The Lagrange function is in which stands for the Lagrange multiplier. Observation: A crucial observation is...

Transcribed Image Text:

The Lagrange function is in which stands for the Lagrange multiplier. Observation: A crucial observation is that there are two arguments (separated by the one comma) in the utility function. The first argument is c and - despite the fact that c again appears the second L(c,n,2)=uc, n. cv+2[w.n-c], V argument is ncv. Thus, in the ensuing analysis, please write the marginal utility function with second argument as u V respect to the first argument as u, c,n.cv and the marginal utility function with respect to the (c.n.cl c,n.c- on the wiki V Because c appears twice in the utility function, it directly affects not only the marginal utility for c (which is natural), but also directly affects the marginal utility for n. Formally, the latter effect arises from the Chain Rule. (Please note that, from formal mathematics, there is nothing incorrect/inconsistent in a function in which the same variable appears in multiple arguments.) 1 Note that there are no typos in the utility function. Also note that the utility function is stated in abstract form - in other words, there is no particular utility function provided. You may not assume any particular utility function and are not allowed to use any numerical values in your analysis. 2 Keeping in mind that any variable raised to the power zero is equal to one (for example, see the 'Zero exponent' section site on Exponentiation), the utility function evaluated at V = 0 is " (.4-0) = { 0,-0 ) = (0,1 U c, n.c U c,n.c-0 uc,n.cl-v =u(c,n1)=u(c,n). Note that you are permitted to view this site while completing the Problem 1 continued a. (8 points) Based on the Lagrange function stated above, compute the first-order condition with respect to n. Please clearly display the first-order condition by drawing a box around it. (Note: Be careful in using the Chain Rule.) b. (9 points) Based on the Lagrange function stated above, compute the first-order condition with respect to c. Please clearly display the first-order condition by drawing a box around it. (Note: Be careful in using the Chain Rule.) The Lagrange function is in which stands for the Lagrange multiplier. Observation: A crucial observation is that there are two arguments (separated by the one comma) in the utility function. The first argument is c and - despite the fact that c again appears the second L(c,n,2)=uc, n. cv+2[w.n-c], V argument is ncv. Thus, in the ensuing analysis, please write the marginal utility function with second argument as u V respect to the first argument as u, c,n.cv and the marginal utility function with respect to the (c.n.cl c,n.c- on the wiki V Because c appears twice in the utility function, it directly affects not only the marginal utility for c (which is natural), but also directly affects the marginal utility for n. Formally, the latter effect arises from the Chain Rule. (Please note that, from formal mathematics, there is nothing incorrect/inconsistent in a function in which the same variable appears in multiple arguments.) 1 Note that there are no typos in the utility function. Also note that the utility function is stated in abstract form - in other words, there is no particular utility function provided. You may not assume any particular utility function and are not allowed to use any numerical values in your analysis. 2 Keeping in mind that any variable raised to the power zero is equal to one (for example, see the 'Zero exponent' section site on Exponentiation), the utility function evaluated at V = 0 is " (.4-0) = { 0,-0 ) = (0,1 U c, n.c U c,n.c-0 uc,n.cl-v =u(c,n1)=u(c,n). Note that you are permitted to view this site while completing the Problem 1 continued a. (8 points) Based on the Lagrange function stated above, compute the first-order condition with respect to n. Please clearly display the first-order condition by drawing a box around it. (Note: Be careful in using the Chain Rule.) b. (9 points) Based on the Lagrange function stated above, compute the first-order condition with respect to c. Please clearly display the first-order condition by drawing a box around it. (Note: Be careful in using the Chain Rule.)