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.aAtle Seierstad,Ar..66 / 310%68L+Suppose, for instance, that f*(b) is the maximum profit that a firm can obtain from aproduction 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 thecorresponding resource constraint whose right-hand side in (1) is b¡. If the firm could buymore of this resource at a price below A¡ per unit, it could earn more profit by doing so; butif the price exceeds Aj, the firm could increase its overall profit by selling a small enoughquantity of the resource at this price because the revenue from the sale would exceed theprofit 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 theresource j. It is a shadow rather than an actual price because it need not correspond to amarket price. Indeed, the resource may be something unique like a particular entrepreneur'stime 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 anincrease in b; can lead to a decrease in the value function.If db1, ..., dbm are small in absolute value, then, according to the linear approximationformula, 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),fdy marginal utility of income (A) multiplied by the quantity demanded (x;). Intuitively, thisis because, for a small price change, the loss of real income is approximately equal to thechange in price multiplied by the quantity demanded.PROBLEMS FOR SECTION 3 31. (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 problemmax 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 toI = 2 + 6 + xx2 + y?.+z? = 1(b) Replace the constraints by x + y + z = 1.02 and x2 + y? + z? = 0.98. What is theapproximate 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 a12:52 PMR4/18/2021dy

Question

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.aAtle Seierstad,Ar..66 / 310%68L+Suppose, for instance, that f*(b) is the maximum profit that a firm can obtain from aproduction 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 thecorresponding resource constraint whose right-hand side in (1) is b¡. If the firm could buymore of this resource at a price below A¡ per unit, it could earn more profit by doing so; butif the price exceeds Aj, the firm could increase its overall profit by selling a small enoughquantity of the resource at this price because the revenue from the sale would exceed theprofit 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 theresource j. It is a shadow rather than an actual price because it need not correspond to amarket price. Indeed, the resource may be something unique like a particular entrepreneur'stime 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 anincrease in b; can lead to a decrease in the value function.If db1, ..., dbm are small in absolute value, then, according to the linear approximationformula, f*(b+db) – f* (b) ~ (af*(b)/ab¡) db1+ ..+(af* (b)/abm) dbm and, using (9),(10)&#34;ap (q) &#34;Y + ...+ !qp (q) 'Y (q),f – (qp + q),fdy marginal utility of income (A) multiplied by the quantity demanded (x;). Intuitively, thisis because, for a small price change, the loss of real income is approximately equal to thechange in price multiplied by the quantity demanded.PROBLEMS FOR SECTION 3 31. (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 problemmax 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 toI = 2 + 6 + xx2 + y?.+z? = 1(b) Replace the constraints by x + y + z = 1.02 and x2 + y? + z? = 0.98. What is theapproximate 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 a12:52 PMR4/18/2021dy

Accepted Answer

According to the given information, it is required to solve the question (2).

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