ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN: 9780190931919
Author: NEWNAN
Publisher: Oxford University Press
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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?
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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- Question 5: Suppose a brewery uses a Cobb-Douglas production function for his production. He studies the production process and finds the following. An additional machine-hour of fermentation capacity would increase output by 500 bottles per day (i. e. MPK = 500). An additional man-hour of labor would increase output by 1000 bottles per day (i. e. MPL = 1000). The price of a man- hour of labor is $50 per hour. The price of a machine-hour of fermentation capacity is $5 per hour. 1. Is the brewery currently minimizing its cost of production? Check using the minimization condition. 2. It turns out, the brewery is not optimally chossing the factors of production. To lower its production cost, which factor of production should the brewery increase and which factor should he decrease? 3. Suppose that the price of a machine-hour of fermentation capacity rises to $25 per hour. How does this change the answer from part 1?arrow_forward1. A firm’s production function is , where Ldenotes the size of the workforce. Find the value of MPLin 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 where A and n are positive constants. 3. The demand and total cost functions of a good arerespectively and a) Find expressions for TR, (profit) , MR, and MC in terms of Q. b) Solve the equation 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¢1/4 million per floor (c) Specialized costs of GH¢10000x per floor. How many floors should the office complex contain if the average cost per floor is to be minimized? 5. The supply…arrow_forward1/2 Consider a firm with the production function ƒ(x₁, x₂) = x¹/²x₂. The price of the two inputs is ₁ = 2 and w₂ = 1. If x₁ = x₂ = 16, the marginal product of input 1 is When ₁ is increasing and 2 stays the same, the marginal product of input 1 is Constant Decreasing Increasing This production function has Constant returns to scale Decreasing returns to scale O Increasing returns to scale None of the other answers is correct Does the production function have a diminishing technical rate of substitution? No Yesarrow_forward
- Suppose the long-run production function for a competitive firm is f(x1,x2)= min {x1,2x2}. The cost per unit of the first input is w1 and the cost of the second input is w2. .a. Find the cheapest input bundle, i.e. amount of labor and capital, that yields the given output level of y. .b. Draw the conditional input demand functions for labor and capital in the x1-y and x2- y spaces. .c. Write down the formula and draw the graph of the firm’s total cost function as a function of y, using the conditional input demand functions. What is the relationship between the returns to production scale and the behavior of the total costs? .d. Write down the formula and draw the graph of the average cost function, as a function of y. .e. Write down the formula and draw the graph of the marginal cost function, as a function of y.arrow_forwardA firm operates in the short run with the production function Q = 10 K0.5 L 0.5, where Q is output, K is capital, and L is labor. The price of capital is $100 per unit, and the wage rate is $50 per unit of labor. How many units of capital and labor should the firm use to minimize its total cost while producing 400 units of output?arrow_forwardProblem 5 Suppose that a firm has a production function f(K, L) = 12L¹/3 K¹/3 and that w and r are input prices and p is output price. The firm does not change those prices. a) Show that the firm's profit π is a concave function of (K, L). b) Find (K*, L*) that maximizes the profit, as a function of (w,r, p).arrow_forward
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