a)
To find:
MRS = 1+r
a)
Explanation of Solution
Given utility function:
Budget constraint :
The above equations are put in Langrange equation:
Taking the fisrt order derivative and equating it to 0.
Divide the above two equations, we get:
Introduction:
b)
To know:
Price eslaticity of demand of c2.
b)
Explanation of Solution
Substitution effect is always negative which implies that any increase in value of
While income effect is positive in case of normal good, so it implies
However,
It is assumed that substitution effect has more impact than income effect, that is,
Introduction: Envelop theorem states that changes in exogeneous variables must be considered for profit maximizing equations, ignoring the change in endogeneous variable.
c)
To ascertain:
Changes in part b due to change in budget constraint.
c)
Explanation of Solution
Budget constraint is given as:
Rearranging the terms:
The above equation is a slope of budget line.
When
Introduction:
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Chapter 17 Solutions
Microeconomic Theory
- An individual derives utility from consumption spending C and leisurel according to the following utility function: U(C,1)=C"1¹-a where 0>a>1. Leisure time in hours is given by: 1=T-H where T is hours of total time available and H is hours of work. The consumer's real income is given by: C=w (T-1)+N where w is real wage and N is real non-labour income. d) Verify that the second-order conditions for a constrained maximum are met. Reduce your answer (which should only be function a, w, N, and T) to the lowest terms.arrow_forwardSuppose Jack lives for two periods. Period one is his working life, during which he earns income $50,000; period two is his retirement, during which he earns nothing. During retirement he consumes from the savings during his working life. The rate of interest on his savings is 10%. His consumption during his working life is Cw, and his consumption during his retirement life is Cr. Assume that Jack's utility function is a standard utility function exhibiting diminishing marginal rate of substitution between Cw and Cr. His current consumption during the working life is 75% of his earned income. a. Using the intertemporal choice model, draw a well labeled graph that details all the information discussed above. Notably, indicate the slope of the budget constraint, the intercepts of the budget constraint on both axes, the value of current and future consumptions, and the current savings. Keep in mind that no tax has been imposed on the saver, yet. b. Now the government taxes interest…arrow_forwardConsider an individual who receives utility from consumption, c, and leisure, l. The individual has L time to allocate to work, n, and leisure. The individual’s consumption is a function of how much he works. In particular, c = root n. The individual’s maximization problem is max U =ln(c)+θl subject to c = √n n+l=L where θ > 0. Solve the maximization problem. Hint: Substitute both constraints into the objective function.arrow_forward
- Assume that consumption and leisure are perfect complements, that is, the consumer always desires a consumption bundle where the quantities of consumption and leisure are equal, that is, C=L 1) (Denote the total hours of time available by h, the real wage by w, the real dividend income from firms by pi (π), and the lump-sum tax by T. Write down the consumer’s budget constraint. 2) Determine the consumer’s optimal choice of consumption and leisure. 3) Assume that there is an increase in w . Show how the consumer’s optimal consumption bundle changes. Explain with reference to income and substitution effectsarrow_forwardQUESTION 1 Qx0.65Qy(1-0.65) and the budget 127 = 6Qx + 6Qy find the CHANGE in optimal consumption of Y if the price of X increases by a factor of For the utility function U = 1.8. Please enter your response as a positive number with 1 decimal and 5/4 rounding (e.g. 1.15 = 1.2, 1.14 = 1.1).arrow_forwardConsider the problem of a consumer who chooses between consuming goods and enjoying leisure in the current and future periods. Denote the consumption and leisure in the current period as C and l, and the consumption and leisure in the future period as C′ and l′, respectively. The preference is summarized by the following utility function: U(C,C′,l,l′)=lnC+ψlnl+β(lnC′ +ψlnl′). This individual is endowed with h units of time in each period. Wage rate per unit of labour time is w and w′ in the current and future period. In addition, the consumer receives profit transfer π and π′ and pays lump-sum taxes T and T′ in the current and future periods. Denote the saving in the current period as Sp. Answer the following questions. Derive the life-time budget constraint of this consumer. Set up the consumer’s problem. Solve for consumption (C and C′), leisure (l and l′), and saving (Sp). How does an increase in wage rate w affect C, Sp, and l?arrow_forward
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- Assume a consumer has current-period income y = 200, future-period income y′ = 150, current and future taxes t = 40 and t′ = 50, respectively, and faces a market real interest rate of r = 0.05, or 5% per period. The consumer would like to consume according to the following utility function: U (c, c′ ) = ln(c) + ln(c′ ). Show mathematically the lifetime budget constraint for this consumer. Find the optimal consumption in the current and future periods and optimal saving. Suppose that instead of r = 0.05 the interest rate is r = 0.1. Repeat parts (a) and (b). Does the substitution effect or the income effect dominate?arrow_forwardSeung's utility function is given by U - C^(1/2), where C is consumption and C^(1/2) is the square root of consumption. She makes $50,625 per year and enjoys jumping out of airplanes. There's a 5% chance that in the next year, she will break both legs, incur medical costs of $30,000, and lose an additional $5,000 from missing work. a. What is Seung's expected utility without insurance? b. Suppose Seung can buy insurance that will cover the medical expenses but not the forgone part of her salary. How much would an actuarially fair policy cost, and what is the expected utility if she buys it? Policy cost: $___ Expected utility: ___ c. Suppose Seung can buy insurance that will cover her medical expenses and foregone salary. How much would such a policy cost if it's actuarially fair, and what is her expected utility if she buys it? Policy cost: $___ Expected Utility: ___arrow_forwardParis has a utility function over berries (denoted by B) and chocolate (denoted by C) as follows: U(B, C) = 2ln(B) + 4ln(C) The price of berries and chocolate is PB and pc, respectively. Paris's income is m. 1. What preferences does this utility function represent? 2. Find the MRSBC as a function of B and C assuming B is on the x-axis. 3. Find the optimal bundle B and C as a function of income and prices using the tangency condition. 4. What is the fraction of total expenditure spent on berries and chocolate out of total income, respectively? 5. Now suppose Paris has an income of $600. The price of a container of berries is $10 and the price of a chocolate bar is $10. Find the numerical answers for the optimal bundle, by plugging the numbers into the solution you found in Q3.3.arrow_forward