ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN: 9780190931919
Author: NEWNAN
Publisher: Oxford University Press
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Transcribed Image Text:(a) For the cost function C(w1, w2, y) = 2y²w} w, calculate the Allen elasticity
of substitution between the two inputs at the cost-minimizing input point
(xf(w1, w2, y), a(w1, w2, y)).
(b) Consider the production function f(r, y, z) = Vry + rz+ yz. Find the scale
elasticity SE at (x, y, z) = (1,2, 3), (5, 1,6), (6, 6, 6) and determine if the pro-
duction function is IRTS, CRTS, or DRTS locally at each point.
(c) A profit maximizing firm in the market operates where the production exhibits
decreasing return to scale (DRTS). Is this market in its long-run equilibrium?
Justify your answer.
(d) Suppose that there are the infinite number of potential firms that produce the
identical output good y under the cost function C(y) = + 3. Assume free
entry and exit. Find the long-run equilibrium output price p, the amount of
the output that each firm in the market produces in the long-run equilibrium,
and the value of profit that each firm earns in the equilibrium.
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- Suppose that a firm’s production function is Q =10 K^(3/4)L^(1/4). The cost of a unit of labor is $1 and the cost of a unit of capital is $3. The manager of this firm is interested in finding the following information: The manager now wants to know the K/L ratio to produce anyoutput level at the minimum total cost. Represent graphically the expansion path in the long run.arrow_forward*Find the elasticity of scale and the elasticity of substitution for the CES production function f(x1,x2) = (x{ + x%)¯,where 0 #p<1.arrow_forwardWhen the quantity of a variable input is increased from 4 to 6 units, the total output increases from 85 units to 105 units. The marginal product of the variable input is 20 units.arrow_forward
- (a) Suppose we have the following production function: Q=K¹/4L³/4 Suppose K is fixed in the short- run at 16. State the firm's short-run cost minimization problem given the fixed input. Discuss.< (b) Let r = $200 and w = $300. Derive the short-run cost function so that we have costs as a function of Q: eg C(Q) = ? Show your work/reasoning.< (c) Show that average costs first fall then rise with greater output. Note: look at Q in the range 0 to 100. You can do this in a table or mathematically. Why does AC fall then rise? < (d) State the firm's profit maximizing problem.< (e) Let P = $600. Find optimal output where r = $200 and w = $300. You can do this in a table or mathematically. What are profits? < (f) A break-even price is one where the best a firm can do is earn zero profits. That is, total revenues equal total costs. How is the break-even price related to average costs? Explain. < (g) What is the break-even price if r = $200 and w = $300? Explain/show. <arrow_forwardLaw of diminishing returns......arrow_forwardConsider the following equation Q = f(K, L) = K0.7 L0.3 Find (a) MRTS of labor for capital (b) Show that the production function above is homothetic (c) Returns to scale (d) Equation representing an isoquant and show that it is convex to the origin (e) Elasticity of substitution (f) Output elasticities of factorsarrow_forward
- Find the elasticity of scale and the elasticity of substitution for the CES production function: 1 1 f(x₁, x₂) = (x³ + x2)³. Solution: We first calculate the marginal products: 2 fx₁ = 3 + = x1fx1 -+ 2 2 10 - ² ( x² + x ) ( + x^²) = ( + + + + ² ) 15 ²0 x² -2/3 x₂ + x2fxz Elasticity of scale = _1₁_+_*__*(+)´<°¸«d«)*;»_ f(x1, f(x1, ‹2)² = -2/3 r2/3 = TRS = t (where TRS = =t). ⇒ r = t³/² ⇒ ln(r) = ln (t) and o = -2/3 dln (r) dln (t) To get elasticity of substitution, we first need TRS and denote r = 1 x3 X1 TMS - F (+4+2)0 = fx₁ fx₂ 2 1 + x²) x₂² MIN x2 3 -2/3 (x² -2/3 2 2/3 2/3 x1 -2/3 = r²/3 x¹/3 + 1/3 = 1.arrow_forwardwhich of the following functions satisfy the following properties: i) Increasing in both inputs. ii) Diminishing marginal products. iii) Constant returns to scale. iii) Complementarity between capital and labor.arrow_forwardSubpart 1-3arrow_forward
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