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SM 2. (a) Solve the problem max x +4y +z subject to x+ y +z? = 216 and x (b) Change the first constraint to x? + y? + z? = 215 and the seca Estimate the corresponding change in the maximum value by us %3D

SM 2. (a) Solve the problem max x +4y +z subject to x+ y +z? = 216 and x (b) Change the first constraint to x? + y? + z? = 215 and the seca Estimate the corresponding change in the maximum value by us %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can you help with question 2? Note (10) is attached for reference. 

%20Mathematics%20for%20Economic%20Analysis%20by%20Knut%20Sydsaeter,Peter%20Hammond,Atle%20Seierstad, Arne%20Strom%20(z-lib.a Atle Seierstad,Ar.. 66 / 310 %68L + Suppose, for instance, that f*(b) is the maximum profit that a firm can obtain from a production process when b1, ..., bm are the available quantities of m different resources. Then af* (b)/ab; is the marginal profit that the firm can earn per extra unit of resource j, which is therefore the firm's marginal willingness to pay for this resource. Equation (9) tells us that this marginal willingness to pay is equal to the Lagrange multiplier 2; for the corresponding resource constraint whose right-hand side in (1) is b¡. If the firm could buy more of this resource at a price below A¡ per unit, it could earn more profit by doing so; but if the price exceeds Aj, the firm could increase its overall profit by selling a small enough quantity of the resource at this price because the revenue from the sale would exceed the profit from production that is sacrificed by giving up the resource. In economics, the number ; (b) is referred to a shadow price (or demand price) of the resource j. It is a shadow rather than an actual price because it need not correspond to a market price. Indeed, the resource may be something unique like a particular entrepreneur's time for which there is not even a market that could determine its actual price. Note that the Lagrange multipliers for problem (1) may well be negative, so that an increase in b; can lead to a decrease in the value function. If db1, ..., dbm are small in absolute value, then, according to the linear approximation formula, f*(b+db) – f* (b) ~ (af*(b)/ab¡) db1+ ..+(af* (b)/abm) dbm and, using (9), (10) "ap (q) "Y + ...+ !qp (q) 'Y (q),f – (qp + q),f dy
Transcribed Image Text:%20Mathematics%20for%20Economic%20Analysis%20by%20Knut%20Sydsaeter,Peter%20Hammond,Atle%20Seierstad, Arne%20Strom%20(z-lib.a Atle Seierstad,Ar.. 66 / 310 %68L + Suppose, for instance, that f*(b) is the maximum profit that a firm can obtain from a production process when b1, ..., bm are the available quantities of m different resources. Then af* (b)/ab; is the marginal profit that the firm can earn per extra unit of resource j, which is therefore the firm's marginal willingness to pay for this resource. Equation (9) tells us that this marginal willingness to pay is equal to the Lagrange multiplier 2; for the corresponding resource constraint whose right-hand side in (1) is b¡. If the firm could buy more of this resource at a price below A¡ per unit, it could earn more profit by doing so; but if the price exceeds Aj, the firm could increase its overall profit by selling a small enough quantity of the resource at this price because the revenue from the sale would exceed the profit from production that is sacrificed by giving up the resource. In economics, the number ; (b) is referred to a shadow price (or demand price) of the resource j. It is a shadow rather than an actual price because it need not correspond to a market price. Indeed, the resource may be something unique like a particular entrepreneur's time for which there is not even a market that could determine its actual price. Note that the Lagrange multipliers for problem (1) may well be negative, so that an increase in b; can lead to a decrease in the value function. If db1, ..., dbm are small in absolute value, then, according to the linear approximation formula, f*(b+db) – f* (b) ~ (af*(b)/ab¡) db1+ ..+(af* (b)/abm) dbm and, using (9), (10) "ap (q) "Y + ...+ !qp (q) 'Y (q),f – (qp + q),f dy
marginal utility of income (A) multiplied by the quantity demanded (x;). Intuitively, this is because, for a small price change, the loss of real income is approximately equal to the change in price multiplied by the quantity demanded. PROBLEMS FOR SECTION 3 3 1. (a) Solve the problem max 100 – x² - y² – z² subject to x + 2y + z = a. (b) Compute the optimal value function f*(a) and verify that (9) holds. SM 2. (a) Solve the problem max x + 4y + z subject to x² + y² + z² = 216 and x+ 2y +3z = 0 %3D (b) Change the first constraint to x² + y² + z² = 215 and the second to x + 2y + 3z = 0.1. Estimate the corresponding change in the maximum value by using (10). 3. (a) Solve the problem max e +y +z subject to I = 2 + 6 + x x2 + y?. +z? = 1 (b) Replace the constraints by x + y + z = 1.02 and x2 + y? + z? = 0.98. What is the approximate change in optimal value of the objective function? SM 4. (a) Solve the utility maximizing problem (assuming m > 4) max U(x1, x2) = In(1+ x1) + In(1 + x2) subject to 2x1 + 3x2 = m (b) With U*(m) as the indirect utility function, show that dU* /dm = . Show a 12:52 PM R 4/18/2021 dy
Transcribed Image Text:marginal utility of income (A) multiplied by the quantity demanded (x;). Intuitively, this is because, for a small price change, the loss of real income is approximately equal to the change in price multiplied by the quantity demanded. PROBLEMS FOR SECTION 3 3 1. (a) Solve the problem max 100 – x² - y² – z² subject to x + 2y + z = a. (b) Compute the optimal value function f*(a) and verify that (9) holds. SM 2. (a) Solve the problem max x + 4y + z subject to x² + y² + z² = 216 and x+ 2y +3z = 0 %3D (b) Change the first constraint to x² + y² + z² = 215 and the second to x + 2y + 3z = 0.1. Estimate the corresponding change in the maximum value by using (10). 3. (a) Solve the problem max e +y +z subject to I = 2 + 6 + x x2 + y?. +z? = 1 (b) Replace the constraints by x + y + z = 1.02 and x2 + y? + z? = 0.98. What is the approximate change in optimal value of the objective function? SM 4. (a) Solve the utility maximizing problem (assuming m > 4) max U(x1, x2) = In(1+ x1) + In(1 + x2) subject to 2x1 + 3x2 = m (b) With U*(m) as the indirect utility function, show that dU* /dm = . Show a 12:52 PM R 4/18/2021 dy
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