Economics Exam > Economics Questions > A production function for a perfectly competi...
Start Learning for Free
A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the function C=c(q) when W1=W2=100 2. Show that the cost function in it's general form is constant with homogeneity and shephard's lemma properties of cost function?
Most Upvoted Answer
A production function for a perfectly competitive firm is given as Q=A...
Derivation of Cost Function C(q) when W1 = W2 = 100:
To derive the cost function C(q) when W1 = W2 = 100, we first need to express the cost function C in terms of the output Q. Given that C = ∑WiXi, we can substitute the values of W1 and W2 into the cost function.
C = 100X1 + 100X2
C = 100(X1 + X2)
C = 100Q
Therefore, the cost function C(q) when W1 = W2 = 100 is C = 100Q.
Constant with Homogeneity and Shepard's Lemma Properties:
Constant with Homogeneity:
The cost function C(q) exhibits constant returns to scale as it is proportional to the output Q. This property is known as homogeneity, where if all input prices are multiplied by a constant factor, the cost function is also multiplied by the same factor.
Shepard's Lemma:
Shepard's lemma states that the cost function can be derived from the production function by taking the partial derivative of the production function with respect to each input and multiplying it by the price of that input. Mathematically, this can be represented as:
∂C/∂Wi = Wixi
In the given production function Q=AX1^(1/2)X2^(1/2), we can calculate the partial derivatives with respect to X1 and X2:
∂Q/∂X1 = (1/2)AX1^(-1/2)X2^(1/2) = (1/2)Q/X1
∂Q/∂X2 = (1/2)AX1^(1/2)X2^(-1/2) = (1/2)Q/X2
Multiplying these partial derivatives by the input prices W1 and W2, we get:
∂C/∂W1 = 100X1 = W1X1
∂C/∂W2 = 100X2 = W2X2
Therefore, the cost function satisfies Shepard's lemma as the partial derivatives of the cost function with respect to input prices are equal to the product of input prices and input quantities.
|
Explore Courses for Economics exam
|
|
A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the function C=c(q) when W1=W2=100 2. Show that the cost function in it's general form is constant with homogeneity and shephard's lemma properties of cost function?
Question Description
A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the function C=c(q) when W1=W2=100 2. Show that the cost function in it's general form is constant with homogeneity and shephard's lemma properties of cost function? for Economics 2024 is part of Economics preparation. The Question and answers have been prepared according to the Economics exam syllabus. Information about A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the function C=c(q) when W1=W2=100 2. Show that the cost function in it's general form is constant with homogeneity and shephard's lemma properties of cost function? covers all topics & solutions for Economics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the function C=c(q) when W1=W2=100 2. Show that the cost function in it's general form is constant with homogeneity and shephard's lemma properties of cost function?.
A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the function C=c(q) when W1=W2=100 2. Show that the cost function in it's general form is constant with homogeneity and shephard's lemma properties of cost function? for Economics 2024 is part of Economics preparation. The Question and answers have been prepared according to the Economics exam syllabus. Information about A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the function C=c(q) when W1=W2=100 2. Show that the cost function in it's general form is constant with homogeneity and shephard's lemma properties of cost function? covers all topics & solutions for Economics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the function C=c(q) when W1=W2=100 2. Show that the cost function in it's general form is constant with homogeneity and shephard's lemma properties of cost function?.
Solutions for A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the function C=c(q) when W1=W2=100 2. Show that the cost function in it's general form is constant with homogeneity and shephard's lemma properties of cost function? in English & in Hindi are available as part of our courses for Economics.
Download more important topics, notes, lectures and mock test series for Economics Exam by signing up for free.
Here you can find the meaning of A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the function C=c(q) when W1=W2=100 2. Show that the cost function in it's general form is constant with homogeneity and shephard's lemma properties of cost function? defined & explained in the simplest way possible. Besides giving the explanation of
A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the function C=c(q) when W1=W2=100 2. Show that the cost function in it's general form is constant with homogeneity and shephard's lemma properties of cost function?, a detailed solution for A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the function C=c(q) when W1=W2=100 2. Show that the cost function in it's general form is constant with homogeneity and shephard's lemma properties of cost function? has been provided alongside types of A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the function C=c(q) when W1=W2=100 2. Show that the cost function in it's general form is constant with homogeneity and shephard's lemma properties of cost function? theory, EduRev gives you an
ample number of questions to practice A production function for a perfectly competitive firm is given as Q=AX1^(1/2)X2^(1/2) where Q is the output (in tons) and Xi are quantities of inputs used in the production of Q (in tons). The cost of producing Q is given as C=summation of WiXi where i=1,2, where Wi are per unit input prices of input 1 and 2 respectively. 1. Derive the function C=c(q) when W1=W2=100 2. Show that the cost function in it's general form is constant with homogeneity and shephard's lemma properties of cost function? tests, examples and also practice Economics tests.
|
|
Explore Courses for Economics exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup