# Thread: Economics Calculus help!

1. ## Economics Calculus help!

Consider a market with the following supply and demand functions:

QD = a0 – a1PD a0, a1 > 0

QS = b0 + b1PS b0, b1 > 0

(a) (8 marks) Find the equilibrium quantity and price as a function of the parameters (use any method you like). Are there any additional restrictions that you must impose on the parameters for this model to make sense?

(b) (12 marks) Now assume the government imposes a per unit tax t on the market. One economic view of government (made famous by noble prize winner James Buchanan) is the Leviathan view of government. This approach assumes government wants to become as large as possible, subject to a utility constraint (if the utility of citizens falls too low, people will rise up and constrain the
Leviathan). Thus, government will try to maximize tax revenues subject to the utility constraint. Here we will consider a simple case were the utility constrain is not binding and can be ignored. Suppose the Leviathan government is trying to maximize its revenues from a specific tax t in a market with the demand and supply functions given above. Find an expression for the tax revenue (e.g. tQ) maximizing t in terms of the parameters of the model.

and this:

Consider the following general form of a constant elasticity of substitution production function:

You may find this question easier to answer if you let α1 = δ1/ρ and α2 = (1−δ)1/ρ. By all means, leave your answer in terms of α1 and α2. Assume a firm is trying to minimize the cost of producing any given y. Costs are given by

C = wL + rK.

Find the firm’s cost minimizing demand function for L. The cost minimizing demand for K is determined simultaneously (so you need both FOCs) but since I am sure many of you left this until the last minute, you only have to come up with the expression for L. However, for your own practice, you may want to find the equivalent expression for K on your own. You may assume that nonnegativity constraints on L and K are not binding.

Any help with that? Thanks!

2. Originally Posted by s0urgrapes
Consider a market with the following supply and demand functions:

QD = a0 – a1PD a0, a1 > 0

QS = b0 + b1PS b0, b1 > 0

(a) (8 marks) Find the equilibrium quantity and price as a function of the parameters (use any method you like). Are there any additional restrictions that you must impose on the parameters for this model to make sense?

(b) (12 marks) Now assume the government imposes a per unit tax t on the market. One economic view of government (made famous by noble prize winner James Buchanan) is the Leviathan view of government. This approach assumes government wants to become as large as possible, subject to a utility constraint (if the utility of citizens falls too low, people will rise up and constrain the
Leviathan). Thus, government will try to maximize tax revenues subject to the utility constraint. Here we will consider a simple case were the utility constrain is not binding and can be ignored. Suppose the Leviathan government is trying to maximize its revenues from a specific tax t in a market with the demand and supply functions given above. Find an expression for the tax revenue (e.g. tQ) maximizing t in terms of the parameters of the model.

and this:

Consider the following general form of a constant elasticity of substitution production function:

You may find this question easier to answer if you let α1 = δ1/ρ and α2 = (1−δ)1/ρ. By all means, leave your answer in terms of α1 and α2. Assume a firm is trying to minimize the cost of producing any given y. Costs are given by

C = wL + rK.

Find the firm’s cost minimizing demand function for L. The cost minimizing demand for K is determined simultaneously (so you need both FOCs) but since I am sure many of you left this until the last minute, you only have to come up with the expression for L. However, for your own practice, you may want to find the equivalent expression for K on your own. You may assume that nonnegativity constraints on L and K are not binding.

Any help with that? Thanks!

Some help with part a) :

Solve for price and quantity by treating the two equations as a system. Find the solution algebraically, in terms of the coefficients.

For part b) add a new term to your supply function, call this term tx where t is the tax rate and x represents quantity. (Change this any way you wish to fit the notation you have above.)

I hope this helps.

3. Originally Posted by s0urgrapes
[FONT=&quot]Consider a market with the following supply and demand functions:

QD = a0 – a1PD a0, a1 > 0

QS = b0 + b1PS b0, b1 > 0

(a) (8 marks) Find the equilibrium quantity and price as a function of the parameters (use any method you like). Are there any additional restrictions that you must impose on the parameters for this model to make sense?
In Equilibrium $Q_D=Q_S$ and $P_D=P_S$ This occurs at a price $P$ which is a solution of:

$a_0-a_1P=b_0+b_1P$

Then use either of the original equations to find the supply/demand quantity at this price.

CB

4. Originally Posted by apcalculus

For part b) add a new term to your supply function, call this term tx where t is the tax rate and x represents quantity. (Change this any way you wish to fit the notation you have above.)

I hope this helps.
tx is the tQ in the question correct?

economically profit is the area:

difference between priceEquilibrium_withTax and PriceOnOldSupply_atEquilibQuant_tax
times the QuantAtEquilib_withTax

so with the above, does tQ eual the following????:

tQ= QuantAtEquilib_withTax(priceEquilibrium_withTax-PriceOnOldSupply_atEquilibQuant_tax)

Upper right corner of TR block is the newEquilibriumWithTax

so now how do you MAX this tQ.......?

thx