of marginal utility to
price for every commodity. Specifically, in the equilibrium or optimum, these ratios
should have the common value  , measuring the comparative-static effect of the
constraint constant on the optimal value of the objective function. Hence, we have in
the present context   U B ; that is, the optimal value of the Lagrange multiplier
can be interpreted as the marginal utility of money (budget money) when the
consumer’s utility is maximized.
Alternatively, we have
Ux
p
 x in terms of indifference curves. An indifference
U y py
curve is defined as the locus of the combination of x and y that will yield a constant
level of U. This means that on an indifference curve we must
find dU  U x dx  U y dy  0 , implying dy / dx  U x / U y . Hence, if we plot an
indifference curve in the xy plane, its slope, dy / dx , must be equal to the negative of
the marginal utility ratio U x / U y . Conversely, since U x / U y is the negative of the
indifference-curve slope, it must represent the marginal rate of substitution between
the two goods. As for p x / p y , this ratio represents the negative of the slope of the
budget constraint. The budget constraint can be written alternatively as
p
B
y
 x x . Hence, a consumer must allocate the budget such that the slope of the
py py
budget line (on which the consumer must remain) is equal to the slope of some
indifference curve.
If the bordered Hessian in the present problem is positive, i.e., if
0
px
H  px
U xx
p y U yx
py
U xy  2 p x p yU xy  ( p y ) 2 U xx  ( p x ) 2 U yy  0 , then the stationary
U yy
value of U will assuredly be a maximum.
Comparative-Static Analysis
In the current model, the prices p x and p y are exogenous, as is the amount of the
budget, B. If we assume the satisfaction of the second-order sufficient condition, we
can analyze the comparative-static properties of the model of the basis of the
first-order condition, viewed as a set of equations F j  0 (j = 1, 2, 3). Consequently,
the implicit-function theorem is applicable, and we may express the optimal values of
the endogenous variables as implicit functions of the exogenous variables:
   ( p x , p y , B) ,
x  x( p x , p y , B) ,
y  y ( p x , p y , B) .
To find these derivatives, we must express the first-order condition as
B  px x  p y y  0 ,
U x ( x, y)   px  0 ,
U y ( x, y )   p y  0 .
By taking the total differential of each identity in turn (allowing every variable to
change), and noting that U xy  U yx , we then obtain the following linear system
 p x d x  p y d y  xdp x  ydp y  dB ,
 p x d   U xx d x  U xy d y  dp x ,
 p y d   U yx d x  U yy d y  dp y .
To study the effect of a change in the budget size, let dp x  dp y  0 , but keep
dB  0 . Then, after dividing the above equations through by dB, and interpreting each
ratio of differentials as a partial derivative, we can write the matrix equation as
 0

 px
 p y
 px
U xx
U yx
 p y   B   1


U xy    x B    0  ,
 
U yy   y B   0 
By Cramer’s rule, we can solve for all three comparative-static derivatives, that is,
0
 x  1
  
 px
 B  J  p
y
 1  px
0
y 1
  
 px
 B  J  p
y
 px
1
U xx
0 
U yx
0
0
U xy 
0
U yy
1  p x U xy
,
J  p y U yy
1  p x U xx
.
J  p y U yx
The rest of the impacts on demands of changes in prices can be done in a similar
fashion.
3. A firm uses capital K, labor L and land T to produce Q units of a commodity
where Q  K 2/ 3  L1/ 2  T 1/ 2 . The price of output is p, the price of capital r, the price
of labor w and the price of land q.
(a) Find the profit maximizing combination of inputs used by the firm. Check the
second order conditions.
(b) If Q* denotes the optimal value of output produced by the firm and K * the
optimal amount of capital used, then show that Q* / r  K * / p .
「Solution」
The first-order condition is given by

 ( 2 / 3) pK 1 / 3  r  0 ,
K

 (1 / 2) pL1 / 2  w  0 ,
L

 (1 / 3) pT 2 / 3  q  0 .
T
It follows that K * 
8 p3
p2
1 p 3/ 2
*
*
,
,
and
.
L

T

3/ 2
27 r 3
4w 2
27 q
To check the second-order conditions, we need the matrix of second-order derivatives.
We have
 92 pK 4 / 3

H 
0

0



0   0 . Hence, we can conclude that the
 92 T 5 / 3 
0
 14 pK
0
3 / 2
0
principal minors are alternating in sigh which is what we need for the solution to be a
maxima.
To show that Q* / r  K * / p , we need to compute Q * . Using
L* 
p2
4w 2
Thus,
,
and
T* 
Q *
8 p2
.

r
9 r3
1 p 3/ 2
3/ 2
27 q
On
the
,
we
other
have
hand,
Q* 
K* 
8 p3
,
27 r 3
4 p2
p
1


2
9 r
2w
3
K *
8 p2
.

p
9 r3
This
p
q
.
shows
that Q* / r  K * / p .
4. A firm produces a single commodity that it can sell at a constant price of p. The
cost function is given by C (q)  aq  bq 2 where a, b  0 . The firm also has to
pay a tax of t for every unit sold by it. Assume that p  a  t .
(a) How much should the firm sell q* to maximize profits? What is the
corresponding level of profit  * ?
(b) Show that  * / p  q* . How would you interpret this economically?
Write out the objective function, then use the same technique as the above ones.
5. Let the consumer’s utility function be given by U ( x1 , x2 )  ( x1  c1 ) ( x2  c2 )1 ,
where c1 , c2  0 and 0    1 . Let the prices of the two goods be p1 and p2
and the consumer’s income be M. Solve for the optimal quantity of x1 and x2 .
See the solution to Question 3.
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