In light of the diagram associated with question 18 , if the budget for this consumer is 48, th

In light of the diagram associated with question 18 , if the...

Key Concepts

Relative Price Analysis

Relative price analysis involves comparing the prices of different goods to determine the opportunity cost of consuming one good over another. This concept is crucial for understanding how consumers make choices between goods when faced with a limited budget, as it influences the consumption bundle that maximizes their satisfaction.

Consumer Equilibrium

Consumer equilibrium occurs when a consumer has allocated their income in such a way that they maximize their total utility given their budget constraint. At this point, the rate at which they are willing to substitute one good for another (marginal rate of substitution) equals the ratio of the prices of the goods, ensuring that no reallocation of spending can increase utility further.

Budget Constraint

A budget constraint represents all the combinations of goods and services that a consumer can purchase given their income and the prices of those goods. It is a foundational concept in consumer theory because it defines the limits within which consumers choose to maximize their satisfaction. Understanding the budget line helps in analyzing trade-offs and opportunity costs associated with different consumption bundles.

Transcript

00:01 In problem 13 we want to calculate the equilibrium point for the domain function here and supply function here where x is a number of units x we have here x and we have here y for the domain which starts at 4 goes something like that and we have the supply which starts at 8 the domain starts at 4 squared and the domain starts at it the domain starts at it the law starts at 8 which goes something like that which here is the equilibrium point we want to get the equilibrium point for part a they have the same x and y values which means we can equate both functions and then solve the equation to get the x value of the equilibrium point for the demand we have x minus 4 squared equals x squared plus 2x plus 8 for the left hand side we have x squared minus 8x plus 16 and for the right hand side we get all terms in the left hand side by a negative sign minus x squared minus 2x minus 8 equals 0 we have cancelled out x squared and x squared and then we have minus 8 minus 2 minus 10 x and we have plus 6 we have plus 16 minus 8 which is plus it equal 0 this means the equilibrium is at x equals 8 divided by 10 or 0 .8 units to get the unit price we can substitute in either the demand or the supply functions we have the demand 0 .8 minus 4 squared equals 3 .2 squared 10 .24 dollars this means the aquarium point is at 0 .8 number of units and the price is 10 .24 dollars for part we want to calculate the consumer surplus which is the area under the demand until the equilibrium point this area can it can be calculated as consumer surplus equals the integration for the demand function from zero to the equilibrium point to 0 .8 for the demand which is x minus four squared d x then we subtract the area of this rectangle rectangular which is the quantity multiplied by the price of the equilibrium point equals x minus 4 cubed divided by 3 we substitute for x from 0 to 0 .8 minus the quantity multiplied by the price we have here the quantity and here is the price 0 .8 multiplied by 10 .24 we substitute first by x equals 0 .8 we get 3 minus 3 .2 cubed divided by 3 then we substitute by x equal 0 we have minus 4 cubed divided by 3 minus 8 .192 this equals minus or plus 4 cubed 2 2 .2 dollars for bar c we want to calculate the produce surplus which is the area above the supply curve until the equilibrium point this area can be calculated as the area of this rectangular minus the integration the area of this rectangle which is the quantum multiply by the price for the equilibrium point minus the integration of the supply curve to get the area under the curve from 0 to a current point of the supply which is x squared plus 2x plus 8 d x...

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