Problem Set 1. A firm's production function is º=50L – 0.012 , where L denotes the size of the workforce. Find the value of MP, in the case when: (a) L=1, (b) L=10, (c) L=100, (d) L=1000 Does the law of diminishing marginal productivity apply to this particular function? 2. Show that the price elasticity of demand is constant for demand functions of the form A P = Q" where A and n are positive constants.

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Chapter1: Making Economics Decisions
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Problem Set
1. A firm's production function is º=50L-0.01Ľ' , where L
denotes the size of the workforce. Find the value of MP
in the case when:
(a) L=1, (b) L=10, (c) L=100, (d) L=1000
Does the law of diminishing marginal productivity apply
to this particular function?
2. Show that the price elasticity of demand is constant for
demand functions of the form
A
P =
Q"
where A and n are positive constants.
3. The demand and total cost functions of a good are
respectively 4P+Q-16=0 and
ТС %3D 4 + 20 —
10
20
a) Find expressions for TR, (profit) 1 , MR, and MC in
terms of Q.
b) Solve the equation
dn
= 0
ÕP
and hence determine the value of Q which maximizes
profit.
c) Verify that, at the point of maximum profit, MR=MC.
4. The cost of building an office complex, x floors high, in a
prime location in Accra is made up of three components:
(a) GH¢10 million for the land
(b) GH¢'/, million per floor
(e) Specialized costs of GH¢10000× per floor.
How many floors should the office complex contain if
the average cost per floor is to be minimized?
5. The supply and demand equations of a good are
respectively given by 3P-Q. = 3 and 2P+Q, =14
The government decides to impose a tax, t, per unit. Find
the value of t (in Ghana cedis) which maximizes the
governments total tax revenue on the assumption that
equilibrium conditions prevail in the market.
6. A firm's demand function for a certain good is given by
P = 100e-0.10. Its total cost function is TC = 100e-010 + 50 . What
output level maximizes the firm's profit?
Transcribed Image Text:Problem Set 1. A firm's production function is º=50L-0.01Ľ' , where L denotes the size of the workforce. Find the value of MP in the case when: (a) L=1, (b) L=10, (c) L=100, (d) L=1000 Does the law of diminishing marginal productivity apply to this particular function? 2. Show that the price elasticity of demand is constant for demand functions of the form A P = Q" where A and n are positive constants. 3. The demand and total cost functions of a good are respectively 4P+Q-16=0 and ТС %3D 4 + 20 — 10 20 a) Find expressions for TR, (profit) 1 , MR, and MC in terms of Q. b) Solve the equation dn = 0 ÕP and hence determine the value of Q which maximizes profit. c) Verify that, at the point of maximum profit, MR=MC. 4. The cost of building an office complex, x floors high, in a prime location in Accra is made up of three components: (a) GH¢10 million for the land (b) GH¢'/, million per floor (e) Specialized costs of GH¢10000× per floor. How many floors should the office complex contain if the average cost per floor is to be minimized? 5. The supply and demand equations of a good are respectively given by 3P-Q. = 3 and 2P+Q, =14 The government decides to impose a tax, t, per unit. Find the value of t (in Ghana cedis) which maximizes the governments total tax revenue on the assumption that equilibrium conditions prevail in the market. 6. A firm's demand function for a certain good is given by P = 100e-0.10. Its total cost function is TC = 100e-010 + 50 . What output level maximizes the firm's profit?
7. An individual's utility function is given by
U = 1000x, + 450x, + 5x,x, – 2x – x
where is the amount of leisure measured in hours per
week and is income earned measured in cedis per week.
Determine the value of the marginal utilities, when *=
138 and = 500. Hence estimate the change in utility if
!!
the individual works for an extra hour, which increases
earned income by GH¢15 per week. Does the law of
diminishing utility hold for this function?
8. A firm's total cost function is given by
TC = 3Q° +2Q,Q, +7Q
Where 9 and l: denote the number of items of goods 1
and 2, respectively that are produced. Using the
substitution method, find the values of and 2: which
minimize costs if the firm is committed to producing 40
goods of either type in total.
9. A monopolistic producer of two goods, 1 and 2, has a
joint total cost function
TC = 10Q + QQ, +10Q,
where 9 and 2: denote the quantity of items of goods 1
and 2, respectively that are produced. If P, and P, denote
the corresponding prices then the demand equations are
P = 50 -Q + Q,
P, = 30 + 20, - Q,
Using the Lagrange multiplier approach, find the
maximum profit if the firm is contracted to produce a
total of 15 goods of either type. Estimate the new optimal
profit if the production quota rises by 1 unit.
Transcribed Image Text:7. An individual's utility function is given by U = 1000x, + 450x, + 5x,x, – 2x – x where is the amount of leisure measured in hours per week and is income earned measured in cedis per week. Determine the value of the marginal utilities, when *= 138 and = 500. Hence estimate the change in utility if !! the individual works for an extra hour, which increases earned income by GH¢15 per week. Does the law of diminishing utility hold for this function? 8. A firm's total cost function is given by TC = 3Q° +2Q,Q, +7Q Where 9 and l: denote the number of items of goods 1 and 2, respectively that are produced. Using the substitution method, find the values of and 2: which minimize costs if the firm is committed to producing 40 goods of either type in total. 9. A monopolistic producer of two goods, 1 and 2, has a joint total cost function TC = 10Q + QQ, +10Q, where 9 and 2: denote the quantity of items of goods 1 and 2, respectively that are produced. If P, and P, denote the corresponding prices then the demand equations are P = 50 -Q + Q, P, = 30 + 20, - Q, Using the Lagrange multiplier approach, find the maximum profit if the firm is contracted to produce a total of 15 goods of either type. Estimate the new optimal profit if the production quota rises by 1 unit.
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