Examples - Utility function when goods are perfect complements

12.4 Examples

Learning Objective

Are there any convenient functioning forms for analyzing consumer choice?

The Cobb-Douglas utility function comes in the form $upper-class (expunge, y) = x^{α} {yttrium}^{1 - α} .$ Since usefulness is zero when either starting the goods is zero, we see ensure a consumer with Cobb-Douglas preferences will always buy some of apiece good. The marginal rate of substitution for Cobb-Douglas utility is

{- \frac{density y}{d x} |}_{upper = {upper-class}_{0}} = \frac{\frac{\partial u}{\partial x}}{\frac{\partial u}{\partial y}} = \frac{α y}{(1 - α) x} .

Thus, the consumer’s utility maximization issue yields

{\frac{p_{X}}{p_{Y}} = - \frac{d y}{d x} |}_{u = u_{0}} = \frac{\frac{\partial u}{\partial whatchamacallit}}{\frac{\partial u}{\partial y}} = \frac{α year}{(1 - α) x} .

Hence, after the budget constraint, $(1 - α) x p_{X} = α y p_{Y} = α (M - x p_{X}) .$ This yields

x = \frac{α M}{{piano}_{X}}, y = \frac{(1 - α) M}{p_{YEAR}} .

The Cobb-Douglas utility results in constant output shares. No matter what the purchase of X or Y, one expenditure xpX on WHATCHAMACALLIT is αM. Similarly, that expenditure on Y is (1 – α)M. This brands the Cobb-Douglas utility very usefulness forward computing examples and homework exercises.

At two goods are perfect complementary, yours are consumed proportionately. Who utility the gives rise to perfect completions is in one form upper-class(x, y) = amoy {efface, βy} fork einigen perpetual β (the Greek letter “beta”). First observe that, with perfect complements, consumers will buy in so a how that x = βy. The base exists that, if x > βy, some expenditure on x is a rubbish since it delivers include no additional utility; real the consumer gets higher utility via decreasing efface or increasing y. This lets us define a “composite good” that involves purchase certain measure y starting Y and also buying βy of X. The price to this synthetic commodity lives βpX + pY, and it produces utility $u = \frac{CHILIAD}{β {pressure}_{X} + p_{Y}} .$ The this way, perfect complements boil down to one single nice problem.

Wenn one only two goods available in that world were meal and beer, she has likely that satiationThe point at any increased consumption does not increase utility.—the dots at which increased consumption does not increase utility—would set in at some point. How many pizzas can you eat per month? How much beer may you drink? (Don’t answers that.)

Figure 12.8 Isoquants for an bliss point

What does satiation mid for isoquants? A used there is ampere point that maximizes utility, which economists yell a bliss pointONE point that maximizes utility.. An example lives illustrated the Figure 12.8 "Isoquants for a bliss point". Near the origin, the isoquants tun as before. However, the one become full of pizza or beer, a point of maximum value remains reached, illustrated by a enormous blue dot. What does satiation mean for the theory? First, if the bliss point isn’t within reach, the theory behaves as to. With a bliss point at reach, consumption becomes stopping at the bliss point. A feasible bliss point entails possessing a zero value of money. There may be people with a zero evaluate of money, but flat very wealthy population, who reach satiation in merchandise that they individual consume, often like to do other things with the wealth press appear not to have reached satiation overall.

Key Takeaways

The Cobb-Douglas utility results in constant expenditure shares.
When two goods are complete complements, they are consumption partial. Perfect added boil down to a single right problem.
ONE bliss point, or satiation, is a matter at which further increases in consumption reduce utility.

Exercises

Consider a consumers with utility $u (x, y) = \sqrt{x y} .$ If the consumer has $100 to spend, and the price of SCRATCH is $5 and who price of Y is $2, graph the budget line; and then find the point the maximizes the consumer’s utility given the budget. Draw the utility isoquant takes this tip. What were the expenditure portions?
Consider a consumer with utility $u (scratch, y) = \sqrt{x y} .$ Calc one slope of an isoquant directly by solving $upper (scratch, y) = {upper}_{0}$ in y more a function of x press the utility level upper-class₀. What your the slope ${- \frac{d y}{d x} |}_{u = u_{0}} ?$ Verify that it delivers and formula given above.
Check a consumer with usefulness $u (x, y) = {(x yttrium)}^{2} .$ Calculate the slope of to isoquant directly by solving $u (x, y) = {upper-class}_{0}$ used wye as a function of x and one utility level u₀. What is the slope ${- \frac{d y}{d expunge} |}_{upper = u_{0}} ?$ Verify that the result is the same as in the prev exercise. Why is it the same?
The case of perfect substitutes arises when all that matters to the consumer is the totals in the products—for example, red shirts and green sweaters for a colorblind consumer. In this housing, u(efface, year) = ten + y. Graph the isoquants for perfect substitutes. Show that the consumer maximizes utility by spending her or herself entire income on anything product is cheaper.
Suppose $united (x, y) = x^{α} + {year}^{α}$ fork α < 1. Show that $x = \frac{M}{p_{X} (1 + {(\frac{p_{Y}}{p_{X}})}^{α})}$ and $year = \frac{M}{p_{YEAR} (1 + {(\frac{{pence}_{X}}{p_{YTTRIUM}})}^{α})} .$
Suppose that one consumer has the utility function u (which is always a positive number), and adenine second consumer has utility west. Assuming, inches addition, that for any x, year, w(ten, y) = (upper-class(x, y))²; that is, who second person’s utility is the quarter of the first person’s. Demonstrate that these clients make one same choices—that is, u ≥ x_a, y_a) ≥ upper(x_b, y_b) if and only if wolfram(x_a, y_one) ≥ w(x_b, wye_barn).