MICRO-1:02 PM Mon Apr 882 \%eclass.yorku.caSpecifi

1:02 PM Mon Apr 8

82 \%

eclass.yorku.ca Specifically, assume the firm should ideally take action

a_{0}

. To take that action someone needs to put in work hours

h

which have an effectiveness of

\alpha

. Work hours are costly at

\frac{1}{2} c h^{2}

. If the hours are put in by a worker with intrinsic motivation, that worker derives satisfaction from the work according to

\gamma \alpha h

. The first possible scenario is of a single owner of the firm who gets all the output but also needs to put in all the hours. In this case, the owner solves the following optimization problem Scenario 1:

\max _{h}-\left(\alpha h-a_{0}\right)^{2}-\frac{1}{2} c h^{2}

If the work is done by an employee, the firm needs to pay that employee an hourly wage

w

for every hour worked and the employee then chooses how many hours to work. So without any consideration for intrinsic motivation, the two parties would solve the following two optimization problems Scenario 2:

\begin{array}{ll} \max _{w}-\left(\alpha h-a_{0}\right)^{2}-w h &amp; \text { (firm) } \\ \max _{h} w h-\frac{1}{2} c h^{2} &amp; \text { (employee) } \end{array}

Finally, if the employee has intrinsic motivation, the optimization problems change to Scenario 3:

\begin{array}{ll} \max _{w}-\left(\alpha h-a_{0}\right)^{2}-w h &amp; \text { (firm) } \\ \max _{h} w h+\gamma \alpha h-\frac{1}{2} c h^{2} &amp; \text { (employee) } \end{array}

When we are asking what role intrinsic motivation plays, we are asking what the difference is between scenario 2 and 3 in the number of hours the employee works, the wage the firm pays, the overall pay-off to the employee, and the overall profit the firm realizes. We start by analyzing scenario 1 . We will then solve scenarios 2 and 3 , and finally compare those two outcomes. 1. How many hours would a worker-owner facing the optimization problem in scenario 1 optimally work? What profit would that worker-owner realize? To answer both questions, find and solve the first-order-condition. Find the profit by substituting the optimal number of hours

h_{1}^{*}

into the objective function. 2. Next consider scenario 2. How many hours does the worker work? What wage does the firm pay? What pay-off does the worker receive? What profit does the firm realize? To answer these questions: a. Assume the firm announces a particular wage

w

which the employee now takes as given. Solve the employee's optimization problem as a function of the wage:

h_{2}^{*}(w)

. b. Now consider the firm's optimization problem. The firm knows that whatever wage

w

it chooses, the employee will respond by working

h_{2}^{*}(w)

hours. Naturally, the firm takes this into account. So substitute

h_{2}^{*}(w)

into

h

in the objective function and solve for the optimal wage

w_{2}^{*}

.

Answer

The worker-owner would optimally work

h_{1}^{*} = \frac{\alpha a_{0}}{\alpha^{2} + \frac{1}{2}c}

hours and realize a profit of

\pi_{1}^{*} = -\left(\alpha h_{1}^{*} - a_{0}\right)^{2} - \frac{1}{2} c \left(h_{1}^{*}\right)^{2}

Solution

a

To find the optimal number of hours

h_{1}^{*}

, we take the derivative of the objective function with respect to

h

and set it to zero (first-order condition)

\( \frac{d}{dh}\left[-\left(\alpha h - a_{0}\right)^{2} - \frac{1}{2} c h^{2}\right] = -2\alpha(\alpha h - a_{0}) - c h = 0 \)

b

Solving the first-order condition for

h

gives us the optimal number of hours

h_{1}^{*}

\( h_{1}^{*} = \frac{\alpha a_{0}}{\alpha^{2} + \frac{1}{2}c} \)

c

To find the profit

\pi_{1}^{*}

, we substitute

h_{1}^{*}

back into the objective function

\( \pi_{1}^{*} = -\left(\alpha h_{1}^{*} - a_{0}\right)^{2} - \frac{1}{2} c \left(h_{1}^{*}\right)^{2} \)

Key Concept

Optimization and First-Order Condition

Explanation

In microeconomics, to find the optimal level of an activity, we set the derivative of the objective function to zero and solve for the variable of interest. This is known as the first-order condition for optimization.

Answer

The worker works

h_{2}^{*}(w) = \frac{w}{c}

hours, the firm pays a wage

w_{2}^{*}

that maximizes its profit, the pay-off to the worker is

U_{2}^{*} = w_{2}^{*} h_{2}^{*}(w_{2}^{*}) - \frac{1}{2} c \left(h_{2}^{*}(w_{2}^{*})\right)^{2}

, and the profit the firm realizes is

\pi_{2}^{*} = -\left(\alpha h_{2}^{*}(w_{2}^{*}) - a_{0}\right)^{2} - w_{2}^{*} h_{2}^{*}(w_{2}^{*})

. The exact value of

w_{2}^{*}

depends on the parameters

\alpha, a_{0},

and

c

and requires further calculation.

Solution

a

To solve the employee's optimization problem, we take the derivative of the employee's objective function with respect to

h

and set it to zero

\( \frac{d}{dh}\left[wh - \frac{1}{2} c h^{2}\right] = w - c h = 0 \)

b

Solving the first-order condition for

h

gives us the optimal number of hours

h_{2}^{*}(w)

as a function of the wage

w

\( h_{2}^{*}(w) = \frac{w}{c} \)

c

The firm's optimization problem requires substituting

h_{2}^{*}(w)

into its objective function and then finding the wage

w

that maximizes its profit. This involves taking the derivative of the firm's profit function with respect to

w

and setting it to zero

The exact value of \( w_{2}^{*} \) requires further calculation based on the firm's profit function after substitution.

Key Concept

Worker's Labor Supply Decision and Firm's Wage Setting

Explanation

The worker decides how many hours to work based on the wage rate, aiming to maximize utility, which in this case is the wage income minus the cost of effort. The firm sets the wage to maximize its profit, taking into account the worker's response to the wage rate.

How does intrinsic motivation impact the optimization problem in scenario 3 compared to scenario 2 in terms of the number of hours worked by the employee and the overall profit realized by the firm?

2. Next consider scenario 2. How many hours does the worker work? What wage does the firm pay? What pay-off does the worker receive? What profit does the firm realize? To answer these questions: a. Assume the firm announces a particular wage w which the employee now takes as given. Solve the employee's optimization problem as a function of the wage: h (w). b. Now consider the firm's optimization problem. The firm knows that whatever wage wit chooses, the employee will respond by working h (w) hours. Naturally, the firm takes this into account. So substitute h½ (w) into h in the objective function and solve for the optimal wage Wz. c. Find the actual number of hours the worker works by calculating h° (W%). d. Find the realized pay-off and realized profit by substituting we and hi (W) into the two respective objective functions. 3. Now, consider scenario 3. How many hours does the worker work? What wage does the firm pay? What pay-off does the worker receive? What profit does the firm realize? To answer these questions: a. Assume the firm announces a particular wage w which the employee now takes as given. Solve the employee's optimization problem as a function of the wage: h3 (W). b. Now consider the firm's optimization problem. The firm knows that whatever wage wit chooses, the employee will respond by working h (w) hours. Naturally, the firm takes this into account. So substitute hg(w) into h in the objective function and solve for the optimal wage W3. c. Find the actual number of hours the worker works by calculating h3 (wñ). d. Find the realized pay-off and realized profit by substituting Ws and h(Ws) into the two respective objective functions.

Answer

The number of hours the worker works, the wage the firm pays, the pay-off the worker receives, and the profit the firm realizes in scenarios 2 and 3 depend on the optimization problems of both the employee and the firm, which are influenced by the wage rate and the intrinsic motivation of the employee.

Solution

a

Employee's Optimization in Scenario 2: The employee maximizes utility as a function of wage

w

, which can be represented as

U(h(w), w)

. The employee chooses

h(w)

to maximize utility given the wage

b

Firm's Optimization in Scenario 2: The firm maximizes profit by choosing wage

W_2

knowing the employee's response function

h(w)

. The firm's profit function is

\pi(W_2) = P \cdot f(h(W_2)) - W_2 \cdot h(W_2)

, where

P

is the price of the output and

f(h)

is the production function. The firm solves for

W_2

that maximizes

\pi(W_2)

c

Actual Hours Worked in Scenario 2: The actual number of hours worked by the employee is

h^*(W_2)

, which is obtained by substituting the optimal wage

W_2

into the employee's response function

h(w)

d

Realized Pay-off and Profit in Scenario 2: The realized pay-off for the worker is

U(h^*(W_2), W_2)

and the realized profit for the firm is

\pi(W_2)

after substituting

W_2

and

h^*(W_2)

into their respective objective functions

a

Employee's Optimization in Scenario 3: Similar to Scenario 2, the employee maximizes utility as a function of wage

w

, but with intrinsic motivation factored in, which may alter the utility function to

U(h_3(w), w)

. The employee chooses

h_3(w)

to maximize this adjusted utility

b

Firm's Optimization in Scenario 3: The firm maximizes profit by choosing wage

W_3

knowing the employee's response function

h_3(w)

which includes intrinsic motivation. The profit function is

\pi(W_3) = P \cdot f(h_3(W_3)) - W_3 \cdot h_3(W_3)

. The firm solves for

W_3

that maximizes

\pi(W_3)

c

Actual Hours Worked in Scenario 3: The actual number of hours worked by the employee is

h_3^*(W_3)

, which is obtained by substituting the optimal wage

W_3

into the employee's response function

h_3(w)

d

Realized Pay-off and Profit in Scenario 3: The realized pay-off for the worker is

U(h_3^*(W_3), W_3)

and the realized profit for the firm is

\pi(W_3)

after substituting

W_3

and

h_3^*(W_3)

into their respective objective functions

Key Concept

Optimization Problems in Labor Economics

Explanation

In labor economics, the employee's optimization problem involves choosing the number of hours to work to maximize utility, which depends on the wage rate and possibly intrinsic motivation. The firm's optimization problem involves choosing the wage rate to maximize profit, taking into account the employee's response in terms of hours worked. Intrinsic motivation can alter the employee's utility function and thus affect both the employee's and the firm's optimization outcomes.

4:06 PM Mon Apr 8

24 \%

eclass.yorku.ca Consider the following scenario: A monopolist airline owns a single airplane. The airplane has 100 seats and is used by the airline to serve a single route between XYZ town and ABC city. There are two types of consumers

A

and

B

that the airline can sell tickets to. The airline can provide service at four different quality levels, and the two types of consumers value the level of services differently. This table shows the happiness that consumers derive from each level of service, expressed in \-terms, and the airline's per-seat cost of providing the service. \begin{tabular}{|c|c|c|c|c|} \hline Quality & 1 & 2 & 3 & 4 \\ \hline Type A & \

12

&

\$ 9

&

\$ 6

&

\$ 3

\\ \hline Type B &

\$ 7

&

\$ 6

&

\$ 5

&

\$ 4

\\ \hline Cost &

\$ 2.50

&

\$ 2

&

\$ 1.50

&

\$ 1

\\ \hline \end{tabular} Note that the two types on consumers coincide in their preferences: Both types prefer quality level 1 over level 2 over level 3 , and value level 4 the least. Among the prices offered by the airline, each consumer will choose the ticket that maximizes the consumer's overall happiness, i.e., the difference between the happiness derived from flying and the price paid for the plane ticket. If the price for an airline ticket exceeds the happiness that the consumer derives from the flying, then the consumer will not buy a ticket and travel by car instead. 1. If the monopolist airlines only faces consumers of type A, what quality level would it offer and at what price? 2. If the monopolist airlines only faces consumers of type B, what quality level would it offer and at what price? Now, the airline knows that there are only 25 consumers of type A. (But there are as many consumers of type B as the airline would like.) The airline then faces three options: I: Only sell airplane tickets to type A consumers. 3. Under this option, what quality level would the monopolist airline offer and at what price? What profit would the airline generate? II: Sell airplane tickets to both types of consumers at one uniform price. 4. Under this option, what quality level would the monopolist airline offer and at what price? What profit would the airline generate? III: Offer two quality levels, say "business" and "economy," at different prices chosen such that different type consumers self-select into buying different quality levels. The following questions lead you through the steps of constructing such a pricing schedule.

This is an interesting scenario you've presented with the monopolist airline and the different types of consumers A and B. Let's break it down step by step. First, if the monopolist airline only faces consumers of type A, it would offer quality level 1 at a price of

12, as this is the level that maximizes consumer happiness for type A consumers. Next, if the monopolist airline only faces consumers of type B, it would offer quality level 1 at a price of

7, as this is the level that maximizes consumer happiness for type B consumers. Now, considering the option of only selling airplane tickets to type A consumers, the monopolist airline would offer quality level 1 at a price of

12 to maximize profit. The profit generated would be the difference between the revenue from selling 25 tickets at

12 each and the cost of providing the service. For the option of selling airplane tickets to both types of consumers at one uniform price, the monopolist airline would still offer quality level 1 at a price that maximizes overall profit. The profit generated would be the difference between the revenue from selling all available tickets at the uniform price and the cost of providing the service. Lastly, for the option of offering two quality levels (business and economy) at different prices, the airline would need to carefully consider the pricing strategy to ensure that different types of consumers self-select into buying different quality levels based on their preferences. I hope this breakdown helps you navigate through the different options and make informed decisions for the monopolist airline in this scenario. Let me know if you have any more questions or need further clarification!

AskSia

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