Microeconomic Homework Help

Microeconomic Homework Help For any help regarding Microeconomic Homework Help Visit : https://www.economicshomeworkhelper.com/ , Email : info@economicshomeworkhelper.com or call us at - +1 678 648 4277 economicshomeworkhelper.com

Problems 1. Suppose that a consumer with utility U(xl, x2) is given an iniitial endowment F1, F2 so that his total budget set is plYi, P2Y2 (a) Derive a formula for the slutsky equation with endowments (b) Suppose that pi,p2 are such that consumption is exactly equal to the initial en-dowment.

Show that &I. = -2= in this special case. Op2 Bpi 2. You are given the following partial information about a consumer's purchases. he consumes only two goods.:

Over what range of quantitites of good 2 consumed in year 2 would you conclude: (b) That his behaviour is inconsistent (i.e. it contradicts the weak axiom of revealed preference) (b) That the consumer's consumption bundle in year 1 is revealed preferred to that in year 2? (c) That the conumer's consumption bundle in year 2 is revealed preferred to that in (d) That there is insufficient information to justify (a),(b),(c) (e) That good 1 is an inferior good (at some price) for this consumer? Assume that the weak axiom is satisfied (f) That good 2 is an inferior good (at some price) for this consumer? Assume that the weak axiom is satisfied. economicshomeworkhelper.com

3. Recall that for demand functions to come from a utility maximization function: • They must be homogeneous of degree 0. That is x(tpbtp2,tru) = x(pi,p2,rn) • They must satisfy the budget constraint p • x = m • The slutsky matrix must be symmetric.

• The slutsky matrix is equal to the matrix of hicksian demand functions:

(a) Suppose that we have 2 goods and consider the demand functions x(p,m) defined by:

Determine a and $ (b) *Suppose that we have 2 goods and the hicksian demand function for good one 12.1(p,u) P2 (p, u) = —u P1

1. Given that 5a= 191 , show that for P2 > 0, h2(p, u) = In (a) u + g(p2, u) Opi in where g is an arbitrary function.

2. Suppose that when pl = p2, h1(P,u) = u, h2(p, u) = 0. Determineeconomicshomeworkhelper.com h2(p,u).

4. Consider the utility function ln(x) + y. (a) Optional: Solve for x(p,m), y(p,m), v(p,m), e(p,u), Xh (p ,rn), yh (p ,m) . Follow the solutions from PS2 if necessary. (b) Suppose that there are 5 people in the economy each with endowments i = 1, 2, 3, 4, 5.

1. Suppose that m1 > py Vi. Construct the aggregate demand function for x and y.What properties do the individual demand functions have that simplifty this problem?

2. Now suppose that m1 < py, m2 < py, > py for i = 3, 4, 5. Construct the aggregate demand for x1,x2 5. Suppose that an agent is strictly risk-averse who has an initial wealth of w but who runs a risk of a loss of D dollars. The probability of loss is 7r. It is possible, however, for the decision maker to buy insurance. One unit of insurance costs q dollars and pays 1 dollar if the loss occurs. Thus, if a units of insurance are bought, the wealth of the individual will be w — aq if there is no loss and to - aq — D + a if the loss occurs. The utility maximizer thus solves:

(a) Assume that at the optimum a > 0. Find the FOC for the problem. (b) Suppose that insurance is actuarially fair in the sense of it being equal to the expected cost of insurance. Find a* economicshomeworkhelper.com

(c) **Optional: Suppose that the agent is insuring his car and once a week he goes to white castle for burgers. There is no drive through and his car must be pushed in order for him to restart it which requires him some effort e. If he turns off the car (and thus expends effort) he has a probability g(e) of his car being stolen while in the restaurant. if he does not spend the effort he has a probability of ir (0) of his car being stolen where 71-(0) > ir(e).

1. Suppose that he is able to get the amount of insurance as in part (b) above. Show that he will always choose to leave his car running.

2. Let u(x) = ln(x). If the agent lets his car run while in white castle his utility is:

If the agent turns off his car his utility is:

Show that for the agent to choose to turn off his car, a" < D. 3. Draw the function ln(x). Why is the agents utility lower in part ii than in part b?

economicshomeworkhelper.com

Solutions 1. (a) Taking the derivative of x wrt to pi yields:

(b) When ; = xi, the last term drops out of the above formula. However we know:

Then the marshallian demand functions must have symmetric cross partials. (a) The weak axiom of revealed preference is vioalted if 1) Bundle 1 is reveal preferred to bundle 2 2) Bundle 2 is reveal preferred to bundle 1.

For Bundle 1 to be reveal preferred to 2, it must be that bundle 2 was avail-able. Thus: economicshomeworkhelper.com 100.120 + 100y < 100.100 + 100 • 100

For Bundle 2 to be reveal preferred to 2, bundle 1 must be availabe when bundle 2 was chosen.

Tis requires: 100 • 100 + 80 • 100 < 100 . 120 + 80 • y These are both satisfied when y E [75,80] (b) Only condition one must hold here, thus for y < 80 (c) Only condition two must hold here, thus y > 75 (d) One of the three conditions hold for any budget level (e) Suppose that y < 75. Then: and Here real wealth decreased from year 1 to year 2 and the relative price of good 1 increases.

However, the demand for good 2, y decreases because y < 75 < 100. This means that the wealth effect of good 1 must be negative. (It might be easier to think about this going the other way from year 2 -> year 1. Then income increases and the relative price of good 1 decreases but output decreases). (f) If y E [80, 100] : 100.100+80.100 < 100.120+80.y and 100 • 120 + 100y > 100.100 + 100 • 100 economicshomeworkhelper.com

Hence the real wealth increases from year 1 to year 2 and the relative price of good 2 decreases. But the demand for good 2, y, decreases since y < 100. This means that the wealth effect of good 2 must be negative and hence an inferior good. (a)Marshallian demand functions are homogeneous of degree 0, thus:

Thus a = 1. The slutsky matrix is also symmetric. Taking the off terms of the slutsky matrix we have:

Combining the first two terms yields:

Which reduces to:

economicshomeworkhelper.com

economicshomeworkhelper.com

economicshomeworkhelper.com

economicshomeworkhelper.com

economicshomeworkhelper.com

economicshomeworkhelper.com