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Equlibrium Level of Output in Given Market

I. Each of 5000 consumers that are buyers of goods X and Y per period has the utility function:

(see attachment)

where X is the number of units of good X consumed and Y represents all other goods that the consumer purchases. The income of each consumer is \$1000. You may assume the price of Y is \$1 per unit of other goods consumed. Keep in mind that there are 5000 consumers in this market and the utility function given above is for a single, representative consumer.

On the supply side of this market, the short run production function for each of 5 firms producing good X is:

(see attachment)

where L and K are the number of labor and capital units, respectively, used by each firm per period. The wage rate paid to labor is \$166.66667 (i.e. 166 & 2/3) per unit and the price of capital is \$0.25 per unit. Assume there are 15,625 units of capital (which includes machinery) being used by each firm.

Using this information and assuming there are 5 firms in this market, determine the equilibrium level of output in this market, the equilibrium price paid by consumers in this market, the profits to the industry, and the amount of labor used by the industry.

II. (Continuing I.) If a 10% tax is imposed on the buyers in this market (assume there are still 5 firms), determine the equilibrium level of output in this market, the equilibrium price paid by consumers in this market, the profits to the industry, the amount of labor used by the industry and the deadweight loss, if any, due to the tax.

III. (Continuing I.) If a 5% tax is imposed on the sellers in this market (assume there are still 5 firms), determine the equilibrium level of output in this market, the equilibrium price paid by consumers in this market, the profits to the industry, the amount of labor used by the industry and the deadweight loss, if any, due to the tax.

Solution Preview

I. Each of 5000 consumers that are buyers of goods X and Y per period has the utility function:

U = 56X0.3Y0.7

where X is the number of units of good X consumed and Y represents all other goods that the consumer purchases. The income of each consumer is \$1000. You may assume the price of Y is \$1 per unit of other goods consumed. Keep in mind that there are 5000 consumers in this market and the utility function given above is for a single, representative consumer.

On the supply side of this market, the short run production function for each of 5 firms producing good X is:

X = 16à?L1/2K1/3

where L and K are the number of labor and capital units, respectively, used by each firm per period. The wage rate paid to labor is \$166.66667 (i.e. 166 & 2/3) per unit and the price of capital is \$0.25 per unit. Assume there are 15,625 units of capital (which includes machinery) being used by each firm.

Using this information and assuming there are 5 firms in this market, determine the equilibrium level of output in this market, the equilibrium price paid by consumers in this market, the profits to the industry, and the amount of labor used by the industry.

1. first, let's consider the individual consumers:
Their utility maximization condition is Marginal Utility of X over Marginal Utility of Y should be equal to the price ratio of X and Y, i.e.,
MUx / MUy = P/1
(0.3 * 56X-0.7Y0.7 )/ (0.7 * 56X0.3Y-0.3) = P
3Y / 7X = P
or Y = 7/3 * X P
substitute into the budget constraint:
P*X + 1*Y = 1000
XP + 7/3 * X P= 1000
10/3* X P= 1000
XP = 300
Or X = 300 / P
Since there are 5000 consumers, the overall demand of X is:
Xa = 5000 X = 1500,000 / P

Assuming the 5 firms are of exactly the same production condition and share the market equally, then each firm's output should be: ...

Solution Summary

This solution determines the equilibrium price paid by consumers, the profits to the industry, and the amount of labour used. It uses step by step calculations and explanations to aid with understanding of the problem.

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