1. Consider a representative individual who receives utility from units of food
(F) and gasoline (G). The individual’s utility function is U(F,G) = F G.
a. Graph the indifference curve associated with this utility function when U (F,G) =
4000. Specifically, for each of the following values of F: 20, 40, 60, 80, 100 and
200, solve for the corresponding value of G that would keep the consumer on the
U=4000 indifference curve. Plot these points and connect them with a smoothed line. Please label all axes clearly.
From given utility function, U(F,G) = FG and a given quantity of G, then we can find a value of F as shown in this following diagram;
Therefore, we can plot indifference ...view middle of the document...
As we know that at optimal point the slope of 2 equation must be equal, thus slope of indifference curve = slope of constraint curve.
Indifference curve;
0 = ∂U∂F dF + δUδG dG
Thus, slope will equal to dGdF = ∂U∂F∂U∂G
Constraint equation; PFF + PGG = m
Rearrange the equation; G = - PfPg F + mPg
Thus, slope will equal to - PfPg
Therefore,
∂U∂FPF=∂U∂GPG ……………… 1.
In this question,
From U(F,G) = FG and Pf = 1, and Pg= 3, and substitute into equation ,1we then have F = 3G .………………2.
From constraint equation; F + 3G = 1,000 ………………..3.
We substituted 2 into 3, 6G = 1000
G = 166.67 Unit
F = 500 Unit
By conclusion, this consumer has to consume 166.67 units of Gasoline and 500 units of Food in order to maximize utility.
d. Over the last six months, the price of gasoline has risen from $3 to $4 per unit.
How much gasoline will the consumer purchase now? How much food? Are gasoline and food complements or substitutes?
From; ∂U∂FPF=∂U∂GPG
Pg = 4, Pf = 1 F = 4G
Insert F = 4G into F + 4G = 1,000; G = 125
F = 500
When there is a change in Gasoline price from $3 to $4 per unit, the consumer will consume 500 Units of food and 125 Units of Gasoline to maximize utility. Meanwhile as we can see that when price of Gasoline changed, demand for gasoline has changed but demand for food hasn’t changed, therefore we can say that food and gasoline are neither substitutes nor compliments.
e. Plot the consumer’s demand curve, given that her budget is $1000 and that food
costs $1 per unit. (You may do this by deriving the demand curve algebraically,
or by calculating the quantities of gasoline demanded at the following prices: $1,
$2, $3, $4, $5, $6, finding the corresponding quantities demanded, plotting them,
and connecting them with a smooth line).
As the question asked us to plot the demand curve of gasoline therefore we have to rearrange the equation in a way that it depicts the relationship between price of gasoline and its quantity demanded.
From the equation of ∂U∂FPF=∂U∂GPG
As U(F,G) = FG; GPf = FPg
Pf =1 ; F = G . Pg ………………… 1.
Insert 1. into F+ G . Pg = 1000; G . Pg = 500
G = 500Pg
Then, based on this equation we can plot demand curve for Gasoline as follow;
f. Calculate the approximate elasticity of demand at a price of $4.
From last question;
Assume that price of Gasoline is changing from $4 to $5, therefore elastricity of demand can be calculated as follow;
From elasticity formula; ε= ∆ x / x∆Px/Px
= ∆x∆Px . Pxx
= (125-100)(4-5) . 4125
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