Calculus Problem Solving for Intermediate Microeconomics Assignment Help

How does Descartes find certainty in the *Meditations*?

Problem 5 – Production economy general equilibrium (35 points)

Robert has utility function u (c, `) = c` over consumption, c, and leisure, `. Robert is endowed

with 16 hours of leisure. Let the price of consumption be p = 1. Robert can sell his time in the

labor market at hourly wage, w. The equilibrium we will consider implies zero firm profits, so labor

income is the only source of income for consumers. Thus, Robert’s budget line can be written by

c + w` = 16w. Production of the consumption good is done by the Acme corporation according to

production function Q = F (L) = AL, where A > 0 is some constant that captures firm productivity.

There are no fixed costs. The acme corporation sells the output in the consumption market at the

price p, and hires labor at wage w. We will in the following work out the general equilibrium.

2

1. For given wage w and consumption good price p, work out Robert’s optimal consumption and

leisure choice (c⇤, `⇤). (Hint: Combine the budget line equation with MRS = p/w). What is

Robert’s supply of hours to the labor market, h⇤ = 16`⇤.

State the expressions as functions

of the real wage, wˆ = w/p.

2. For a given wage and productivity A, determine Acme’s cost function, C (Q), and the associated

input level for a given output requirement, L(Q).

3. Verify that Acme Corp’s marginal cost curve is flat. That is, Acme’s supply curve is flat. At

what price level? Notice that Acme is willing to supply any level of output at this price level

and that its profits are exactly zero regardless of what output level it chooses.

4. There are a total of 100 individuals in the economy all identical to Robert. State the market

demand for consumption, QD as a function of p and w. Combine with the flat supply curve

to determine the equilibrium price that equates market consumption demand with supply (it

will be a function of w). This equilibrium price equation is effectively also an equilibrium

condition on the real wage, wˆ = w/p. What is the equilibrium level of the real wage?

5. Set A = 2. State the equilibrium market demand of the consumption good, QD. Acme will

be supplying exactly the same quantity. Verify that the labor market is also clearing.

6. In equilibrium, how much labor is Robert supplying to the market? How much is he consuming?

7. Acme has come up with a brilliant new production process which has increased the productivity

parameter to A = 3. Determine the new general equilibrium. What is the equilibrium

wage? How much labor is Robert supplying? How much is he consuming

Problem 1 – Market clearing curves (16 points)

Consider the markets for ice cream (good 1) and pie (good 2). Let the market demand for ice cream

be given by QD

1 (p1, p2) = 200

2p1

4p2 and market demand for pie is given by QD

2 (p1, p2) =

100

4p2

2p1. The supply of is ice cream is given by QS

1 (p1)=4p1

6 and supply of pie is given

by QS

2 (p2)=2p2

6.

1. Determine the market clearing curve for ice cream. Illustrate it in a diagram with the price

of pie (p2) on the x-axis and the price of ice cream (p1) on the y-axis. You obtain the market

clearing curve by the ice cream market clearing equation QD

1 (p1, p2) = QS

1 (p1) and rearrange

this equation to state p1 as a function of p2.

2. Determine the market clearing curve for pie. Illustrate it in a diagram with the price of pie

(p2) on the x-axis and the price of ice cream (p1) on the y-axis. In this case use the market

clearing equation for the pie market, QD

2 (p1, p2) = QS

2 (p2) and rearrange this equation to

state p1 as a function of p2.

3. Find the general equilibrium for the two markets as the intersection of the two curves. Also

solve for (p⇤

1, p⇤

2) using the two curves (you have two equations and two unknowns. Numbers

are not designed to be nice and round.). What are the quantities consumed of ice cream and

pie?

4. Suppose the supply of ice cream increases and is expressed by, QS

1 (p1)=9p1

6. What are

the new equilibrium prices and quantities in the two m

Problem 4 – Endowment equilibrium (20 points)

Suppose an economy consists of 11 people: 10 farmers and one miner. All individuals have identical

preferences, u (f,g) = ln(f)+ 2 ln(g), where f is a kilo of food, and g is a gram of gold. Each farmer

is endowed with a 50 kilos of food and no gold. The miner is endowed with no food and one gram

of gold. The farmers and the miner can trade food and gold at prices pf and pg. In the following,

normalize the price of food to one, pf = 1.

1. For given prices (pf = 1 and pg), solve for a farmer’s optimal demand for food and gold given

that the farmer has an endowment that has value 50.

2. For given prices (pf = 1 and pg), solve for the miner’s optimal demand for food and gold given

that the miner has an endowment that has value 1 ⇥ pg.

3. Determine the equilibrium price of gold, p⇤

g. How does the value of the miner’s endowment

compare to that of the farmers altogether.

4. Suppose instead of 10 farmers, there are 100 farmers. In equilibrium, how valuable is the

miner’s 1 gram of gold in terms of food, then?

urve.

Problem 3 – Firm Upstream and firm Downstream (24 point)

Following the example in class, consider two firms, U and D. Both firms have cost functions

C (Q)=0.5Q2. Firm U’s product sells at price pU = 6 whereas firm D’s product sells at price

pD = 10. Firm U’s production pollutes the river that both firm’s are situated by. The pollutio

heavier, the more U produces. Firm D is downstream and is negatively affected by the pollution.

To capture this negative impact, say firm D faces unavoidable fixed cost of F C (QU )=0.5Q2

U ,

where QU is firm U’s quantity choice.

1. Suppose firm U is given the right to freely pollute.

(a) In the absence of any negotiation between the firms, what are their optimal output levels?

What are their profits?

(b) Now, introduce the possibility of a negotiation between the firms. Suppose they will split

the net joint profit gains from a negotiated deal evenly. Define a contract between the

two firms by output levels Q⇤

U , Q⇤

D, and a transfer between them, T (specifically, say

that T is the transfer from D to U). Determine the optimal contract.

2. Suppose firm D is given the right to a clean river (no pollution). Thus, firm U will be forced

to not pollute in the absence of any agreement otherwise.

(a) In the absence of any negotiation between the firms, what are their optimal output levels?

What are their profits?

(b) As in question 1.b, introduce the possibility of a negotiation between the firms. Again,

assume the net joint profit gains are split evenly. Determine the optimal contract.

kets?

Problem 2 – Contract curve (5 points)

Kate and Humphrey have initial endowments of goods 1 and 2, eL = (8, 2) and eH = (2, 8). Kate’s

preferences are represented by utility function uL(x1, x2) = x1x2 and Humphrey’s preferences are

represented by utility function uH(x1, x2) = x2

1×2. Denote by xL = (xL

1 , xL

2 ) Lauren’s equilibrium

consumption bundle. Humprey’s consumption bundle is xH = (xH

1 , xH

2 ). In equilibrium it must

that xL

1 + xH

1 = eL

1 + eH

1 = 10 and xL

2 + xH

2 = eL

2 + eH

2 = 2 + 8 = 10. Determine all equlibrium

allocations, (xL, xH) that are on the contract